The Economic Journal

Barro, Robert, J.

doi: 10.1111/ecoj.12247pmid: N/A

Abstract In a country panel since 1960, the estimated annual convergence rate for GDP is 1.7%, conditional on time‐varying explanatory variables. With country fixed effects, the estimated convergence rate is misleadingly high. With data starting in 1870, country fixed effects are reasonable and the estimated convergence rate is 2.6%. Combining the two estimates suggests conditional convergence close to the ‘iron‐law’ rate of 2%. With post‐1960 data, estimation without country fixed effects reveals positive effects of GDP and schooling on law and order and democracy – consistent with the modernisation hypothesis. With post‐1870 data, estimation without or with country fixed effects indicates modernisation. According to the ‘iron law of convergence’, countries eliminate gaps in levels of real per capita GDP at a rate around 2% per year.1 Convergence at a 2% rate implies that it takes 35 years for half of an initial gap to vanish and 115 years for 90% to disappear. Convergence‐rate parameters are important to pin down because they provide guidance on how fast countries like China and India are likely to catch up to richer countries. The convergence rate may also reveal how fast a poor African country could develop or how rapidly North Korea could catch up to South Korea, and so on. Empirically, the iron law takes the form of unconditional or absolute convergence in some samples of economies; those that are reasonably homogeneous in terms of long‐run or steady‐state characteristics. For example, a roughly 2% convergence rate emerged for per capita personal income in a long‐term panel of US states in Barro and Sala‐i‐Martin (1992).2 This convergence was absolute in the sense of not having to be conditioned on a set of variables that capture differences in long‐run positions. The results implied – in accordance with the data – that it would take the US South about a century after the Civil War to get close in per capita income to the rest of the country. Applying these results to East versus West Germany suggested that a short time frame for convergence was not a realistic expectation.3 And, looking forward to the potential reunification of North and South Korea, the iron law presents a pessimistic outlook on how rapidly the large gap in per capita product could be eliminated. In a collection of heterogeneous economies that differ substantially in terms of long‐run properties, the 2% convergence rate holds in a conditional sense. That is, convergence applies only with an allowance for differences in constant or slowly varying cross‐economy characteristics, such as saving rates or fertility rates or quantity of human capital or institutional quality or colonial history or geographical features. For example, a convergence rate around 2% per year appeared in a cross section of 98 countries in Barro and Sala‐i‐Martin (1992, Table 3), after conditioning on an array of variables that differed by country.4 Because of the conditioning variables, these results were more pessimistic than the iron‐law convergence rate would suggest. Poor places – for example, many sub‐Saharan African countries, North Korea, Burma, Bolivia or Venezuela – might not converge at all if key underlying variables, such as the quality of human capital and institutions, were not improved. The present study assesses convergence behaviour within two empirical contexts. One data set comprises the large number of countries with observations for many variables since 1960. Another data set exploits recent advances in long‐term national‐accounts information. These data go back to 1870 but apply to fewer countries and variables. In both contexts, the distinction between absolute and conditional convergence is important. And, within the context of conditional convergence, a key technical issue is whether the cross‐country growth regressions include country fixed effects. Many analyses of economic growth stress effects from the quality of institutions, gauged particularly by maintenance of the rule of law and democracy.5 A prominent feature of this analysis is two‐way causation between economic development and institutional quality. Specifically, according to the ‘modernisation hypothesis’, economic development spurs the introduction and maintenance of higher quality institutions, including well‐functioning representative democracy.6 The validity of the modernisation thesis is important for its own sake – particularly for understanding how democracy and rule of law evolve – as well as for assessing institutional determinants of economic growth. I use the post‐1960 and post‐1870 data sets to assess the modernisation hypothesis. From an econometric standpoint, the analysis of modernisation parallels the study of convergence in per capita GDP. Both analyses involve convergence rates and an array of explanatory variables that determine long‐run positions. And empirical inferences in both contexts are sensitive to the treatment of country fixed effects. 1. Country Fixed Effects Cross‐country empirical findings on convergence and modernisation depend on whether the panel regressions include country fixed effects. Although the incorporation of these fixed effects into cross‐country panel regressions has become almost routine,7 the merits are not straightforward, because they involve a trade‐off between two forces, highlighted by Nerlove (2000). The Appendix brings out details, using Monte Carlo methods. This analysis expands on Kiviet (1995, Tables 1–3), Judson and Owen (1999, Tables 1–2) and Hauk and Wacziarg (2009, Table 4) to allow for sample lengths (such as 50 or 140 years) and convergence rates (around 0.02 per year) that match up with data on economic growth. To fix ideas, consider cross‐country panel regressions for the growth rate of per capita GDP. Country fixed effects are attractive as a way to allow for unobserved, persistent country characteristics that influence long‐run per capita GDP and are also correlated with observed per capita GDP. That is, rich countries tend to have prospered because they possess persistently favourable characteristics, such as high‐quality institutions, that lead to high steady‐state per capita GDP. From this omitted‐variables perspective, the exclusion of country fixed effects tends to bias upward the estimated effect of lagged GDP on current GDP and, thereby, bias downward the estimated convergence rate. One familiar example of this effect is the tendency to estimate an absolute convergence rate near zero in a panel of heterogeneous countries. However, the bias may be small if the framework without country fixed effects includes a rich set of explanatory variables so that little remains of omitted variables that are conditionally correlated with per capita GDP. The second force involves the Hurwicz (1950)‐type bias in the estimated coefficient of a lagged dependent variable. In panels that are small in the time dimension, Nickell (1981), Arellano and Bond (1991), Kiviet (1995) and Nerlove (2000), among others, show that this force biases down the fixed‐effects estimator (based on least‐squares‐with‐dummy‐variables) for the coefficient of the lagged dependent variable.8 On this ground, the estimated convergence rate tends to be overestimated (because the persistence in the level of the dependent variable is underestimated). Nickell (1981, p. 1422) provides a formula for the Hurwicz‐type bias in the least‐squares‐with‐dummy‐variables estimate for the coefficient of a lagged dependent variable. The Nickell specification includes cross‐sectional fixed effects, no other explanatory variables (X variables), and – unlike Hurwicz – a large (effectively infinite) number of cross sections. The Nickell analysis also treats the initial value of the dependent variable as given; specifically, independent of the fixed effects. To get a simplified version of the Nickell formula for the Hurwicz (1950) type bias, let β denote the magnitude of the convergence rate per year. The coefficient of the lagged dependent variable, labelled ρ in Nickell's analysis, is then ρ = e−βτ, where τ is the period length in years. Let T be the sample length in years (differing from Nickell's notation). Nickell's formula for the proportionate bias in the estimated β can then be expressed, if β > 0, as [(β^−β)/β]≈2(e−βT−1+βT)β2T2−2(e−βT−1+βT)>0,(1) where the approximation uses βτ ≪ 1 (so that, in effect, the convergence that occurs within an observation period is small). An important implication of the Nickell formula in (1) is that the proportionate bias depends only on the product βT. Therefore, a change in the period length, τ, with T held fixed, does not affect the bias. (The Monte Carlo analyses in the Appendix and the empirical findings discussed later accord with this result.) In particular, the bias depends on the overall length of the sample, T, not the number of periods, T/τ, over which the data are observed (given the condition βτ ≪ 1). This finding means that the bias cannot be reduced by raising the frequency of observation; for example, from 10 years to 5 years or 1 year. It also does not help to have a large number of cross sections (since this number is already taken to be infinity in Nickell's analysis). Quantitatively, if β = 0.02 per year, Nickell's formula in (1) generates an upward bias of 0.056 when T = 50 years and 0.018 when T = 140 years (corresponding to the long‐term sample used later). Hence, although the bias approaches zero as T approaches infinity, the bias tends to be large in samples of realistic length. However, these results are only suggestive, because Nickell's formula depends on a number of unrealistic assumptions, particularly about the exclusion of X variables and the treatment of the initial value of the dependent variable as given, independent of the fixed effects. Monte Carlo results related to these issues are in the Appendix. When country fixed effects are included and the time dimension of the sample is moderate – say 20–50 years – the panel effectively comprises multiple cases of moderate length. Since the regression for each country features a Hurwicz (1950) type bias, this bias applies also to the overall panel. Therefore, although the Nickell (1981) extension to include a large number of cross sections matters quantitatively, that extension does not alter the essential source of the bias. In any event, I refer subsequently to the Hurwicz–Nickell bias when considering the estimated coefficient of a lagged dependent variable. If country fixed effects are excluded, the observations are effectively stacked in the time and country dimensions. That is, with 151 countries and 10 time periods (as in the subsequent analysis of data since 1960), the roughly 1,500 observations amount to a large sample in the relevant time dimension, and the Hurwicz–Nickell bias is negligible. In other words, the bias arises because of the inclusion of country fixed effects. These conjectures are confirmed by the Monte Carlo studies described in the Appendix. Inclusion of country fixed effects also affects the estimated coefficients and, especially, standard errors of explanatory variables – X variables – other than lagged dependent variables. Coefficients on country variables that are constant (such as geographical features and colonial history) cannot be estimated at all and variables that have little within‐country time variation cannot be estimated with precision. In effect, the inclusion of country fixed effects throws out much of the information in isolating the effects of X variables on growth rates or other variables. Problems in estimating coefficients of X variables in a fixed‐effects context apply to the recent debate about the modernisation hypothesis in Glaeser et al. (2004, 2007) and Acemoglu et al. (2005, 2008). The failure in the Acemoglu et al. studies to find statistically significant effects on democracy from per capita GDP and education depends, in their main analysis, on the inclusion of country fixed effects. This result is not surprising because, with country fixed effects, it is challenging to estimate statistically significant coefficients on X variables that do not have a lot of independent variation over time within countries. In contrast, without country fixed effects, as in the Glaeser et al. studies and Barro (1999), the typically substantial cross‐sectional variations in the X variables make it easier to isolate statistically significant effects. The perspective changes in the context of panel data observed for over a century. In this setting, the econometric problems posed by the inclusion of country fixed effects are less serious. In particular, the Hurwicz–Nickell bias is smaller in this context, as confirmed by (1) and the Monte Carlo studies in the Appendix. 2. The Framework of Conditional Convergence This Section summarises well‐known implications of the neoclassical growth model and its extensions for empirical analyses of conditional convergence. The model features a production function, Y=A×F(K,L),(2) where F(∙) satisfies the usual neoclassical properties, including constant returns to scale in capital, K, and labour, L. Output per worker, y ≡ Y/L, depends on capital per worker, k ≡ K/L: y=f(k).(3) The economy is closed, so that saving and investment coincide. In the baseline model, there is no government sector but extensions allow for government purchases and taxes. The labour force, measured as worker‐hours per year, is related in a fixed way to population, which grows at rate n. The labour force is fully employed and, therefore, corresponds to labour input, L, in (2). Given these assumptions, the quantities y and k, measured per worker in (3), can be interpreted as quantities per capita. In the baseline model, n is constant over time within an economy but may differ across economies. In the baseline model, the gross saving rate (which equals the ratio of gross investment to GDP) is a constant, s (as in Solow, 1956; Swan, 1956), which may differ across economies. The first major extension of the Solow–Swan framework was to endogenise the saving rate. In a setting based on Ramsey (1928), Cass (1965) and Koopmans (1965), the representative household determines the optimal saving rate at each point in time in the context of an iso–elastic, time‐additive utility function. The equilibrium value of s then varies over time within economies along a transition path to the steady state. Barro and Sala‐i‐Martin (2004, ch. 2) show that, if F(∙) is Cobb‐Douglas, the transitional behaviour of the saving rate is monotonic – always rising towards its steady‐state value, always falling towards its steady‐state value, or always constant at its steady‐state value. With rising (falling) s, the convergence rate is slower (faster) than in the model with fixed s. With alternative utility functions, the transition may not be monotonic; for example, the transition path for s may be hump‐shaped. Across economies, differences in preference and other parameters can shift the path of s when considered in relation to GDP per capita, y. An economy with higher s at a given y tends to grow faster. Just as the Ramsey‐Cass‐Koopmans analysis endogenised the fixed and exogenous saving rate of Solow–Swan, the model can be extended along the lines of Malthus (1798) to endogenise the population growth rate, n.9 Barro and Sala‐i‐Martin (2004) allow for choices over time of the fertility rate for a given mortality rate. In the case where child‐rearing costs rise linearly with k (because, as in Becker (1991), child‐rearing is intensive in parental time, which is valued in accordance with the wage rate), n falls monotonically during the transition to the steady state. This pattern reduces the convergence rate compared to the setting in which n is fixed. However, the allowance for goods costs of child‐rearing can generate a non‐monotonic pattern in which fertility first rises and later falls as per capita GDP, y, increases. Across economies, differences in preference and other parameters (including costs related to the rearing of children) can shift the path of n when considered in relation to y. An economy with lower n at a given y tends to grow faster. In the baseline model, the productivity factor, A, can differ across economies. This factor may rise over time due to exogenous technical progress. The usual steady‐state properties depend on technical progress taking the labour‐augmenting form. In this setting, effective labour input, L^ ⁠, replaces L in (1), L^ is the multiple ext of L, and x ≥ 0 is the rate of technological progress. In this context, the results for the growth‐rate transition go through in terms of quantities per effective labour; that is, as y^≡Y/L^ and k^≡K/L^ ⁠. In the Cobb–Douglas case, labour‐augmenting technical progress is equivalent to exogenous growth in A in (1). In extensions, differences in A (or in the path of A) within or across economies can reflect the quality of institutions, including maintenance of property rights and the efficiency of the tax system. That is, the economy's response to weak property rights or high marginal income tax rates is essentially equivalent to its response to a reduction in productivity, A. The research on endogenous growth due to endogenous technical change began with Romer (1987, 1990), Grossman and Helpman (1991, chs. 3 and 4) and Aghion and Howitt (1992).10 In these models, technological advances derive from purposeful and successful R&D activity. In ‘varieties’ models (Romer, 1987, 1990; Grossman and Helpman, 1991, ch. 3), discoveries of new types of intermediate inputs raise productivity by (metaphorically) expanding the number of inputs employed in production. In ‘quality‐ladders’ models (Aghion and Howitt, 1992; Grossman and Helpman, 1991, ch. 4), innovations raise the quality of intermediate inputs within sectors (or, equivalently, improve the efficiency of production processes). From the perspective of the Solow–Swan model, the theories of technological progress effectively endogenise the growth rate of the productivity factor, A. Endogenous growth theory can be viewed, accordingly, as another extension of the Solow–Swan model to endogenise a key parameter; in this case, A (i.e. the time path of A), rather than s or n. Endogenous growth theory has not played a major role in empirical studies of the determinants of economic growth across countries. The main empirical applications of the theory have involved effects of R&D outlays. This research builds on the (pre‐endogenous‐growth) approach described by Griliches (1973), which starts by computing total factor productivity (TFP) growth rates as Solow residuals in a growth‐accounting framework.11 The TFP growth rates can then be related econometrically to measures of R&D expenditures. This methodology was applied to US firms and industries by Griliches and Lichtenberg (1984) and Griliches (1988). Coe and Helpman (1995) applied this framework to aggregate data for OECD countries and reported large positive effects of R&D outlays on economic growth.12 However, a problem with this approach is that – in the absence of good instruments – a positive relation between R&D spending and economic growth can reflect reverse causation from growth opportunities to R&D, rather than effects of R&D and technological progress on growth. One reason that endogenous growth theory has not played a major role in cross‐country growth analysis is that the theory may apply mostly to the worldwide average growth rate. This perspective would apply if international trade, foreign investment and the flow of ideas lead to rapid diffusion of technology across countries. The speed of diffusion of technology was assessed theoretically in Nelson and Phelps (1966), who stressed the role of human capital, and in subsequent models summarised in Barro and Sala‐i‐Martin (2004). Caselli and Coleman (2001) found empirically for developing countries that imports of computers and other high‐tech equipment – viewed as a proxy for technology absorption – were spurred by increased imports from technologically advanced countries. Technological diffusion was also higher when the home country had higher levels of school attainment at secondary and higher levels and better institutional quality. A further extension of the neoclassical growth model, introduced by Mankiw et al. (1992), distinguishes human from physical capital. One way that this extension affects the dynamics of economic growth involves the greater difficulty in adjusting human capital, H, compared to physical capital, K. In this case, the growth rate of y tends to be higher at a given y if the ratio H/K is higher. For an economy below its steady‐state position, a higher H/K (perhaps generated from a war that destroyed much more physical than human capital) means that subsequent growth focuses on K, which is easier than H to expand rapidly. Suppose that L is the number of workers and h is human capital per worker, so that quality‐adjusted labour input is H = hL. The input H then replaces L in (1). Assume, further, that h relates to years of schooling along Mincerian lines, so that h=eλS, where S is (average) years of schooling and λ is the rate of return on schooling (if the cost of schooling is the income foregone by not employing human capital in production). If the production function in (1) is Cobb–Douglas with exponents a on K and b on H, we can derive: log(H/K)=(1/a)×log(A)−(1/a)×log(y)+λ×(1+b/a)×S.(4) Therefore, for given A, a higher S signals a higher H/K at a given y, which predicts higher economic growth. However, for a given H/K, a higher S would signal a lower A at a given y, which predicts lower economic growth. In the empirical analysis, I assume that, although S and y are observable, A (corresponding to total factor productivity) cannot be measured (because K cannot be measured accurately). In this case, the overall effect of S on economic growth, for given y, is ambiguous. In the empirical analysis, the estimated effect on growth from the level of S turns out to be small and typically statistically insignificantly different from zero. 3. Cross‐country Growth Regressions The preceding Section implies that the growth rate of real per capita GDP, Dyit, for country i at date t can be written as Dyit=Φ(yit,sit,nit,Ait,…),(5) where the negative sign under yit reflects convergence, conditional on the other variables. These other influences include variables related to the saving rate, sit, the population growth rate, nit and the level of productivity, Ait. In the main analysis, I estimate (5) using per capita growth rates of GDP averaged over five‐year periods. Sometimes the analysis includes constant terms that are specific to countries (country fixed effects). The estimation always includes constants for each time period (time effects) – therefore, the analysis does not attempt to explain variations over time of world average growth rates. One well‐known problem with cross‐country growth regressions is endogeneity of some of the X variables. For example, it is unclear whether good institutions cause economic growth or are a reaction to rising living standards – or, perhaps, that GDP and institutional quality are responses to common influences. Previous studies have proposed instruments to deal with this problem. Examples are gravity variables such as country size and trade restrictions that influence international trade (Lee, 1993); ethnolinguistic fractionalisation (Mauro, 1995); population density and settler mortality at the time of colonial settlement (Engerman and Sokoloff, 1994;13 Acemoglu et al., 2001, 2002); the form of legal origins (La Porta et al., 1998); absolute degrees latitude and primary language (Hall and Jones, 1999); the presence of state religion (Barro and McCleary, 2003); and physical characteristics of islands (Feyrer and Sacerdote, 2009). One problem is that the proposed instruments typically do not vary over time in the sample within countries and, therefore, do not help when country fixed effects are included. A more basic issue, if one allows for the multidimensional set of X variables that matters for economic growth, is that there are never enough convincing instruments to allow for full identification. The present empirical analysis emphasises panel least‐squares but compares these results with two‐stage least‐squares estimates based on lagged values of the X variables as instruments. For example, in considering the growth rate from 2005 to 2010, the average of the investment ratio from 2005 to 2009 enters into the regression equation but the investment ratio for 2005 is on the instrument list. This use of lagged X values as instruments helps to deal with endogeneity and also to alleviate problems of temporary measurement error.14 Another well‐known issue is the robustness of the results with respect to which X variables are included and in what functional form. Barro and Sala‐i‐Martin (2004, ch. 12) discuss the Bayesian model‐averaging approach to this problem. This technique effectively weights each possible specification by the fits to the dependent variable, taken to be the growth rate of per capita GDP from 1960 to 1996. This method was applied to 67 X variables that have been proposed in the empirical growth literature, using data for 88 countries. The conclusion is that only five of the variables have posterior inclusion probabilities above 0.5 and 18 have probabilities above 0.1. However, with 67 variables considered, many are conceptually similar, so that low inclusion probabilities are not surprising. My view is that pinpointing precisely which X variables matter for growth is impossible. However, what is feasible is interpreting the results in terms of broad influences that matter for growth; for example, quality of institutions, openness to markets and so on. In addition, one can interpret results on conditional convergence – gauged empirically by the estimated coefficient on the log of lagged per capita GDP – holding fixed an array of X variables. These results on conditional convergence tend not to be highly sensitive to exactly which X variables are included. 4. Empirical Results Table 1 contains empirical results for the cross‐country panel. The dependent variable is each country's growth rate of real per capita GDP over 10 five‐year intervals from 1960–5 to 2005–10. Note that the growth rates are expressed per year, not per five‐year period. The sample of countries was chosen based on the availability of data on GDP at least by 1970, so that each country has at least 40 years of data. This sample, corresponding to columns 1 and 2, comprises 151 countries. Subsequent estimation in columns 3 and 4 includes an array of X variables. In this case, the criterion of data availability at least by the 1970–5 period led to the selection of 89 countries, listed in Table 2. Estimation in Table 1, columns 1–4, is by panel least‐squares15 and includes time effects. Column 1 of Table 1 includes as a right‐hand side variable only the five‐year lag of the log of per capita GDP. The estimated coefficient is positive (indicating divergence rather than convergence) and statistically significantly different from zero. However, the coefficient is small in magnitude, in the sense of indicating divergence at a rate of only 0.2% per year. This result reproduces the typical pattern whereby absolute convergence of per capita GDP does not appear in a heterogeneous collection of countries. From the standpoint of (5), the interpretation is that the five‐year lag of the log of per capita GDP is positively correlated with determinants of the steady‐state position that raise the long‐run level of per capita GDP. This omitted‐variables effect offsets the convergence force and leads to an estimated coefficient on the lagged log of per capita GDP that is positive but close to zero. The Monte Carlo analysis in the Appendix (Table A1, line 11) reproduces this kind of result. Column 2 adds country fixed effects, which are jointly highly statistically significant. The estimated coefficient of the log of lagged per capita GDP is now significantly negative, −0.0335 (SE = 0.0039).16 This result suggests conditional convergence (conditional on individual constants for each country) at a rate around 3.4% per year. An interpretation is that each country's fixed effect proxies for the influences of the various determinants of long‐run per capita GDP that appear in (5) – at least to the extent that these determinants do not vary much over time within countries. Therefore, the estimated coefficient on the log of lagged per capita GDP now picks up the predicted conditional convergence, indicated by the negative sign in (5). However, as noted before, there is a tendency to overestimate the convergence speed in the presence of country fixed effects because of the Hurwicz–Nickell bias in the estimated coefficient of a lagged dependent variable. The Monte Carlo analysis (Table A1, lines 1 and 2) supports this interpretation.17 Table 1 Growth‐rate Regressions for Cross‐country Panel Five‐year Periods: 1960–5, … , 2005–10 (All Equations Estimated by OLS and Include Time Effects) . (1) . (2) . (3) . (4) . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) 0.0024* −0.0335** −0.0170** −0.0458** (0.0010) (0.0039) (0.0021) (0.0045) 1/(life expectancy at birth) – – −3.09** −1.03 (0.58) (1.04) Log(fertility rate) – – −0.0277** −0.0301** (0.0043) (0.0073) Law and order (rule of law) – – 0.0157** 0.0051 (0.0054) (0.0088) Investment ratio – – 0.031* 0.053** (0.012) (0.020) Female school years – – 0.0024 0.0054 (0.0014) (0.0033) Male school years – – −0.0028 −0.0066* (0.0015) (0.0029) Government consumption ratio – – −0.026 −0.090* (0.023) (0.042) Openness ratio – – 0.0056* 0.0175* (0.0025) (0.0071) Terms‐of‐trade change – – 0.117** 0.113** (0.026) (0.027) Democracy indicator – – 0.029 −0.015 (0.015) (0.019) Democracy squared – – −0.028* 0.009 (0.014) (0.017) Inflation rate – – −0.0180** −0.0213** (0.0042) (0.0041) R2 0.059 0.298 0.329 0.501 SE of regression 0.0371 0.0339 0.0242 0.0221 No. countries; observations 151; 1,430 151; 1,430 89; 841 89; 841 . (1) . (2) . (3) . (4) . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) 0.0024* −0.0335** −0.0170** −0.0458** (0.0010) (0.0039) (0.0021) (0.0045) 1/(life expectancy at birth) – – −3.09** −1.03 (0.58) (1.04) Log(fertility rate) – – −0.0277** −0.0301** (0.0043) (0.0073) Law and order (rule of law) – – 0.0157** 0.0051 (0.0054) (0.0088) Investment ratio – – 0.031* 0.053** (0.012) (0.020) Female school years – – 0.0024 0.0054 (0.0014) (0.0033) Male school years – – −0.0028 −0.0066* (0.0015) (0.0029) Government consumption ratio – – −0.026 −0.090* (0.023) (0.042) Openness ratio – – 0.0056* 0.0175* (0.0025) (0.0071) Terms‐of‐trade change – – 0.117** 0.113** (0.026) (0.027) Democracy indicator – – 0.029 −0.015 (0.015) (0.019) Democracy squared – – −0.028* 0.009 (0.014) (0.017) Inflation rate – – −0.0180** −0.0213** (0.0042) (0.0041) R2 0.059 0.298 0.329 0.501 SE of regression 0.0371 0.0339 0.0242 0.0221 No. countries; observations 151; 1,430 151; 1,430 89; 841 89; 841 Notes * Significant at 5% level. ** Significant at 1% level. The dependent variable is the annual growth rate of real per capita GDP for the ten five‐year periods: 1960–5, … , 2005–10. The sample criterion in columns 1–2 is to include countries only if they have data starting by the 1970–5 period (151 countries). The equations in columns 3–4 also require data on the array of X variables (89 countries, shown in Table 2). Lagged per capita GDP, the reciprocal of life expectancy at birth, the total fertility rate and female and male years of school attainment for persons aged 15 and over are five‐year lags (for 1960, … , 2005). The ratios of investment and government consumption to GDP, the openness ratio, the indicator for law and order and the democracy indicator are five‐year averages of values lagged one to five years. The growth rate of the terms of trade and the inflation rate are for the same periods as the dependent variable. Standard errors of coefficient estimates are in parentheses. For calculating standard errors, the error terms are allowed to be correlated over time within countries. Joint p‐values for the two democracy variables are 0.14 in column 3 and 0.56 in column 4. Definitions and sources. PPP‐adjusted real per capita GDP is from Penn World Tables (www.pwt.econ.upenn.edu), version 7.0, in units of 2005 international dollars. Data for 2010 are from version 7.1. Also from version 7.0 are the ratios to GDP of investment (private plus public) and government consumption and the openness ratio (exports plus imports relative to GDP). These ratio variables use current‐price information. Life expectancy at birth and the total fertility rate are from the World Bank's World Development Indicators (WDI). The law‐and‐order indicator is from Political Risk Services, International Country Risk Guide. The data were converted from seven categories to a 0–1 scale, with 1 representing the highest maintenance of law and order. Average years of school attainment for females and males aged 15 and over at various levels of schooling are from Barro and Lee (2013), with data available at www.barrolee.com. These data are at five‐year intervals. The terms‐of‐trade change (growth rates over five years of export prices relative to import prices) is from International Monetary Fund, International Financial Statistics and WDI. This variable is interacted with the openness ratio. The democracy indicator is the political rights variable from Freedom House (www.freedomhouse.org). The data were converted from seven categories to a 0–1 scale, with 1 representing the highest rights. Data on an analogous concept for 1960 and 1965 are from Bollen (1980). The inflation rate (averaged over five‐year intervals) is calculated from retail price indexes from International Monetary Fund, International Financial Statistics and WDI. Open in new tab Table 1 Growth‐rate Regressions for Cross‐country Panel Five‐year Periods: 1960–5, … , 2005–10 (All Equations Estimated by OLS and Include Time Effects) . (1) . (2) . (3) . (4) . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) 0.0024* −0.0335** −0.0170** −0.0458** (0.0010) (0.0039) (0.0021) (0.0045) 1/(life expectancy at birth) – – −3.09** −1.03 (0.58) (1.04) Log(fertility rate) – – −0.0277** −0.0301** (0.0043) (0.0073) Law and order (rule of law) – – 0.0157** 0.0051 (0.0054) (0.0088) Investment ratio – – 0.031* 0.053** (0.012) (0.020) Female school years – – 0.0024 0.0054 (0.0014) (0.0033) Male school years – – −0.0028 −0.0066* (0.0015) (0.0029) Government consumption ratio – – −0.026 −0.090* (0.023) (0.042) Openness ratio – – 0.0056* 0.0175* (0.0025) (0.0071) Terms‐of‐trade change – – 0.117** 0.113** (0.026) (0.027) Democracy indicator – – 0.029 −0.015 (0.015) (0.019) Democracy squared – – −0.028* 0.009 (0.014) (0.017) Inflation rate – – −0.0180** −0.0213** (0.0042) (0.0041) R2 0.059 0.298 0.329 0.501 SE of regression 0.0371 0.0339 0.0242 0.0221 No. countries; observations 151; 1,430 151; 1,430 89; 841 89; 841 . (1) . (2) . (3) . (4) . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) 0.0024* −0.0335** −0.0170** −0.0458** (0.0010) (0.0039) (0.0021) (0.0045) 1/(life expectancy at birth) – – −3.09** −1.03 (0.58) (1.04) Log(fertility rate) – – −0.0277** −0.0301** (0.0043) (0.0073) Law and order (rule of law) – – 0.0157** 0.0051 (0.0054) (0.0088) Investment ratio – – 0.031* 0.053** (0.012) (0.020) Female school years – – 0.0024 0.0054 (0.0014) (0.0033) Male school years – – −0.0028 −0.0066* (0.0015) (0.0029) Government consumption ratio – – −0.026 −0.090* (0.023) (0.042) Openness ratio – – 0.0056* 0.0175* (0.0025) (0.0071) Terms‐of‐trade change – – 0.117** 0.113** (0.026) (0.027) Democracy indicator – – 0.029 −0.015 (0.015) (0.019) Democracy squared – – −0.028* 0.009 (0.014) (0.017) Inflation rate – – −0.0180** −0.0213** (0.0042) (0.0041) R2 0.059 0.298 0.329 0.501 SE of regression 0.0371 0.0339 0.0242 0.0221 No. countries; observations 151; 1,430 151; 1,430 89; 841 89; 841 Notes * Significant at 5% level. ** Significant at 1% level. The dependent variable is the annual growth rate of real per capita GDP for the ten five‐year periods: 1960–5, … , 2005–10. The sample criterion in columns 1–2 is to include countries only if they have data starting by the 1970–5 period (151 countries). The equations in columns 3–4 also require data on the array of X variables (89 countries, shown in Table 2). Lagged per capita GDP, the reciprocal of life expectancy at birth, the total fertility rate and female and male years of school attainment for persons aged 15 and over are five‐year lags (for 1960, … , 2005). The ratios of investment and government consumption to GDP, the openness ratio, the indicator for law and order and the democracy indicator are five‐year averages of values lagged one to five years. The growth rate of the terms of trade and the inflation rate are for the same periods as the dependent variable. Standard errors of coefficient estimates are in parentheses. For calculating standard errors, the error terms are allowed to be correlated over time within countries. Joint p‐values for the two democracy variables are 0.14 in column 3 and 0.56 in column 4. Definitions and sources. PPP‐adjusted real per capita GDP is from Penn World Tables (www.pwt.econ.upenn.edu), version 7.0, in units of 2005 international dollars. Data for 2010 are from version 7.1. Also from version 7.0 are the ratios to GDP of investment (private plus public) and government consumption and the openness ratio (exports plus imports relative to GDP). These ratio variables use current‐price information. Life expectancy at birth and the total fertility rate are from the World Bank's World Development Indicators (WDI). The law‐and‐order indicator is from Political Risk Services, International Country Risk Guide. The data were converted from seven categories to a 0–1 scale, with 1 representing the highest maintenance of law and order. Average years of school attainment for females and males aged 15 and over at various levels of schooling are from Barro and Lee (2013), with data available at www.barrolee.com. These data are at five‐year intervals. The terms‐of‐trade change (growth rates over five years of export prices relative to import prices) is from International Monetary Fund, International Financial Statistics and WDI. This variable is interacted with the openness ratio. The democracy indicator is the political rights variable from Freedom House (www.freedomhouse.org). The data were converted from seven categories to a 0–1 scale, with 1 representing the highest rights. Data on an analogous concept for 1960 and 1965 are from Bollen (1980). The inflation rate (averaged over five‐year intervals) is calculated from retail price indexes from International Monetary Fund, International Financial Statistics and WDI. Open in new tab Column 3 includes, instead of country fixed effects, an array of time‐varying X variables for each country.18 These variables proxy for the various growth determinants, apart from the log of lagged per capita GDP, that enter into (5). Unlike in column 1, the estimated coefficient of the log of lagged per capita GDP is significantly negative, −0.0170 (SE = 0.0021), and indicates convergence at about 1.7% per year. This convergence is conditional in the sense of holding for given values of the X variables (which are themselves time‐varying). The details on the X variables used in the estimation are in the notes to Table 1. The estimated coefficients of the X variables in column 3 can be viewed mostly as effects on long‐run or steady‐state positions for each country. For example, with respect to institutional quality, the results suggest that a country's long‐run economic position is enhanced by better maintenance of law and order (and the rule of law).19 Greater democracy (gauged by the level and square of the Freedom House indicator of political rights20) tends initially to be positively associated with growth but the sign switches for higher values of democracy. The break point between marginal effects being positive or negative occurs roughly at the halfway mark between full dictatorship (value zero) and full representative democracy (value one). Analogous relations between democracy and economic growth were reported in Barro (1997). Table 2 Sample of 89 Countries Used in Regressions in Table 1, columns 3–4 Country . Starting period . Country . Starting period . Algeria 1960–5 Kenya 1960–5 Argentina 1960–5 Luxembourg 1960–5 Australia 1960–5 Malawi 1965–70 Austria 1960–5 Malaysia 1960–5 Bahrain 1970–5 Mali 1965–70 Bangladesh 1965–70 Malta 1970–5 Belgium 1960–5 Mexico 1960–5 Bolivia 1965–70 Morocco 1960–5 Botswana 1965–70 Netherlands 1960–5 Brazil 1960–5 New Zealand 1960–5 Cameroon 1965–70 Nicaragua 1960–5 Canada 1960–5 Niger 1960–5 Chile 1960–5 Norway 1960–5 China 1960–5 Pakistan 1960–5 Colombia 1960–5 Panama 1965–70 Congo, Republic 1960–5 Papua New Guinea 1960–5 Costa Rica 1960–5 Paraguay 1960–5 Cyprus 1960–5 Peru 1965–70 Denmark 1960–5 Philippines 1960–5 Dominican Republic 1960–5 Portugal 1960–5 Ecuador 1960–5 Senegal 1960–5 Egypt 1960–5 Sierra Leone 1965–70 El Salvador 1960–5 Singapore 1965–70 Finland 1960–5 South Africa 1960–5 France 1960–5 South Korea 1965–70 Gabon 1965–70 Spain 1960–5 Gambia 1965–70 Sri Lanka 1960–5 Germany 1970–5 Sudan 1970–5 Ghana 1960–5 Sweden 1960–5 Greece 1960–5 Switzerland 1960–5 Guatemala 1960–5 Syria 1970–5 Guyana 1970–5 Taiwan 1960–5 Haiti 1960–5 Tanzania 1970–5 Honduras 1960–5 Thailand 1960–5 Hungary 1970–5 Togo 1965–70 Iceland 1960–5 Trinidad 1960–5 India 1960–5 Tunisia 1965–70 Indonesia 1965–70 Turkey 1965–70 Ireland 1960–5 Uganda 1965–70 Israel 1970–5 UK 1960–5 Italy 1960–5 Uruguay 1965–70 Ivory Coast 1960–5 US 1960–5 Jamaica 1960–5 Venezuela 1965–70 Japan 1960–5 Zambia 1965–70 Jordan 1965–70 Country . Starting period . Country . Starting period . Algeria 1960–5 Kenya 1960–5 Argentina 1960–5 Luxembourg 1960–5 Australia 1960–5 Malawi 1965–70 Austria 1960–5 Malaysia 1960–5 Bahrain 1970–5 Mali 1965–70 Bangladesh 1965–70 Malta 1970–5 Belgium 1960–5 Mexico 1960–5 Bolivia 1965–70 Morocco 1960–5 Botswana 1965–70 Netherlands 1960–5 Brazil 1960–5 New Zealand 1960–5 Cameroon 1965–70 Nicaragua 1960–5 Canada 1960–5 Niger 1960–5 Chile 1960–5 Norway 1960–5 China 1960–5 Pakistan 1960–5 Colombia 1960–5 Panama 1965–70 Congo, Republic 1960–5 Papua New Guinea 1960–5 Costa Rica 1960–5 Paraguay 1960–5 Cyprus 1960–5 Peru 1965–70 Denmark 1960–5 Philippines 1960–5 Dominican Republic 1960–5 Portugal 1960–5 Ecuador 1960–5 Senegal 1960–5 Egypt 1960–5 Sierra Leone 1965–70 El Salvador 1960–5 Singapore 1965–70 Finland 1960–5 South Africa 1960–5 France 1960–5 South Korea 1965–70 Gabon 1965–70 Spain 1960–5 Gambia 1965–70 Sri Lanka 1960–5 Germany 1970–5 Sudan 1970–5 Ghana 1960–5 Sweden 1960–5 Greece 1960–5 Switzerland 1960–5 Guatemala 1960–5 Syria 1970–5 Guyana 1970–5 Taiwan 1960–5 Haiti 1960–5 Tanzania 1970–5 Honduras 1960–5 Thailand 1960–5 Hungary 1970–5 Togo 1965–70 Iceland 1960–5 Trinidad 1960–5 India 1960–5 Tunisia 1965–70 Indonesia 1965–70 Turkey 1965–70 Ireland 1960–5 Uganda 1965–70 Israel 1970–5 UK 1960–5 Italy 1960–5 Uruguay 1965–70 Ivory Coast 1960–5 US 1960–5 Jamaica 1960–5 Venezuela 1965–70 Japan 1960–5 Zambia 1965–70 Jordan 1965–70 Open in new tab Table 2 Sample of 89 Countries Used in Regressions in Table 1, columns 3–4 Country . Starting period . Country . Starting period . Algeria 1960–5 Kenya 1960–5 Argentina 1960–5 Luxembourg 1960–5 Australia 1960–5 Malawi 1965–70 Austria 1960–5 Malaysia 1960–5 Bahrain 1970–5 Mali 1965–70 Bangladesh 1965–70 Malta 1970–5 Belgium 1960–5 Mexico 1960–5 Bolivia 1965–70 Morocco 1960–5 Botswana 1965–70 Netherlands 1960–5 Brazil 1960–5 New Zealand 1960–5 Cameroon 1965–70 Nicaragua 1960–5 Canada 1960–5 Niger 1960–5 Chile 1960–5 Norway 1960–5 China 1960–5 Pakistan 1960–5 Colombia 1960–5 Panama 1965–70 Congo, Republic 1960–5 Papua New Guinea 1960–5 Costa Rica 1960–5 Paraguay 1960–5 Cyprus 1960–5 Peru 1965–70 Denmark 1960–5 Philippines 1960–5 Dominican Republic 1960–5 Portugal 1960–5 Ecuador 1960–5 Senegal 1960–5 Egypt 1960–5 Sierra Leone 1965–70 El Salvador 1960–5 Singapore 1965–70 Finland 1960–5 South Africa 1960–5 France 1960–5 South Korea 1965–70 Gabon 1965–70 Spain 1960–5 Gambia 1965–70 Sri Lanka 1960–5 Germany 1970–5 Sudan 1970–5 Ghana 1960–5 Sweden 1960–5 Greece 1960–5 Switzerland 1960–5 Guatemala 1960–5 Syria 1970–5 Guyana 1970–5 Taiwan 1960–5 Haiti 1960–5 Tanzania 1970–5 Honduras 1960–5 Thailand 1960–5 Hungary 1970–5 Togo 1965–70 Iceland 1960–5 Trinidad 1960–5 India 1960–5 Tunisia 1965–70 Indonesia 1965–70 Turkey 1965–70 Ireland 1960–5 Uganda 1965–70 Israel 1970–5 UK 1960–5 Italy 1960–5 Uruguay 1965–70 Ivory Coast 1960–5 US 1960–5 Jamaica 1960–5 Venezuela 1965–70 Japan 1960–5 Zambia 1965–70 Jordan 1965–70 Country . Starting period . Country . Starting period . Algeria 1960–5 Kenya 1960–5 Argentina 1960–5 Luxembourg 1960–5 Australia 1960–5 Malawi 1965–70 Austria 1960–5 Malaysia 1960–5 Bahrain 1970–5 Mali 1965–70 Bangladesh 1965–70 Malta 1970–5 Belgium 1960–5 Mexico 1960–5 Bolivia 1965–70 Morocco 1960–5 Botswana 1965–70 Netherlands 1960–5 Brazil 1960–5 New Zealand 1960–5 Cameroon 1965–70 Nicaragua 1960–5 Canada 1960–5 Niger 1960–5 Chile 1960–5 Norway 1960–5 China 1960–5 Pakistan 1960–5 Colombia 1960–5 Panama 1965–70 Congo, Republic 1960–5 Papua New Guinea 1960–5 Costa Rica 1960–5 Paraguay 1960–5 Cyprus 1960–5 Peru 1965–70 Denmark 1960–5 Philippines 1960–5 Dominican Republic 1960–5 Portugal 1960–5 Ecuador 1960–5 Senegal 1960–5 Egypt 1960–5 Sierra Leone 1965–70 El Salvador 1960–5 Singapore 1965–70 Finland 1960–5 South Africa 1960–5 France 1960–5 South Korea 1965–70 Gabon 1965–70 Spain 1960–5 Gambia 1965–70 Sri Lanka 1960–5 Germany 1970–5 Sudan 1970–5 Ghana 1960–5 Sweden 1960–5 Greece 1960–5 Switzerland 1960–5 Guatemala 1960–5 Syria 1970–5 Guyana 1970–5 Taiwan 1960–5 Haiti 1960–5 Tanzania 1970–5 Honduras 1960–5 Thailand 1960–5 Hungary 1970–5 Togo 1965–70 Iceland 1960–5 Trinidad 1960–5 India 1960–5 Tunisia 1965–70 Indonesia 1965–70 Turkey 1965–70 Ireland 1960–5 Uganda 1965–70 Israel 1970–5 UK 1960–5 Italy 1960–5 Uruguay 1965–70 Ivory Coast 1960–5 US 1960–5 Jamaica 1960–5 Venezuela 1965–70 Japan 1960–5 Zambia 1965–70 Jordan 1965–70 Open in new tab Other findings in Table 1, column 3, are that countries' long‐run positions are positively associated with a lower mortality rate (gauged by the reciprocal of life expectancy at birth), a lower fertility rate and higher female relative to male school attainment for persons aged 15 and over.21 The schooling effect does not relate to the overall level of human capital in the sense of total years of attainment. In fact, the estimated effect from a general increase in attainment – where average years of female and male schooling rise by equal amounts – differs insignificantly from zero, a result that is consistent with the conceptual analysis of human capital presented earlier. A reasonable interpretation is that an expansion in female relative to male attainment signals an improvement more generally in political and social arrangements that support economic growth. The results in Table 1, column 3, show a significantly positive effect from greater international openness (exports plus imports relative to GDP) and from a higher growth rate of a country's terms of trade (entered as an interaction with the international‐openness variable). The estimates also reveal a significantly positive coefficient for the ratio of investment to GDP and a significantly negative coefficient for the inflation rate. However, instrumental‐variable estimates discussed later suggest that these last results may reflect reverse causation from growth to investment (positive) or inflation (negative), rather than the reverse. Finally, the estimated coefficient on the ratio of government consumption to GDP is negative but not statistically significantly different from zero. As discussed before, the Hurwicz–Nickell bias in the estimated coefficient of a lagged dependent variable is likely to be small in the context of column 3, where country fixed effects are absent. The magnitude of the estimated coefficient of the log of lagged per capita GDP would tend to be underestimated if there are still important omitted determinants of long‐run per capita GDP that are conditionally correlated with per capita GDP. However, given the substantial list of growth determinants included, this omitted‐variables bias may be small (unlike in column 1). Therefore, it is possible that OLS without country fixed effects can deliver accurate estimates of the convergence rate, as confirmed in the Appendix by Monte Carlo analysis (Table A2, lines 9–12). Table 1, column 4, allows for country fixed effects along with the X variables. The fixed effects are still jointly highly statistically significant. The estimated coefficient of the log of lagged per capita GDP, −0.0458 (SE = 0.0045), is negative, statistically significant, and larger in magnitude than in columns 2 or 3. The indicated conditional convergence rate is around 4.6% per year. However, as discussed before, the convergence rate tends to be overestimated because of the Hurwicz–Nickell bias (as confirmed by the Monte Carlo studies considered in the Appendix; see Table A2, lines 1 and 3).22 Another effect from the introduction of country fixed effects is that the standard errors of the estimated coefficients of the X variables are mostly much higher in column 4 than in column 3. This pattern arises because, with country fixed effects included, only within‐country variation over time is used to identify the coefficients. Consequently, some of the estimated coefficients that were statistically significantly different from zero in column 3 are no longer statistically significant in column 4. As an example, although the estimated coefficient on the indicator for the maintenance of law and order was significantly positive in column 3, this variable no longer has significant explanatory power in column 4. The likely explanation is that there is insufficient within‐country variation in the measured institutional quality to isolate a statistically significant effect on economic growth. Table 3 contains additional panel regressions for growth rates. Column 1 parallels Table 1, column 3, but uses two‐stage least squares instead of OLS. The instrument list replaces the five‐year lag of the log of per capita GDP with the six‐year lag and replaces averages of several variables lagged 1‐to–5 years with five‐year lagged values. Thus, the idea is to use longer lags of some of the X variables as instruments. The results in Table 3, column 1, are similar in most respects to the OLS results (Table 1, column 3), except that the estimated coefficients of the investment ratio and the inflation rate are no longer statistically significantly different from zero. These results are likely to reflect the joint short‐run determination of GDP with investment and inflation, whereby the typical pattern is for the investment ratio to be procyclical and the inflation rate to be countercyclical. Instrumenting with five‐year lags eliminates most of this high‐frequency interaction, and the remaining partial association of economic growth with investment and inflation turns out to be weak. The bottom line is that the OLS results on these estimated coefficients (in Table 1, column 3) cannot be reliably interpreted as isolating causation from the investment ratio or the inflation rate to GDP growth.23 An important finding in Table 3, column 1, is that the estimated coefficient on lagged per capita GDP, −0.0173 (SE = 0.0022), is virtually the same as that in Table 1, column 3. Thus, the estimated convergence rate – around 1.7% per year – is robust to using longer lags of some of the X variables as instruments. This robustness in the estimated coefficient of lagged GDP suggests that endogeneity of the X variables may not be a central issue for the purpose of estimating convergence rates. Table 3, column 2, uses 10‐year averages for growth rates, with corresponding adjustments in the definitions of the X variables. The results are similar to those from the five‐year case in Table 1, column 3. In particular, the estimated convergence rates per year are close: −0.0166 (SE = 0.0020) for the 10‐year case, versus −0.0170 (SE = 0.0021) for the five‐year. The same general conclusions hold with annual data on GDP growth rates. For the convergence rate, in a system without country fixed effects, the OLS estimate of the convergence coefficient is −0.0167 (SE = 0.0023), similar to those in Table 1, column 3, and Table 3, column 2.24 The results show that changing the time dimension of the sample by observing the data more or less frequently – shifting, say, from 10‐year to 5‐year to 1‐year periods – has minor implications, particularly for the estimated convergence rate. As discussed before, this finding is consistent with the formula for the Hurwicz–Nickell bias derived by Nickell (1981) and shown in (1). Table 3 Additional Growth‐rate Regressions for Cross‐country Panel (All Equations Include Time Effects and Exclude Country Fixed Effects) . (1) . (2) . (3) . (4) . . 2SLS . OLS 10‐year periods . OLS two sets of fixed effects . OLS polity variable . Log(lagged per capita GDP) −0.0173** −0.0166** −0.0745** −0.0150** (0.0022) (0.0020) (0.0066) (0.0022) 1/(life expectancy at birth) −3.32** −3.29** −2.32 −2.81** (0.63) (0.58) (1.43) (0.60) Log(fertility rate) −0.0303** −0.0242** −0.0298** −0.0260** (0.0045) (0.0042) (0.0099) (0.0044) Law and order (rule of law) 0.0159** 0.0173** 0.0149 0.0121* (0.0059) (0.0053) (0.0122) (0.0054) Investment ratio 0.005 0.031* 0.051* 0.023 (0.014) (0.012) (0.024) (0.013) Female school years 0.0025 0.0014 0.0066 0.0013 (0.0014) (0.0014) (0.0043) (0.0014) Male school years −0.0030* −0.0019 −0.0059 −0.0017 (0.0015) (0.0014) (0.0039) (0.0015) Government consumption ratio −0.031 −0.014 −0.112* −0.037 (0.026) (0.023) (0.052) (0.024) Openness ratio 0.0072** 0.0042 0.0224** 0.0053 (0.0028) (0.0027) (0.0085) (0.0028) Terms‐of‐trade change 0.129** 0.129* 0.116** 0.134** (0.027) (0.038) (0.029) (0.027) Democracy indicator 0.033 0.036* −0.005 0.022 (0.018) (0.016) (0.021) (0.018) Democracy squared −0.035* −0.032* 0.002 −0.023 (0.017) (0.014) (0.019) (0.016) Inflation rate −0.0088 −0.0171** −0.0135** −0.0172** (0.0091) (0.0055) (0.0050) (0.0042) R2 0.322 0.448 0.616 0.316 SE of regression 0.0243 0.0179 0.0206 0.0237 No. countries; observations 89; 820 89; 407 89; 841 86; 755 . (1) . (2) . (3) . (4) . . 2SLS . OLS 10‐year periods . OLS two sets of fixed effects . OLS polity variable . Log(lagged per capita GDP) −0.0173** −0.0166** −0.0745** −0.0150** (0.0022) (0.0020) (0.0066) (0.0022) 1/(life expectancy at birth) −3.32** −3.29** −2.32 −2.81** (0.63) (0.58) (1.43) (0.60) Log(fertility rate) −0.0303** −0.0242** −0.0298** −0.0260** (0.0045) (0.0042) (0.0099) (0.0044) Law and order (rule of law) 0.0159** 0.0173** 0.0149 0.0121* (0.0059) (0.0053) (0.0122) (0.0054) Investment ratio 0.005 0.031* 0.051* 0.023 (0.014) (0.012) (0.024) (0.013) Female school years 0.0025 0.0014 0.0066 0.0013 (0.0014) (0.0014) (0.0043) (0.0014) Male school years −0.0030* −0.0019 −0.0059 −0.0017 (0.0015) (0.0014) (0.0039) (0.0015) Government consumption ratio −0.031 −0.014 −0.112* −0.037 (0.026) (0.023) (0.052) (0.024) Openness ratio 0.0072** 0.0042 0.0224** 0.0053 (0.0028) (0.0027) (0.0085) (0.0028) Terms‐of‐trade change 0.129** 0.129* 0.116** 0.134** (0.027) (0.038) (0.029) (0.027) Democracy indicator 0.033 0.036* −0.005 0.022 (0.018) (0.016) (0.021) (0.018) Democracy squared −0.035* −0.032* 0.002 −0.023 (0.017) (0.014) (0.019) (0.016) Inflation rate −0.0088 −0.0171** −0.0135** −0.0172** (0.0091) (0.0055) (0.0050) (0.0042) R2 0.322 0.448 0.616 0.316 SE of regression 0.0243 0.0179 0.0206 0.0237 No. countries; observations 89; 820 89; 407 89; 841 86; 755 Notes * Significant at 5% level. ** Significant at 1% level. See the notes to Table 1. Column 1 corresponds to column 3 of Table 1, except that the instrument list replaces some of the explanatory variables with longer lags. For the log of lagged real per capita GDP, the instruments are the values for 1959, 1964, … , 2004. For the ratios of investment and government consumption to GDP, the openness ratio, the indicators for law‐and‐order and democracy, and the inflation rate, the instruments are five‐year lags. The p‐value for the two democracy variables jointly is 0.094. Column 2 corresponds to column 3 of Table 1, except that the dependent variable is the growth rate of per capita GDP for the five ‘10‐year’ periods 1960–70, … , 2000–10. The regressors are defined analogously for the 10‐year periods. Column 3 corresponds to column 4 of Table 1 but has one set of fixed effects for the five five‐year growth‐rate observations from 1960–5 to 1980–5 and another set for the five observations from 1985–90 to 2005–10. Column 4 is the same as column 3 of Table 1, except that the democracy indicator (the Freedom House measure of political rights) is replaced by the Polity measure of democracy less autocracy. The joint p‐value for the two Polity variables is 0.35. The sample drops to 86 countries because of missing Polity data for Iceland, Luxembourg and Malta. Sources. The Polity indicator is for democracy less autocracy (converted from a −10 to +10 scale to a 0–1 scale, with 1 representing highest democracy), from Polity IV (www.systemicpeace.org). Other sources are given in the notes to Table 1. Open in new tab Table 3 Additional Growth‐rate Regressions for Cross‐country Panel (All Equations Include Time Effects and Exclude Country Fixed Effects) . (1) . (2) . (3) . (4) . . 2SLS . OLS 10‐year periods . OLS two sets of fixed effects . OLS polity variable . Log(lagged per capita GDP) −0.0173** −0.0166** −0.0745** −0.0150** (0.0022) (0.0020) (0.0066) (0.0022) 1/(life expectancy at birth) −3.32** −3.29** −2.32 −2.81** (0.63) (0.58) (1.43) (0.60) Log(fertility rate) −0.0303** −0.0242** −0.0298** −0.0260** (0.0045) (0.0042) (0.0099) (0.0044) Law and order (rule of law) 0.0159** 0.0173** 0.0149 0.0121* (0.0059) (0.0053) (0.0122) (0.0054) Investment ratio 0.005 0.031* 0.051* 0.023 (0.014) (0.012) (0.024) (0.013) Female school years 0.0025 0.0014 0.0066 0.0013 (0.0014) (0.0014) (0.0043) (0.0014) Male school years −0.0030* −0.0019 −0.0059 −0.0017 (0.0015) (0.0014) (0.0039) (0.0015) Government consumption ratio −0.031 −0.014 −0.112* −0.037 (0.026) (0.023) (0.052) (0.024) Openness ratio 0.0072** 0.0042 0.0224** 0.0053 (0.0028) (0.0027) (0.0085) (0.0028) Terms‐of‐trade change 0.129** 0.129* 0.116** 0.134** (0.027) (0.038) (0.029) (0.027) Democracy indicator 0.033 0.036* −0.005 0.022 (0.018) (0.016) (0.021) (0.018) Democracy squared −0.035* −0.032* 0.002 −0.023 (0.017) (0.014) (0.019) (0.016) Inflation rate −0.0088 −0.0171** −0.0135** −0.0172** (0.0091) (0.0055) (0.0050) (0.0042) R2 0.322 0.448 0.616 0.316 SE of regression 0.0243 0.0179 0.0206 0.0237 No. countries; observations 89; 820 89; 407 89; 841 86; 755 . (1) . (2) . (3) . (4) . . 2SLS . OLS 10‐year periods . OLS two sets of fixed effects . OLS polity variable . Log(lagged per capita GDP) −0.0173** −0.0166** −0.0745** −0.0150** (0.0022) (0.0020) (0.0066) (0.0022) 1/(life expectancy at birth) −3.32** −3.29** −2.32 −2.81** (0.63) (0.58) (1.43) (0.60) Log(fertility rate) −0.0303** −0.0242** −0.0298** −0.0260** (0.0045) (0.0042) (0.0099) (0.0044) Law and order (rule of law) 0.0159** 0.0173** 0.0149 0.0121* (0.0059) (0.0053) (0.0122) (0.0054) Investment ratio 0.005 0.031* 0.051* 0.023 (0.014) (0.012) (0.024) (0.013) Female school years 0.0025 0.0014 0.0066 0.0013 (0.0014) (0.0014) (0.0043) (0.0014) Male school years −0.0030* −0.0019 −0.0059 −0.0017 (0.0015) (0.0014) (0.0039) (0.0015) Government consumption ratio −0.031 −0.014 −0.112* −0.037 (0.026) (0.023) (0.052) (0.024) Openness ratio 0.0072** 0.0042 0.0224** 0.0053 (0.0028) (0.0027) (0.0085) (0.0028) Terms‐of‐trade change 0.129** 0.129* 0.116** 0.134** (0.027) (0.038) (0.029) (0.027) Democracy indicator 0.033 0.036* −0.005 0.022 (0.018) (0.016) (0.021) (0.018) Democracy squared −0.035* −0.032* 0.002 −0.023 (0.017) (0.014) (0.019) (0.016) Inflation rate −0.0088 −0.0171** −0.0135** −0.0172** (0.0091) (0.0055) (0.0050) (0.0042) R2 0.322 0.448 0.616 0.316 SE of regression 0.0243 0.0179 0.0206 0.0237 No. countries; observations 89; 820 89; 407 89; 841 86; 755 Notes * Significant at 5% level. ** Significant at 1% level. See the notes to Table 1. Column 1 corresponds to column 3 of Table 1, except that the instrument list replaces some of the explanatory variables with longer lags. For the log of lagged real per capita GDP, the instruments are the values for 1959, 1964, … , 2004. For the ratios of investment and government consumption to GDP, the openness ratio, the indicators for law‐and‐order and democracy, and the inflation rate, the instruments are five‐year lags. The p‐value for the two democracy variables jointly is 0.094. Column 2 corresponds to column 3 of Table 1, except that the dependent variable is the growth rate of per capita GDP for the five ‘10‐year’ periods 1960–70, … , 2000–10. The regressors are defined analogously for the 10‐year periods. Column 3 corresponds to column 4 of Table 1 but has one set of fixed effects for the five five‐year growth‐rate observations from 1960–5 to 1980–5 and another set for the five observations from 1985–90 to 2005–10. Column 4 is the same as column 3 of Table 1, except that the democracy indicator (the Freedom House measure of political rights) is replaced by the Polity measure of democracy less autocracy. The joint p‐value for the two Polity variables is 0.35. The sample drops to 86 countries because of missing Polity data for Iceland, Luxembourg and Malta. Sources. The Polity indicator is for democracy less autocracy (converted from a −10 to +10 scale to a 0–1 scale, with 1 representing highest democracy), from Polity IV (www.systemicpeace.org). Other sources are given in the notes to Table 1. Open in new tab Table 3, column 3, goes further in the exploration of the implications of country fixed effects by allowing for two separate sets of country dummy variables – one applying to the first five periods (1960–5, … , 1980–5) and the other to the second five (1985–90, … , 2005–10). The break in the set of country fixed effects from the first half to the second half of the sample is jointly highly statistically significant, as are the fixed effects overall. Not surprisingly, the standard errors of the estimated coefficients of all the X variables are higher than those in Table 1, column 4 (with one set of country fixed effects), which were in turn mostly higher than those in Table 1, column 3 (with no fixed effects). That is, the richer structure of country fixed effects makes it even more difficult to use the within‐country variation to estimate the effects of the X variables. For present purposes, the most interesting result is that the estimated coefficient on the log of lagged per capita GDP, −0.0745 (SE = 0.0066), becomes even larger in magnitude than that in Table 1, column 4, now indicating convergence at about 7.4% per year. A reasonable interpretation is that the extra set of country fixed effects effectively cuts the time dimension of the sample by half (from 50 years to 25 years) and, thereby, intensifies the Hurwicz–Nickell bias. Hence, these findings provide a further warning that country fixed effects can cause serious upward bias in estimated convergence rates. Finally, Table 3, column 4, uses as a democracy indicator the Polity measure of democracy/autocracy, rather than the Freedom House measure of political rights. The overall results are similar to those based on the Freedom House indicator (Table 1, column 3), but the estimated coefficients on the Polity variable and its square are neither individually nor jointly statistically significantly different from zero. My inference from the cross‐country data starting in 1960 is that the most reliable estimates of convergence rates come from systems that exclude country fixed effects but include an array of time‐varying X variables to mitigate the consequences of omitted variables. That is, I would emphasise the results in Table 1, column 3 (based on OLS) or Table 3, column 1 (which uses longer lags of some of the X variables as instruments). However, as I discuss later, the fixed‐effects results seem more reliable in systems for the more limited set of countries that have data over a century or more. 5. Modernisation The modernisation thesis applied to democracy is that economic development – gauged particularly by per capita GDP and education – promotes democratic institutions. This idea was emphasised by Lipset (1959), who credits the concept to Aristotle.25 Glaeser et al. (2007) provide a theoretical rationale for the effect of education on democracy through the channel of higher education motivating greater participation in political and other social activities. The Aristotle–Lipset hypothesis can be extended beyond democracy to apply to institutional quality, including the indicator used earlier for maintenance of law and order. Barro (1999) provided empirical confirmation of the Aristotle–Lipset hypothesis in a cross‐country panel, with stress on the Freedom House measure of political rights. Additional supporting evidence along these lines appears in Glaeser et al. (2004, 2007). However, these results have been challenged by Acemoglu et al. (2005, 2008), who argue that education and per capita GDP do not have statistically significant influences on democracy. As in estimating convergence rates for economic growth, the essence of much of this recent debate about modernisation turns on whether the empirical framework includes country fixed effects. Table 4 contains panel regressions in which the dependent variables are indicators of institutional quality. The estimation method, parallel to Table 1, is panel least‐squares. Columns 1 and 2 use International Country Risk Guide's indicator for maintenance of law‐and‐order (previously described by ICRG as maintenance of the rule of law). Columns 3 and 4 use the Freedom House measure of democracy (political rights), and columns 5 and 6 use the Polity measure of democracy (calculated as differences between the Polity measures of democracy and autocracy). To make the analysis comparable to that for economic growth, the estimation applies to the sample of countries used in Table 1, columns 3 and 4, and listed in Table 2.26 Because of data availability, the time frame for the law‐and‐order indicator is from 1985 to 2010 and that for the Freedom House measure of democracy is from 1970 to 2010. The Polity analysis is limited to 1970–2010 to make the results comparable to those for Freedom House. The framework for the modernisation equations parallels that for economic growth. Thus, for five‐year periods, if Yt is the level of an indicator of institutional quality in year t, then the dependent variable is (Yt − Yt−5)/5, and the lag associated with this dependent variable is Yt−5 The system includes on the right‐hand side a set of X variables, analogous to those used to explain economic growth. In this setting, the coefficient on Yt−5 gives the convergence rate per year over a five‐year period. This convergence is conditional on the X variables, just as in the case of economic growth. In the present study of institutional indicators, the X variables are limited to those stressed in the literature on modernisation – the log of per capita GDP and measures of school attainment (distinguished by gender). The variables used in the regressions – for the log of per capita GDP and the schooling variables – are five‐year lags, as in the equations for economic growth. Table 4, column 1, applies to the law‐and‐order indicator. The estimated convergence coefficient, −0.0577 (SE = 0.0056), is statistically significant and indicates conditional convergence at around 5.8% per year. Therefore, the convergence rate for this measure of institutional quality is notably higher than that for the log of per capita GDP, as computed from the analogous growth‐rate equation in Table 1, column 3. With respect to X variables, the estimated coefficient on the log of per capita GDP (lagged five years) in Table 4, column 1, is significantly positive, 0.0051 (SE = 0.0018). The sum of the estimated coefficients for female and male school attainment (lagged five years) is also significantly positive (p–value = 0.020). However, the estimated coefficient for male schooling is positive and statistically significant (0.0056, SE = 0.0015), whereas that for female schooling is negative and statistically significant (−0.0037, SE = 0.0014). The results reject with a p‐value of 0.000 the joint hypothesis that the coefficient on the GDP variable is zero and the sum of the coefficients for female and male schooling is zero against the alternative that each of these is positive. Hence, the system that excludes country fixed effects provides evidence of modernisation with regard to the overall association between the law‐and‐order indicator and the GDP and schooling variables. Table 4 Regressions for Changes in Indicators of Law and Order and Democracy Five‐year Periods (All Equations Estimated by OLS and Include Time Effects) . (1) . (2) . (3) . (4) . (5) . (6) . . Law and order: 1985–90, … , 2005–10 . Political rights (freedom house): 1970–5, … , 2005–10 . Democracy/autocracy (polity): 1970–5, … , 2005–10 . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Lagged indicator −0.0577** −0.130** −0.0581** −0.1184** −0.0505** −0.1018** (0.0056) (0.010) (0.0052) (0.0086) (0.0046) (0.0090) Log(per capita GDP) 0.0051** −0.0030 0.0040* −0.0051 0.0012 −0.0078 (0.0018) (0.0090) (0.0019) (0.0082) (0.0018) (0.0087) Female school years −0.0037** 0.0056 0.0036* 0.0011 0.0037* 0.0013 (0.0015) (0.0054) (0.0017) (0.0056) (0.0016) (0.0059) Male school years 0.0056** −0.0048 −0.0007 −0.0026 −0.0013 0.0019 (0.0015) (0.0053) (0.0017) (0.0053) (0.0017) (0.0056) p‐value for school‐years variables > 0 0.020 0.82 0.001 0.67 0.003 0.34 p‐value for GDP> 0, school‐years variables > 0 0.000 0.82 0.000 0.67 0.000 0.34 R2 0.392 0.584 0.173 0.356 0.187 0.331 SE of regression 0.0241 0.0223 0.0352 0.0333 0.0339 0.0330 No. countries; observations 89; 445 89; 445 89; 712 89; 712 86; 653 86; 653 . (1) . (2) . (3) . (4) . (5) . (6) . . Law and order: 1985–90, … , 2005–10 . Political rights (freedom house): 1970–5, … , 2005–10 . Democracy/autocracy (polity): 1970–5, … , 2005–10 . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Lagged indicator −0.0577** −0.130** −0.0581** −0.1184** −0.0505** −0.1018** (0.0056) (0.010) (0.0052) (0.0086) (0.0046) (0.0090) Log(per capita GDP) 0.0051** −0.0030 0.0040* −0.0051 0.0012 −0.0078 (0.0018) (0.0090) (0.0019) (0.0082) (0.0018) (0.0087) Female school years −0.0037** 0.0056 0.0036* 0.0011 0.0037* 0.0013 (0.0015) (0.0054) (0.0017) (0.0056) (0.0016) (0.0059) Male school years 0.0056** −0.0048 −0.0007 −0.0026 −0.0013 0.0019 (0.0015) (0.0053) (0.0017) (0.0053) (0.0017) (0.0056) p‐value for school‐years variables > 0 0.020 0.82 0.001 0.67 0.003 0.34 p‐value for GDP> 0, school‐years variables > 0 0.000 0.82 0.000 0.67 0.000 0.34 R2 0.392 0.584 0.173 0.356 0.187 0.331 SE of regression 0.0241 0.0223 0.0352 0.0333 0.0339 0.0330 No. countries; observations 89; 445 89; 445 89; 712 89; 712 86; 653 86; 653 Notes * Significant at 5% level. ** Significant at 1% level. See the notes to Table 1. The dependent variable is (Yt − Yt−5)/5, where Yt is the level of an indicator variable in year t. Columns 1 and 2 use for Yt the indicator for law and order from International Country Risk Guide, observed at the five dates: 1990, 1995, 2000, 2005 and 2010. Columns 3 and 4, use the Freedom House measure of political rights, observed at the eight dates: 1975, … , 2010. (Values for 1970 are interpolated based on data for 1965 (from Bollen, 1980, 1972.) Columns 5 and 6 use the Polity measure of democracy less autocracy, observed at the eight dates: 1975, … , 2010. The explanatory variables apply to 1985, … , 2005 in columns 1–2 and 1970, … , 2005 in columns 3–6. Country fixed effects are included in columns 2, 4 and 6. The p‐value for school‐years variables > 0 is a test of the hypothesis that the sum of the coefficients of the school‐years variables is zero against the alternative of being positive. The p‐value for GDP > 0, school‐years variables > 0 is a test of the hypothesis that the coefficient on the GDP variable is zero and the sum of the coefficients of the school‐years variables is zero against the alternative that each of these is positive. Open in new tab Table 4 Regressions for Changes in Indicators of Law and Order and Democracy Five‐year Periods (All Equations Estimated by OLS and Include Time Effects) . (1) . (2) . (3) . (4) . (5) . (6) . . Law and order: 1985–90, … , 2005–10 . Political rights (freedom house): 1970–5, … , 2005–10 . Democracy/autocracy (polity): 1970–5, … , 2005–10 . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Lagged indicator −0.0577** −0.130** −0.0581** −0.1184** −0.0505** −0.1018** (0.0056) (0.010) (0.0052) (0.0086) (0.0046) (0.0090) Log(per capita GDP) 0.0051** −0.0030 0.0040* −0.0051 0.0012 −0.0078 (0.0018) (0.0090) (0.0019) (0.0082) (0.0018) (0.0087) Female school years −0.0037** 0.0056 0.0036* 0.0011 0.0037* 0.0013 (0.0015) (0.0054) (0.0017) (0.0056) (0.0016) (0.0059) Male school years 0.0056** −0.0048 −0.0007 −0.0026 −0.0013 0.0019 (0.0015) (0.0053) (0.0017) (0.0053) (0.0017) (0.0056) p‐value for school‐years variables > 0 0.020 0.82 0.001 0.67 0.003 0.34 p‐value for GDP> 0, school‐years variables > 0 0.000 0.82 0.000 0.67 0.000 0.34 R2 0.392 0.584 0.173 0.356 0.187 0.331 SE of regression 0.0241 0.0223 0.0352 0.0333 0.0339 0.0330 No. countries; observations 89; 445 89; 445 89; 712 89; 712 86; 653 86; 653 . (1) . (2) . (3) . (4) . (5) . (6) . . Law and order: 1985–90, … , 2005–10 . Political rights (freedom house): 1970–5, … , 2005–10 . Democracy/autocracy (polity): 1970–5, … , 2005–10 . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Lagged indicator −0.0577** −0.130** −0.0581** −0.1184** −0.0505** −0.1018** (0.0056) (0.010) (0.0052) (0.0086) (0.0046) (0.0090) Log(per capita GDP) 0.0051** −0.0030 0.0040* −0.0051 0.0012 −0.0078 (0.0018) (0.0090) (0.0019) (0.0082) (0.0018) (0.0087) Female school years −0.0037** 0.0056 0.0036* 0.0011 0.0037* 0.0013 (0.0015) (0.0054) (0.0017) (0.0056) (0.0016) (0.0059) Male school years 0.0056** −0.0048 −0.0007 −0.0026 −0.0013 0.0019 (0.0015) (0.0053) (0.0017) (0.0053) (0.0017) (0.0056) p‐value for school‐years variables > 0 0.020 0.82 0.001 0.67 0.003 0.34 p‐value for GDP> 0, school‐years variables > 0 0.000 0.82 0.000 0.67 0.000 0.34 R2 0.392 0.584 0.173 0.356 0.187 0.331 SE of regression 0.0241 0.0223 0.0352 0.0333 0.0339 0.0330 No. countries; observations 89; 445 89; 445 89; 712 89; 712 86; 653 86; 653 Notes * Significant at 5% level. ** Significant at 1% level. See the notes to Table 1. The dependent variable is (Yt − Yt−5)/5, where Yt is the level of an indicator variable in year t. Columns 1 and 2 use for Yt the indicator for law and order from International Country Risk Guide, observed at the five dates: 1990, 1995, 2000, 2005 and 2010. Columns 3 and 4, use the Freedom House measure of political rights, observed at the eight dates: 1975, … , 2010. (Values for 1970 are interpolated based on data for 1965 (from Bollen, 1980, 1972.) Columns 5 and 6 use the Polity measure of democracy less autocracy, observed at the eight dates: 1975, … , 2010. The explanatory variables apply to 1985, … , 2005 in columns 1–2 and 1970, … , 2005 in columns 3–6. Country fixed effects are included in columns 2, 4 and 6. The p‐value for school‐years variables > 0 is a test of the hypothesis that the sum of the coefficients of the school‐years variables is zero against the alternative of being positive. The p‐value for GDP > 0, school‐years variables > 0 is a test of the hypothesis that the coefficient on the GDP variable is zero and the sum of the coefficients of the school‐years variables is zero against the alternative that each of these is positive. Open in new tab As usual, this least‐squares estimation does not provide clear evidence of causation; in this case, from GDP and school attainment to law‐and‐order, rather than the reverse. However, using two‐stage least‐squares, with longer lags of the GDP and school‐attainment variables used as instruments, produces only minor changes in the conclusions. This finding applies also to the other equations in Table 4. Table 4, column 2, shows the impact of allowing for country fixed effects, which are jointly highly statistically significant. The results change compared with column 1 in ways familiar from the analysis of economic growth: the estimated convergence rate rises sharply, to around 13% per year, and the standard errors of all the estimated coefficients of the explanatory variables increase substantially. Consequently, none of the estimated coefficients on the three X variables (for the log of per capita GDP, female schooling and male schooling) are individually statistically significantly different from zero. The sum of the two coefficients for the schooling variables is also insignificant (p‐value = 0.82). Hence, the inclusion of country fixed effects eliminates the evidence in favour of modernisation with regard to the law‐and‐order indicator. Results for democracy are in Table 4, columns 3–6. For the equations without country fixed effects, the estimated convergence coefficients are negative and statistically significant for the Freedom House measure (column 3) and the Polity measure (column 5). The estimated rates of convergence are 5–6% per year, similar to that found for the law‐and‐order indicator (column 1). The estimated coefficient on the lagged GDP variable is significantly positive for the Freedom House data (column 3) but not for the Polity data (column 5). The sums of the estimated coefficients on the schooling variables are significantly positive, with p‐values of 0.001 for Freedom House (column 3) and 0.003 for Polity (column 5). The results reject with p‐values of 0.000 the joint hypothesis that the coefficient on the GDP variable is zero and the sum of the coefficients for female and male schooling is zero against the alternative that each of these is positive (columns 3 and 5). Thus, the results without country fixed effects support the modernisation hypothesis with respect to the two measures of democracy. The introduction of country fixed effects again raises the estimated rates of convergence – to 12% per year in column 4 and 10% in column 6. As before, the standard errors of all the estimated coefficients rise substantially. This change makes the estimated coefficients on the GDP and schooling variables individually and jointly insignificant (columns 4 and 6). Hence, results with country fixed effects do not support the modernisation hypothesis with regard to the measures of democracy. Acemoglu et al. (2005, 2008) rely on results with country fixed effects to argue that the modernisation hypothesis is not supported empirically in cross‐country panel data. My inference is that their results signal the dangers from the inclusion of country fixed effects: this procedure reduces the information contained in the panel data (by blowing up the standard errors of the estimated coefficients of X variables) and biases upward the estimated convergence rates.27 I show in the following Section that the results with country fixed effects look different – and supportive of the modernisation hypothesis – in systems estimated over time frames of over a century. For systems estimated with data spanning 25 to 40 years – starting between 1970 and 1985 and ending in 2010 – the most reliable (though imperfect) information on the determinants of institutional quality likely comes from systems that exclude country fixed effects. Thus, in Table 4, I would emphasise the findings in columns 1, 3 and 5. These results support the modernisation hypothesis with regard to the relationships of GDP and schooling to the quality of institutions, gauged by law and order and democracy. 6. Long‐term Panels Until recently, the best long‐term macroeconomic panel data were the per capita GDP series assembled by Maddison (2003). These series constitute a monumental contribution that has been widely used. However, the data have serious shortcomings, discussed and largely rectified in a new data set on per capita GDP and consumer expenditure assembled by Ursúa and myself, described in Ursúa (2011).28 The construction of these data was challenging, described by Ursúa as macroeconomic archaeology, and various methods were implemented to include periods and countries originally missing or inadequately treated in standard sources. The new data set covers 42 countries (40 with nearly continuous annual data) and goes back at least to 1913 and in many cases to 1870 or earlier. For the present analysis, I use a sample of 28 countries selected by the availability of annual data on per capita GDP by 1896 and by the availability of the Polity indicator for democracy. The list of countries is in the notes to Table 5. There is a limited supply of X variables available over this long time frame. My analysis uses the Polity indicator of democracy and recently constructed data since 1870 on average years of school attainment by females and males.29 Table 5 Regressions for Long‐term Panels Five‐year periods: 1870–5, … , 2005–10 Estimation by Panel Least‐squares; All Equations Include Time Effects . (1) . (2) . (3) . (4) . (5) . (6) . . Growth rate of per capita GDP . Polity indicator . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) −0.0044** −0.0258** −0.0090** −0.0262** 0.0068** 0.0108* (0.0013) (0.0041) (0.0020) (0.0041) (0.0022) (0.0045) Female school years – – −0.0009 −0.0026 0.0025 −0.0013 (0.0018) (0.0025) (0.0019) (0.0027) Male school years – – 0.0020 −0.0009 −0.0008 0.0015 (0.0019) (0.0026) (0.0021) (0.0028) Democracy indicator (polity) – – −0.0296 −0.0323 −0.0390** −0.0510** (0.0161) (0.0188) (0.0048) (0.0056) Democracy indicator squared – – 0.0332* 0.0341* – – (0.0144) (0.0168) p‐value for democracy and democracy squared – – 0.025 0.082 – – p‐value for school‐years variables > 0 – – 0.126 1.0 0.028 0.88 p‐value for GDP > 0, school‐years variables > 0 – – – – 0.000 0.056 R2 0.228 0.267 0.211 0.255 0.146 0.183 SE of regression 0.0281 0.0279 0.0267 0.0264 0.0285 0.0285 No. countries; observations 28; 764 28; 764 28; 727 28; 727 28; 700 28; 700 . (1) . (2) . (3) . (4) . (5) . (6) . . Growth rate of per capita GDP . Polity indicator . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) −0.0044** −0.0258** −0.0090** −0.0262** 0.0068** 0.0108* (0.0013) (0.0041) (0.0020) (0.0041) (0.0022) (0.0045) Female school years – – −0.0009 −0.0026 0.0025 −0.0013 (0.0018) (0.0025) (0.0019) (0.0027) Male school years – – 0.0020 −0.0009 −0.0008 0.0015 (0.0019) (0.0026) (0.0021) (0.0028) Democracy indicator (polity) – – −0.0296 −0.0323 −0.0390** −0.0510** (0.0161) (0.0188) (0.0048) (0.0056) Democracy indicator squared – – 0.0332* 0.0341* – – (0.0144) (0.0168) p‐value for democracy and democracy squared – – 0.025 0.082 – – p‐value for school‐years variables > 0 – – 0.126 1.0 0.028 0.88 p‐value for GDP > 0, school‐years variables > 0 – – – – 0.000 0.056 R2 0.228 0.267 0.211 0.255 0.146 0.183 SE of regression 0.0281 0.0279 0.0267 0.0264 0.0285 0.0285 No. countries; observations 28; 764 28; 764 28; 727 28; 727 28; 700 28; 700 Notes * Significant at 5% level. ** Significant at 1% level. The sample criterion is to include countries only if they have GDP data starting by 1896 and also have data for most of the period on years of schooling and the Polity indicator of democracy. This criterion selected 28 countries: Argentina, Australia, Austria, Belgium, Brazil, Canada, Chile, China, Denmark, France, Germany, Italy, Japan, Mexico, Netherlands, New Zealand, Norway, Peru, Portugal, Russia, Spain, Sweden, Switzerland, Turkey, UK, US, Uruguay and Venezuela. Standard errors of coefficient estimates are in parentheses. In calculating standard errors of coefficient estimates, the error terms are allowed to be correlated over time within countries. Columns 1–4: The dependent variable is the annual growth rate of real per capita GDP for the 28 countries for 28 periods: 1870–5, 1875–80, … , 2005–10. For the independent variables, the log of lagged per capita GDP, average years of female and male school attainment for persons aged 15 and over, and the Polity indicator are five‐year lags, referring to 1870, 1875, … , 2005. Columns 5–6: The dependent variable is (Yt − Yt−5)/5, where Yt is the level of the Polity indicator in year t. This variable applies over the same periods as in columns 1–4. The independent variables are defined as above. Sources. GDP is from ‘Barro‐Ursúa macroeconomic data,’ available at www.rbarro.com/data-sets. The source of the Polity indicator of democracy is given in the notes to Table 3. The data at five‐year intervals since 1950 on female and male average years of school attainment for persons aged 15 and over are discussed in the notes to Table 1. Data from 1870 to 1945 at five‐year intervals are unpublished estimates to be described in Barro and Lee (2015). Open in new tab Table 5 Regressions for Long‐term Panels Five‐year periods: 1870–5, … , 2005–10 Estimation by Panel Least‐squares; All Equations Include Time Effects . (1) . (2) . (3) . (4) . (5) . (6) . . Growth rate of per capita GDP . Polity indicator . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) −0.0044** −0.0258** −0.0090** −0.0262** 0.0068** 0.0108* (0.0013) (0.0041) (0.0020) (0.0041) (0.0022) (0.0045) Female school years – – −0.0009 −0.0026 0.0025 −0.0013 (0.0018) (0.0025) (0.0019) (0.0027) Male school years – – 0.0020 −0.0009 −0.0008 0.0015 (0.0019) (0.0026) (0.0021) (0.0028) Democracy indicator (polity) – – −0.0296 −0.0323 −0.0390** −0.0510** (0.0161) (0.0188) (0.0048) (0.0056) Democracy indicator squared – – 0.0332* 0.0341* – – (0.0144) (0.0168) p‐value for democracy and democracy squared – – 0.025 0.082 – – p‐value for school‐years variables > 0 – – 0.126 1.0 0.028 0.88 p‐value for GDP > 0, school‐years variables > 0 – – – – 0.000 0.056 R2 0.228 0.267 0.211 0.255 0.146 0.183 SE of regression 0.0281 0.0279 0.0267 0.0264 0.0285 0.0285 No. countries; observations 28; 764 28; 764 28; 727 28; 727 28; 700 28; 700 . (1) . (2) . (3) . (4) . (5) . (6) . . Growth rate of per capita GDP . Polity indicator . . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . No fixed effects . Country fixed effects . Log(lagged per capita GDP) −0.0044** −0.0258** −0.0090** −0.0262** 0.0068** 0.0108* (0.0013) (0.0041) (0.0020) (0.0041) (0.0022) (0.0045) Female school years – – −0.0009 −0.0026 0.0025 −0.0013 (0.0018) (0.0025) (0.0019) (0.0027) Male school years – – 0.0020 −0.0009 −0.0008 0.0015 (0.0019) (0.0026) (0.0021) (0.0028) Democracy indicator (polity) – – −0.0296 −0.0323 −0.0390** −0.0510** (0.0161) (0.0188) (0.0048) (0.0056) Democracy indicator squared – – 0.0332* 0.0341* – – (0.0144) (0.0168) p‐value for democracy and democracy squared – – 0.025 0.082 – – p‐value for school‐years variables > 0 – – 0.126 1.0 0.028 0.88 p‐value for GDP > 0, school‐years variables > 0 – – – – 0.000 0.056 R2 0.228 0.267 0.211 0.255 0.146 0.183 SE of regression 0.0281 0.0279 0.0267 0.0264 0.0285 0.0285 No. countries; observations 28; 764 28; 764 28; 727 28; 727 28; 700 28; 700 Notes * Significant at 5% level. ** Significant at 1% level. The sample criterion is to include countries only if they have GDP data starting by 1896 and also have data for most of the period on years of schooling and the Polity indicator of democracy. This criterion selected 28 countries: Argentina, Australia, Austria, Belgium, Brazil, Canada, Chile, China, Denmark, France, Germany, Italy, Japan, Mexico, Netherlands, New Zealand, Norway, Peru, Portugal, Russia, Spain, Sweden, Switzerland, Turkey, UK, US, Uruguay and Venezuela. Standard errors of coefficient estimates are in parentheses. In calculating standard errors of coefficient estimates, the error terms are allowed to be correlated over time within countries. Columns 1–4: The dependent variable is the annual growth rate of real per capita GDP for the 28 countries for 28 periods: 1870–5, 1875–80, … , 2005–10. For the independent variables, the log of lagged per capita GDP, average years of female and male school attainment for persons aged 15 and over, and the Polity indicator are five‐year lags, referring to 1870, 1875, … , 2005. Columns 5–6: The dependent variable is (Yt − Yt−5)/5, where Yt is the level of the Polity indicator in year t. This variable applies over the same periods as in columns 1–4. The independent variables are defined as above. Sources. GDP is from ‘Barro‐Ursúa macroeconomic data,’ available at www.rbarro.com/data-sets. The source of the Polity indicator of democracy is given in the notes to Table 3. The data at five‐year intervals since 1950 on female and male average years of school attainment for persons aged 15 and over are discussed in the notes to Table 1. Data from 1870 to 1945 at five‐year intervals are unpublished estimates to be described in Barro and Lee (2015). Open in new tab 6.1. Growth rates of per capita GDP Columns 1–4 of Table 5 use as the dependent variable the growth rate of real per capita GDP over five‐year periods from 1870–5 to 2005–10. These systems include no X variables and, therefore, parallel the results in Table 1, columns 1 and 2. Column 1 excludes country fixed effects and includes as a regressor only the log of the five‐year lag of per capita GDP (along with time effects). Unlike in Table 1, column 1, the estimated coefficient on the lagged log of per capita GDP is significantly negative at the 5% level, −0.0044 (SE = 0.0013). However, the indicated convergence rate is only around 0.4% per year. Again, this low estimate likely reflects omitted X variables. Table 5, column 2, adds country fixed effects. The estimated coefficient of the log of lagged per capita GDP, −0.0258 (SE = 0.0041), shows convergence at around 2.6% per year. I redid the analysis from Table 5, columns 1 and 2, by adding as X variables the Polity indicator of democracy and its square (as in Table 3, column 4) and the measures of average years of schooling for females and males aged 15 and over (as in Table 1, column 3, and Table 3, column 4).30 Without country fixed effects (Table 5, column 3), the estimated convergence rate becomes −0.9% per year, compared to −0.4% in Table 5, column 1. The interpretation is that the inclusion of the X variables lessens the problem of omitted variables and, thereby, tends to raise the estimated convergence rate. In the equations with country fixed effects (Table 5, columns 2 and 4), the estimated convergence rate depends little on whether the X variables are included. That is, the estimated convergence rate in column 4 is around 2.6% per year, similar to that in column 2. In the long samples, the lack of information on the rich array of X variables considered in Table 1, columns 3 and 4, and Table 3, column 4, suggests that the omitted‐variables bias would be substantial in systems that omit country fixed effects. Thus, the estimated convergence rate of less than 1% per year in Table 5, columns 1 and 3, is likely to be a serious underestimate of the true convergence rate. On the other hand, because of the long time series, (1) implies that the Hurwicz–Nickell bias on the coefficient of the log of lagged per capita GDP – in Table 5, columns 2 and 4–is likely to be much less serious than that in Table 1, columns 2 and 4. Therefore, the estimated conditional convergence rate around 2.6% per year in Table 5, columns 2 and 4, may be only a small overestimate of the true convergence rate. We can put the long‐term results from Table 5 together with the shorter term results from Table 1. In Table 1, the best estimate of the conditional convergence rate probably comes from column 3, which excludes country fixed effects and includes an array of X variables. The estimated conditional convergence rate here is around 1.7% per year, with a two‐standard‐error band of 1.3% to 1.9%. This result may be a small underestimate of the true convergence rate – to the extent that the system still omits long‐run growth determinants that are conditionally correlated with per capita GDP. From Table 5, the best estimates are likely to come in columns 2 and 4 from equations with country fixed effects. The estimated convergence rate here is around 2.6% per year, with a two‐standard‐error band of 1.8% to 3.4%. This result may overestimate the true convergence rate – to the extent that the Hurwicz‐Nickell bias still operates in this long‐term sample. When viewed together, the results from the two panel data sets suggest that the conditional convergence rate is, in terms of point estimates, bounded between 1.7% and 2.6% per year. Interestingly, this interval contains the iron‐law rate of 2% per year. 6.2. Polity indicator of democracy Columns 5 and 6 of Table 5 return to the modernisation hypothesis by considering long‐term panel regressions for the Polity indicator of democracy. Column 5 excludes country fixed effects. The estimated convergence rate, based on the estimated coefficient on the lagged Polity indicator, is −0.0390 (SE = 0.0048). This coefficient is lower than that, −0.0505 (SE = 0.0046), found with post‐1970 data in Table 4, column 5. From the standpoint of the modernisation hypothesis, the important results in Table 5, column 5, are the significantly positive coefficient on the log of lagged per capita GDP, 0.0068 (SE = 0.0022) and the significantly positive sum of the estimated coefficients on years of female and schooling (p‐value = 0.028). The GDP and schooling variables are jointly statistically significantly positive with a p‐value of 0.000. This last finding parallels the result for post‐1970 data in Table 4, column 5. Table 5, column 6, adds country fixed effects. As usual, this change raises the magnitude of the estimated convergence rate–to −0.0510 (SE = 0.0056). However, this rate is much lower than that (−0.1108, SE = 0.0090) found with country fixed effects in the post‐1970 data (Table 4, column 6). This result is expected because of the much longer time series employed in Table 5. With the variables observed over 140 years, the Hurwicz–Nickell bias in the presence of country fixed effects should be small (see the Monte Carlo results in the Appendix). In Table 5, column 6, the standard errors of the estimated coefficients of the per capita GDP and schooling variables go up, as before, due to the inclusion of country fixed effects. However, the estimated coefficient of the log of per capita GDP, 0.0108 (SE = 0.0045), remains significantly positive. The sum of the estimated coefficients on the schooling variables is now close to zero. However, the coefficient on GDP and the sum of the coefficients on the schooling variables are jointly statistically significantly positive with a p‐value of 0.056. Hence, even with the inclusion of country fixed effects, the long‐term sample supports the modernisation hypothesis with regard to the overall relationship of GDP and schooling to the Polity indicator for democracy.31 7. Summary Observations For a large panel of countries since 1960, the estimated conditional convergence rate for per capita GDP is around 1.7% per year in systems with a rich array of explanatory variables but no country fixed effects. The results feature statistically significant influences on economic growth from institutional quality, measured by maintenance of law and order and democracy. Analogous settings support the modernisation hypothesis, in the sense of a significantly positive overall relationship of per capita GDP and schooling to indicators for maintenance of law and order and democracy. The inclusion of country fixed effects produces much higher convergence rates in per capita GDP, eliminates statistically significant effects of the institutional measures on economic growth, and removes the statistical support for modernisation. I argue that these problematic findings reflect econometric issues associated with the inclusion of country fixed effects in panels with moderate time dimension. In a panel for a limited number of countries observed since 1870, the estimated conditional convergence rate for per capita GDP in the presence of country fixed effects is about 2.6% per year. This setting also supports the modernisation hypothesis, in the sense of a statistically significant, positive relationship between per capita GDP and democracy. These ‘reasonable’ results with country fixed effects likely arise because the econometric problems posed by country fixed effects may not be serious in samples with long time frames. A combination of the results from the post‐1960 and post‐1870 panels suggests that the conditional convergence rate for per capita GDP is in the neighbourhood of the ‘iron‐law’ rate of 2% per year. The similar findings from these two very different empirical settings suggest that a conditional convergence rate around 2% per year may be a robust empirical regularity. The two settings considered jointly also provide strong support for the modernisation hypothesis. Appendix A. Monte Carlo Analysis of Dynamic Estimation with and without Country Fixed Effects The dynamic model with fixed effects and no time‐varying X variables follows Hurwicz (1950), Nickell (1981), Arellano and Bond (1991), and Nerlove (2000): yit=ηi+γ×yi,t−1+εit,(A.1) where i = 1, … , N represents countries; t = 1, … , T represents periods; yit is per capita GDP or some other country‐time variable; 0 < γ < 1; and εit ~ N(0, σϵ2 ⁠) is an i.i.d. shock. I think of the fixed effect, ηi, as being drawn once for each country at the beginning of time, where ηi ~ N(0, ση2 ⁠) is an i.i.d. disturbance. As stressed by Nerlove (2000), the initial sample value yi0 cannot be viewed as independent of ηi, because yi0 comes from the cumulation of (A.1) from the indefinite past; hence, yi0 depends on ηi and past realisations of the εit. Specifically, we have yi0 ~ N(⁠ ηi/1−λ,σε2/1−λ) ⁠. Therefore, a country that gets a high draw for ηi tends also to have a high yi0. The model with the addition of a time‐varying, exogenous X variable is: yit=ηi+γ×yi,t−1+α×Xit+εit,(A.2) Xit=ρ×Xi,t−1+uit,(A.3) where ηi, γ and εit are defined as before, α is a constant, 0 < ρ < 1 governs the persistence of the X variable, and uit∼N(0,σu2) is an i.i.d. shock. The initial sample value of the X variable is given accordingly by Xi0∼N(0,σu2/1−ρ) ⁠. The initial sample value of y reflects its dependence on ηi and past shocks to εit and uit from (A.2) and (A.3), and this dependence implies a relationship with Xi0 (which depends on past shocks to uit). Specifically, we can write yi0=φ×Xi0+wio,(A.4) where we can derive φ=α/[(1+ρ)(1−γρ)],(A.5) and wi0∼N(0,σw2) ⁠, independently of Xi0, where σw2=σϵ21−γ+α2σu2(1−γρ)2ρ(1−ρ)(1+ρ)2+γ2(1−γ)(1+γ).(A.6) Equations (A.4) and (A.5) imply that, if α > 0, a country that gets a high draw for Xi0 tends also to have a high yi0. If periods are of length τ (years), the persistence coefficient, γ, relates to the convergence rate, β > 0, from γ = e−βτ. I assume that βτ is much less than 1, so that γ ≈ 1 − βτ and, hence, β ≈ (1 − γ)/τ. For annual periods (τ = 1 year), the convergence rate per year is β ≈ 1 − γ. A.1. Model with Country Fixed Effects Consider first the model in (A.1) with a country fixed effect and no X variable. The standard deviation of the time‐series shock, σε, can be normalised to 1, so that ση represents the dispersion of the country (cross‐sectional) shock, relative to the time‐series shock. In Nickell (1981), the analysis applied as the number of cross sections, N, tended to infinity. In my Monte Carlo analysis, N (number of countries) = 20 seems large enough to approximate this asymptotic setting. I use N = 100 below. I contrast the results using OLS without country fixed effects with those using country fixed effects (OLS with dummy variables). As discussed in the text, the estimated coefficients γ and β from the two methods involve a trade‐off between two types of biases. I begin with a heuristic description of the results. For OLS without country fixed effects, the Hurwicz–Nickell bias is unimportant – even if the time dimension, T, is small – but an omitted‐variables bias applies. The omitted variables are the country fixed effects, ηi, which are positively correlated with the yit. This effect biases up the estimated γ and, therefore, biases down the estimated β. This omitted‐variables channel is more important the larger ση. For small enough ση, the omitted‐variables effect is minor, and OLS without country fixed effects produces nearly unbiased estimates. For OLS with country fixed effects, there are no omitted variables,32 but the Hurwicz–Nickell bias may be large. This effect biases down the estimated γ and, therefore, biases up the estimated β. The size of the Hurwicz–Nickell bias depends on the length of the time series, T. If T is small – even 20 or 50 years – the bias is substantial. However, for large enough T, the bias becomes small, so that OLS with country fixed effects produces nearly unbiased estimates. However, even a sample of 140 years (the largest time frame considered in the text) is not sufficient to make the bias negligible. Unlike the case without country fixed effects, the size of ση is unimportant for the bias. Table A1 Monte Carlo Results for Estimated Convergence Rates Model with Country Fixed Effects . (1) . (2) . (3) . (4) . (5) . . β . T . ση . E(β^) (Nickell formula) . mean(β^) (Monte Carlo) . OLS with country fixed effects 1 0.02 20 1 0.165 0.151 (0.014) 2 0.02 50 1 0.076 0.070 (0.006) 3 0.02 100 1 0.046 0.043 (0.004) 4 0.02 100 0.1 0.046 0.043 (0.004) 5 0.02 140 1 0.038 0.036 (0.003) 6 0.10 20 1 0.231 0.206 (0.014) 7 0.10 50 1 0.147 0.140 (0.008) 8 0.10 100 1 0.122 0.120 (0.005) 9 0.10 100 0.1 0.122 0.120 (0.004) 10 0.10 140 1 0.115 0.113 (0.004) OLS without country fixed effects 11 0.02 100 1 – 0.00026 (0.00015) 12 0.02 100 0.1 – 0.0111 (0.0015) 13 0.02 20 0.1 – 0.0126 (0.0029) 14 0.02 100 0.01 – 0.0200 (0.0021) 15 0.10 100 1 – 0.0053 (0.0009) 16 0.10 100 0.1 – 0.0858 (0.0045) 17 0.10 20 0.1 – 0.0884 (0.0092) 18 0.10 100 0.01 – 0.1006 (0.0036) . (1) . (2) . (3) . (4) . (5) . . β . T . ση . E(β^) (Nickell formula) . mean(β^) (Monte Carlo) . OLS with country fixed effects 1 0.02 20 1 0.165 0.151 (0.014) 2 0.02 50 1 0.076 0.070 (0.006) 3 0.02 100 1 0.046 0.043 (0.004) 4 0.02 100 0.1 0.046 0.043 (0.004) 5 0.02 140 1 0.038 0.036 (0.003) 6 0.10 20 1 0.231 0.206 (0.014) 7 0.10 50 1 0.147 0.140 (0.008) 8 0.10 100 1 0.122 0.120 (0.005) 9 0.10 100 0.1 0.122 0.120 (0.004) 10 0.10 140 1 0.115 0.113 (0.004) OLS without country fixed effects 11 0.02 100 1 – 0.00026 (0.00015) 12 0.02 100 0.1 – 0.0111 (0.0015) 13 0.02 20 0.1 – 0.0126 (0.0029) 14 0.02 100 0.01 – 0.0200 (0.0021) 15 0.10 100 1 – 0.0053 (0.0009) 16 0.10 100 0.1 – 0.0858 (0.0045) 17 0.10 20 0.1 – 0.0884 (0.0092) 18 0.10 100 0.01 – 0.1006 (0.0036) Notes β = 1 − γ is the convergence rate in (A.1), T is the length of the time series and ση is the standard deviation of the country fixed effect. The Monte Carlo results in column 5 show the mean and standard deviation of the estimate β^ from 100 iterations of the model in (A.1). The upper part refers to OLS estimation with country fixed effects. The lower part refers to OLS estimation without country fixed effects. All results shown use σε = 1 (for the time‐series shock), N (number of countries) = 100 and an observation period of τ = 1 year. Results are similar with 50 iterations, N = 20 are similar or τ = 5. For example, with country fixed effects, the mean of β^ when β = 0.10, T = 100, ση = 1 and τ = 5 is 0.125 (0.006). Without country fixed effects, the mean of β^ for this case is 0.0047 (0.0008). Column 4 shows the expected value of β^ from the formula from Nickell (1981) in (1). This formula applies to OLS estimation with country fixed effects. Open in new tab Table A1 Monte Carlo Results for Estimated Convergence Rates Model with Country Fixed Effects . (1) . (2) . (3) . (4) . (5) . . β . T . ση . E(β^) (Nickell formula) . mean(β^) (Monte Carlo) . OLS with country fixed effects 1 0.02 20 1 0.165 0.151 (0.014) 2 0.02 50 1 0.076 0.070 (0.006) 3 0.02 100 1 0.046 0.043 (0.004) 4 0.02 100 0.1 0.046 0.043 (0.004) 5 0.02 140 1 0.038 0.036 (0.003) 6 0.10 20 1 0.231 0.206 (0.014) 7 0.10 50 1 0.147 0.140 (0.008) 8 0.10 100 1 0.122 0.120 (0.005) 9 0.10 100 0.1 0.122 0.120 (0.004) 10 0.10 140 1 0.115 0.113 (0.004) OLS without country fixed effects 11 0.02 100 1 – 0.00026 (0.00015) 12 0.02 100 0.1 – 0.0111 (0.0015) 13 0.02 20 0.1 – 0.0126 (0.0029) 14 0.02 100 0.01 – 0.0200 (0.0021) 15 0.10 100 1 – 0.0053 (0.0009) 16 0.10 100 0.1 – 0.0858 (0.0045) 17 0.10 20 0.1 – 0.0884 (0.0092) 18 0.10 100 0.01 – 0.1006 (0.0036) . (1) . (2) . (3) . (4) . (5) . . β . T . ση . E(β^) (Nickell formula) . mean(β^) (Monte Carlo) . OLS with country fixed effects 1 0.02 20 1 0.165 0.151 (0.014) 2 0.02 50 1 0.076 0.070 (0.006) 3 0.02 100 1 0.046 0.043 (0.004) 4 0.02 100 0.1 0.046 0.043 (0.004) 5 0.02 140 1 0.038 0.036 (0.003) 6 0.10 20 1 0.231 0.206 (0.014) 7 0.10 50 1 0.147 0.140 (0.008) 8 0.10 100 1 0.122 0.120 (0.005) 9 0.10 100 0.1 0.122 0.120 (0.004) 10 0.10 140 1 0.115 0.113 (0.004) OLS without country fixed effects 11 0.02 100 1 – 0.00026 (0.00015) 12 0.02 100 0.1 – 0.0111 (0.0015) 13 0.02 20 0.1 – 0.0126 (0.0029) 14 0.02 100 0.01 – 0.0200 (0.0021) 15 0.10 100 1 – 0.0053 (0.0009) 16 0.10 100 0.1 – 0.0858 (0.0045) 17 0.10 20 0.1 – 0.0884 (0.0092) 18 0.10 100 0.01 – 0.1006 (0.0036) Notes β = 1 − γ is the convergence rate in (A.1), T is the length of the time series and ση is the standard deviation of the country fixed effect. The Monte Carlo results in column 5 show the mean and standard deviation of the estimate β^ from 100 iterations of the model in (A.1). The upper part refers to OLS estimation with country fixed effects. The lower part refers to OLS estimation without country fixed effects. All results shown use σε = 1 (for the time‐series shock), N (number of countries) = 100 and an observation period of τ = 1 year. Results are similar with 50 iterations, N = 20 are similar or τ = 5. For example, with country fixed effects, the mean of β^ when β = 0.10, T = 100, ση = 1 and τ = 5 is 0.125 (0.006). Without country fixed effects, the mean of β^ for this case is 0.0047 (0.0008). Column 4 shows the expected value of β^ from the formula from Nickell (1981) in (1). This formula applies to OLS estimation with country fixed effects. Open in new tab The upper part of Table A1 applies to OLS estimation with country fixed effects. Two values of the convergence rate, β = 1 − γ, are considered, 0.02 and 0.10 per year. The emphasis in this part of the table is on the effect of the time dimension, T, on the mean of the estimated β^ ⁠, shown in column 5. The bias is always upward, reflecting the Hurwicz–Nickell channel. For example, if β = 0.02 per year, the mean of β^ when T = 20 (line 1) is 0.151; that is, the upward bias is dramatic. The mean of β^ falls to 0.070 when T = 50 (line 2) and 0.036 when T = 140 (line 5). Hence, although the bias is much reduced compared to that applying to short time series, even T = 140 years is insufficient to make the bias negligible. Changing the standard deviation ση of the fixed effect does not materially impact the results (lines 4 and 9 of the Table). The results for E(⁠ β^ ⁠) from Nickell's (1981) formula, shown in column 4 of Table A1, are similar to the Monte Carlo results for the mean of β^ in column 5. Some differences arise because Nickell's formula applies as the number of cross sections, N, approaches infinity. In addition, Nickell treated the initial sample values, yi0, as given, rather than allowing for a relation with the fixed effect, ηi. This issue is more important in the estimation without country fixed effects (not considered by Nickell, 1981). The lower part of Table A1 applies to OLS estimation without country fixed effects. The focus in this part of the Table is on the impact of the standard deviation, ση, for the fixed effects. The bias in the estimate β^ is always downward, reflecting the omitted‐variables channel. For example, when β = 0.02, T = 100, and ση = 1 (line 11), the mean of β^ in column 5 is 0.0003. Hence, the bias is sharply downward in this case. However, the mean of β^ rises to 0.011 when ση = 0.1 (line 12) and 0.020 when ση = 0.01 (line 14). Thus, because the Hurwicz–Nickell bias is unimportant, the estimate β^ is virtually unbiased for a ση that is small enough to make the omitted‐variables bias unimportant. The time dimension, T, has a nonzero but moderate effect on the results without country fixed effects (lines 13 and 17 of Table A1). For example, reducing T from 100 to 20 (lines 12 and 13) actually lowers the bias: the mean of the estimate β^ goes from 0.0111 to 0.0126 (but the standard error of the estimate roughly doubles). A.2. Model with Country Fixed Effects and an X Variable Table A2 has the results for the model in (A.2) – (A.6), which adds an exogenous X variable. The conditions σε = 1 (for the time‐series shock) and σu = 1 (for the shock to the X variable) are normalisations. The other assumptions are β (convergence rate) = 0.02, ση = 0.1 (for the fixed effect), ρ (persistence coefficient for the X process) = 0.98 and N (number of countries) = 100. Results are shown for α (coefficient of the X variable) equal to 0.05 or 0.25 and T (time‐series length) equal to 100 or 20. Table A2 Monte Carlo Results for Estimated Convergence Rates Model with Country Fixed Effects and X Variable . (1) . (2) . (3) . (4) . (5) . (6) . . β . α . T . include X? . mean(β^) (Monte Carlo) . mean(β^) (Monte Carlo) . OLS with fixed effects 1 0.02 0.05 100 y 0.0315 (0.0025) 0.0534 (0.0034) 2 0.02 0.25 100 y 0.0208 (0.0005) 0.2516 (0.0032) 3 0.02 0.05 20 y 0.1029 (0.0120) 0.0450 (0.0144) 4 0.02 0.25 20 y 0.0288 (0.0034) 0.2465 (0.0135) 5 0.02 0.05 100 n 0.0234 (0.0025) – 6 0.02 0.25 100 n 0.0069 (0.0024) – 7 0.02 0.05 20 n 0.1056 (0.0131) – 8 0.02 0.25 20 n 0.0326 (0.0054) – OLS without fixed effects 9 0.02 0.05 100 y 0.0156 (0.0013) 0.0446 (0.0026) 10 0.02 0.25 100 y 0.0196 (0.0004) 0.2472 (0.0031) 11 0.02 0.05 20 y 0.0166 (0.0024) 0.0470 (0.0040) 12 0.02 0.25 20 y 0.0198 (0.0024) 0.2489 (0.0049) 13 0.02 0.05 100 n 0.0041 (0.0010) – 14 0.02 0.25 100 n 0.0004 (0.0009) – 15 0.02 0.05 20 n 0.0041 (0.0024) – 16 0.02 0.25 20 n −0.0024 (0.0023) – . (1) . (2) . (3) . (4) . (5) . (6) . . β . α . T . include X? . mean(β^) (Monte Carlo) . mean(β^) (Monte Carlo) . OLS with fixed effects 1 0.02 0.05 100 y 0.0315 (0.0025) 0.0534 (0.0034) 2 0.02 0.25 100 y 0.0208 (0.0005) 0.2516 (0.0032) 3 0.02 0.05 20 y 0.1029 (0.0120) 0.0450 (0.0144) 4 0.02 0.25 20 y 0.0288 (0.0034) 0.2465 (0.0135) 5 0.02 0.05 100 n 0.0234 (0.0025) – 6 0.02 0.25 100 n 0.0069 (0.0024) – 7 0.02 0.05 20 n 0.1056 (0.0131) – 8 0.02 0.25 20 n 0.0326 (0.0054) – OLS without fixed effects 9 0.02 0.05 100 y 0.0156 (0.0013) 0.0446 (0.0026) 10 0.02 0.25 100 y 0.0196 (0.0004) 0.2472 (0.0031) 11 0.02 0.05 20 y 0.0166 (0.0024) 0.0470 (0.0040) 12 0.02 0.25 20 y 0.0198 (0.0024) 0.2489 (0.0049) 13 0.02 0.05 100 n 0.0041 (0.0010) – 14 0.02 0.25 100 n 0.0004 (0.0009) – 15 0.02 0.05 20 n 0.0041 (0.0024) – 16 0.02 0.25 20 n −0.0024 (0.0023) – Notes For the model in (A.2) – (A.6), β = 1 − γ is the convergence rate, α is the coefficient on the X variable, ρ is the persistence coefficient for the X variable and T is the length of the time series. The Monte Carlo results in columns 5 and 6 show the mean and standard deviation of the estimates β^ and α^ from 100 iterations of the model. The upper part refers to OLS estimation with country fixed effects. The lower part refers to OLS estimation without country fixed effects. All results normalise to have σε = 1 (for the time‐series shock) and σu = 1 (for the X shock). Other assumptions are ση = 0.1 (for the country fixed effect), ρ (coefficient for the persistence of the X process) = 0.98, N (number of countries) = 100 and use an observation period of τ = 1 year. Column 4 indicates whether the regressions include the X variable. Open in new tab Table A2 Monte Carlo Results for Estimated Convergence Rates Model with Country Fixed Effects and X Variable . (1) . (2) . (3) . (4) . (5) . (6) . . β . α . T . include X? . mean(β^) (Monte Carlo) . mean(β^) (Monte Carlo) . OLS with fixed effects 1 0.02 0.05 100 y 0.0315 (0.0025) 0.0534 (0.0034) 2 0.02 0.25 100 y 0.0208 (0.0005) 0.2516 (0.0032) 3 0.02 0.05 20 y 0.1029 (0.0120) 0.0450 (0.0144) 4 0.02 0.25 20 y 0.0288 (0.0034) 0.2465 (0.0135) 5 0.02 0.05 100 n 0.0234 (0.0025) – 6 0.02 0.25 100 n 0.0069 (0.0024) – 7 0.02 0.05 20 n 0.1056 (0.0131) – 8 0.02 0.25 20 n 0.0326 (0.0054) – OLS without fixed effects 9 0.02 0.05 100 y 0.0156 (0.0013) 0.0446 (0.0026) 10 0.02 0.25 100 y 0.0196 (0.0004) 0.2472 (0.0031) 11 0.02 0.05 20 y 0.0166 (0.0024) 0.0470 (0.0040) 12 0.02 0.25 20 y 0.0198 (0.0024) 0.2489 (0.0049) 13 0.02 0.05 100 n 0.0041 (0.0010) – 14 0.02 0.25 100 n 0.0004 (0.0009) – 15 0.02 0.05 20 n 0.0041 (0.0024) – 16 0.02 0.25 20 n −0.0024 (0.0023) – . (1) . (2) . (3) . (4) . (5) . (6) . . β . α . T . include X? . mean(β^) (Monte Carlo) . mean(β^) (Monte Carlo) . OLS with fixed effects 1 0.02 0.05 100 y 0.0315 (0.0025) 0.0534 (0.0034) 2 0.02 0.25 100 y 0.0208 (0.0005) 0.2516 (0.0032) 3 0.02 0.05 20 y 0.1029 (0.0120) 0.0450 (0.0144) 4 0.02 0.25 20 y 0.0288 (0.0034) 0.2465 (0.0135) 5 0.02 0.05 100 n 0.0234 (0.0025) – 6 0.02 0.25 100 n 0.0069 (0.0024) – 7 0.02 0.05 20 n 0.1056 (0.0131) – 8 0.02 0.25 20 n 0.0326 (0.0054) – OLS without fixed effects 9 0.02 0.05 100 y 0.0156 (0.0013) 0.0446 (0.0026) 10 0.02 0.25 100 y 0.0196 (0.0004) 0.2472 (0.0031) 11 0.02 0.05 20 y 0.0166 (0.0024) 0.0470 (0.0040) 12 0.02 0.25 20 y 0.0198 (0.0024) 0.2489 (0.0049) 13 0.02 0.05 100 n 0.0041 (0.0010) – 14 0.02 0.25 100 n 0.0004 (0.0009) – 15 0.02 0.05 20 n 0.0041 (0.0024) – 16 0.02 0.25 20 n −0.0024 (0.0023) – Notes For the model in (A.2) – (A.6), β = 1 − γ is the convergence rate, α is the coefficient on the X variable, ρ is the persistence coefficient for the X variable and T is the length of the time series. The Monte Carlo results in columns 5 and 6 show the mean and standard deviation of the estimates β^ and α^ from 100 iterations of the model. The upper part refers to OLS estimation with country fixed effects. The lower part refers to OLS estimation without country fixed effects. All results normalise to have σε = 1 (for the time‐series shock) and σu = 1 (for the X shock). Other assumptions are ση = 0.1 (for the country fixed effect), ρ (coefficient for the persistence of the X process) = 0.98, N (number of countries) = 100 and use an observation period of τ = 1 year. Column 4 indicates whether the regressions include the X variable. Open in new tab Results from OLS without country fixed effects, shown in the lower part of Table A2, are straightforward. If the X variable is included in the regression (lines 9–12), so that omitted‐variables effects are minor, the OLS estimates of the convergence rate, β (and also for α, the coefficient of the X variable) have small biases. These results follow because the Hurwicz–Nickell bias is again unimportant when fixed effects are excluded. Moreover, these findings apply for the different values considered for α and T. However, if the X variable is excluded from the regression (lines 13–16), the estimate β^ is seriously biased downward because of the omitted‐variable effect described before. The bottom line is that satisfactory implementation of OLS without country fixed effects depends on the inclusion of the X variable. The results for OLS with country fixed effects are shown in the upper part of Table A2. The conclusions with respect to β^ depend on the time‐series length, T, and the coefficient α on the X variable. For example, if T = 100, the mean of β^ is 0.032 (line 1) when α = 0.05 and the mean is 0.021 (line 2) when α = 0.25. Hence, the bias is small with a 100‐year sample when a lot of the variation in the y variable reflects the observed X variable. However, if T = 20, the mean of β^ is 0.103 (line 3) when α = 0.05 and 0.029 (line 4) when α = 0.25. That is, the Hurwicz–Nickell channel can lead to a strong upward bias when T = 20. When the X variable is excluded, the mean of β^ is 0.023 (line 5) when α = 0.05 and T = 100. That is, the bias can be small with a long time series if α is not too large. However, the mean of β^ is 0.007 (line 6) when α = 0.25. That is, the bias is substantially downward because of the important omitted‐variables effect with this higher value of α. When T = 20, the bias is upward for each value of α because of the Hurwicz–Nickell effect. The mean of β^ is 0.106 (line 7) when α = 0.05 and 0.033 (line 8) when α = 0.25. The results suggest circumstances when the regressions generate estimates of the convergence rate, β, without large bias. The first is for OLS with country fixed effects when the sample is large in the time domain. For example, when β = 0.02, the fixed‐effects model yields a mean of β^ of 0.036 when T = 140 (Table A1, line 5). For the model that adds an exogenous X variable, the mean of β^ is 0.021 when T = 100 and α = 0.25 (Table A2, line 2). Even with the X variable excluded, the mean of β^ is 0.023 when T = 100 and α = 0.05 (line 5). The second case is for OLS without country fixed effects in the X‐model when the (exogenous) X variable is included in the regression. When β = 0.02, this framework yields a mean for β^ between 0.016 and 0.020 (Table A2, lines 9–12). OLS without country fixed effects also does well in the fixed‐effects model when the variation in the fixed effects is small (Table A1, row 14). The results can be used to get reasonable bounds on the convergence rate, β, as carried out in the text. The upper bound came from the estimate with country fixed effects without X variables, based on the long‐term data back to 1870. Because T is large, the upper bound may be reasonably tight (as in Table A1, line 5, or Table A2, line 5). The lower bound came from the estimate without country fixed effects using the shorter term data since 1960 with a rich array of X variables (as in Table A2, lines 9–12). Footnotes 1 " I first heard this term applied to my empirical findings on economic growth by Rudi Dornbusch. However, Larry Summers said that Rudi got the term from him. In any event, the term is reminiscent of the ‘iron law of wages’. According to Wikipedia, this phrase came from Lassalle but Marx and Engels argued that Lassalle got the idea from Malthus's theory of population and the specific terminology from Goethe. 2 " Baumol (1986, Figure 2) reported unconditional convergence from 1870 to 1979 for 16 countries (all subsequently OECD members), using data from Maddison (1982). However, De Long (1988) showed that Baumol's results depended on a sample‐selection issue, whereby only countries that were rich towards the end of the sample (1979) were considered. Unconditional convergence did not hold for an expanded sample of 22 countries that were selected based on per capita income in 1870 (De Long, 1988, Figure 2). This sample‐selection criticism of Baumol's (1986) findings was presented earlier by Romer (1986, pp. 1012–13). Rodrik (2013) finds unconditional convergence in labour productivity across manufacturing industries for recent decades in 118 countries. 3 " Barro (2002) found that the predicted slow convergence between East and West Germany accorded with regional data on GDP per worker through the late 1990s. However, wage rates converged faster because of the German government's transfer and subsidy policies. 4 " In earlier work, Barro (1991) reported conditional convergence for the cross section of 98 countries but did not express the results in terms of a convergence rate. 5 " Knack and Keefer (1995) and Mauro (1995) studied growth effects from rule of law and corruption. Przeworski and Limongi (1993) and Barro (1997, ch. 2) assessed growth effects from democracy. King and Levine (1993) examined effects of financial institutions on economic growth. Glaeser et al. (2004) argued that institutions should be measured by basic legal constraints on the government, rather than political outcomes, which include official corruption and risk of expropriation. 6 " Contributions to the modernisation literature include Aristotle (1932), Lipset (1959), Dahl (1991), and Huntington (1991). Marx (1904) extended the modernisation idea to a predicted collapse of organised religion under capitalism. 7 " This approach applied to economic growth seems to have begun with Knight et al. (1993), Islam (1995) and Caselli et al. (1996). Acemoglu et al. (2005, 2008) advocate the use of country fixed effects in studies of the modernisation hypothesis. 8 " Let yit be the dependent variable and εit the associated error term for country i at time t. (Hurwicz, 1950, dealt with only one country.) The εit are serially uncorrelated, i.i.d. random variables. The Hurwicz (1950) type bias arises because the realised error terms appear in the sample mean of yi,t‐1. That is, a higher εit implies a higher sample mean of yi,t‐1 and, therefore, a lower yit,t‐1 when expressed relative to its sample mean. The implied negative covariance between εit and the deviation of yi,t‐1 from its sample mean biases downward the OLS estimate of the coefficient of yi,t‐1. 9 " Malthus had a model of endogenous population growth, though he sometimes got the signs wrong. Specifically, he predicted a positive effect of real per capita GDP on fertility, whereas, in the modern data, the relation goes strongly in the opposite direction. See Manuelli and Seshadri (2009), who emphasise the distinction between income effects (stressed by Malthus) and substitution effects associated with increases in wage rates. 10 " Other research on endogenous growth – exemplified by Romer (1986), Lucas (1988) and Rebelo (1991) – built on earlier work by Arrow (1962) and Uzawa (1965) and did not provide a theory of technical change. In these models, growth may go on indefinitely because the returns to investment in a broad class of capital goods, which include human capital, may not diminish as economies develop. (This idea goes back to Knight, 1944.) Spillovers of knowledge across producers and external benefits from human capital may be parts of the process because they help to avoid the tendency for diminishing returns to capital. Models of this type are sometimes described as ‘AK models’, because the growth dynamics looks like that in a simple framework with constant returns to capital. 11 " This idea began with Solow (1957) and was extended to allow for changing quality of an array of inputs by Jorgenson and Griliches (1967). See also Barro and Sala‐i‐Martin (2004). 12 " Aghion et al. (2005) have used a model related to endogenous growth theory as the framework for an empirical study of the relation between competition and innovation across firms. 13 " Engerman and Sokoloff (1994) emphasise the density of the indigenous population and the nature of land and climate as influences on institutions and, thereby, on long‐term economic growth. Thus, the exogenous elements in their historical analysis, such as population density at the time of colonial settlement, can be viewed as informal instruments for descriptive regressions. 14 " Because of the strong positive serial correlation in many of the X variables, weak instruments tend not to be a problem in this context. However, this approach relies on serial independence of the error terms and also maintains the assumption that lagged values of the X variables do not directly influence the dependent variable. 15 " Standard errors of coefficient estimates allow each country's error term to be correlated over time. 16 " In this and the subsequent regressions, one can think equivalently of the dependent variable as the log of the level of per capita GDP (observed for 1965, … , 2010), with the five‐year lag of this variable included on the right‐hand side. In this version of the regression from Table 1, column 2, the coefficient on the lagged dependent variable is 0.83 (which equals 1 minus 5 times the estimated convergence coefficient of 0.0335 per year). 17 " Several dynamic panel estimators have been developed to alleviate this problem but none of these approaches seem to be reliable. For example, Blundell and Bond (1998, pp. 115, 116, 120) note that the method of Arellano and Bond (1991) — which uses lagged levels of the dependent variable as instruments in a first‐difference form of the model — suffers from a weak instruments problem and has large finite‐sample bias. The problems are especially serious when the coefficient of the lagged dependent variable in a level form of the annual equation is close to one – as in the growth context — and when the relative variance of the fixed effect is large. I find in the Monte Carlo setting described in Appendix A that the Arellano and Bond (1991) estimator sharply overestimates the convergence rate. In contrast, the Blundell and Bond (1998) estimator (which uses lagged levels and differences of the dependent variable as instruments in a system of equations) turns out to sharply underestimate the convergence rate. These Monte Carlo findings accord with results obtained with the data and functional form from Table 1, column 2. In this setting, the estimated coefficient of the log of lagged per capita GDP (using annual data) is −0.048 (SE = 0.010) with the Arellano and Bond (1991) estimator and 0.005 (SE = 0.001) with the Blundell and Bond (1998) estimator. Another alternative is the bias‐adjustment procedure of Kiviet (1995), made operational by Bruno (2005). One problem is that this approach requires, in the first stage, a consistent estimate of the coefficient of the lagged dependent variable. In practice, Bruno (2005) relies on Arellano and Bond (1991), which is problematic because of the large bias in this estimator. Using the data and functional form from Table 1, column 2, the Bruno (2005) procedure yields an estimated coefficient of the log of lagged per capita GDP of −0.007 (SE = 0.003). 18 " I considered some variables that were constant over time within countries — the absolute value of degrees latitude, land‐locked status and aspects of colonial history and legal origins — but these variables turned out to be unimportant. 19 " The law‐and‐order variable (previously called rule of law) comes from International Country Risk Guide, which is produced by Political Risk Services. These data were first used in academic research by Knack and Keefer (1995). The law‐and‐order indicator starts in 1982 or later. The panel regressions in Table 1, column 3, use each country's first value available for this variable for the periods that start in 1980–5 or earlier. If the sample covers only the five periods that begin with 1985–90, the results are similar to those in column 3. An alternative to the ICRG data is the information on perceived quality of governance assembled by the World Bank (available at www.govindicators.org). However, these data are available only since the late 1990s. 20 " Analogous data from Bollen (1980) were used for 1960 and 1965. 21 " The school‐attainment data are updated values described in Barro and Lee (2013) and available at www.barrolee.com 22 " Hauk and Wacziarg (2009, Table 4, column 1) used a Monte Carlo analysis to evaluate the Arellano and Bond (1991) and Blundell and Bond (1998) estimators in this context. They find, consistent with the results for a model without X variables described in footnote 17, that Arellano and Bond (1991) substantially overestimates the convergence rate, whereas Blundell and Bond (1998) substantially underestimates it. 23 " This conclusion accords with the one reached with respect to the investment ratio in Blomström et al. (1996). 24 " The annual system is problematic because some of the data (school attainment, changes in the terms of trade and the inflation rate) were constructed only over five‐year intervals. Moreover, other data – notably on life expectancy and the fertility rate – are provided at an annual frequency but are not really annual variables. In any event, the annual results (using interpolated values of the variables available at five‐year intervals) show much poorer fits than the systems at five‐year and 10‐year intervals. The R2 value for the annual case is 0.132, and the standard error of the regression is 0.0505. (This system for 89 countries has 3,760 observations in total.) 25 " According to Lipset (1959, p. 75): ‘From Aristotle down to the present, men have argued that only in a wealthy society in which relatively few citizens lived in real poverty could a situation exist in which the mass of the population could intelligently participate in politics and could develop the self‐restraint necessary to avoid succumbing to the appeals of irresponsible demagogues’. 26 " Iceland, Luxembourg, and Malta are excluded in the case of Polity because of missing data. 27 " Acemoglu et al. (2008, p. 820) argue that this bias in the fixed‐effects estimated coefficient of a lagged dependent variable can be mitigated by observing the data at a higher frequency, such as annually: ‘Column 6 [of Table 3] estimates … with OLS using annual observations. This is useful since the fixed effect OLS estimator becomes consistent as the number of observations becomes large. With annual observations, we have a reasonably large time dimension'. The Nickell formula for the bias in (1) implies that this argument is incorrect. The proportionate bias in the estimated coefficient of a lagged dependent variable depends on the overall length of the sample in years (or, actually, on the product βT, where β is the convergence rate per year and T is the sample length in years), not on the frequency of observation of the data. 28 " The data are available as ‘Barro‐Ursúa macroeconomic data’ at www.rbarro.com/data-sets. Full annual time series back to 1913 or earlier (except for a few missing observations around the Second World War) are provided for 40 countries for real per capita GDP and 28 countries for real per capita personal consumer expenditure. The data on the website are expressed as indexes for each country, with the value in 2006 normalised to 100. For the present study, the values were converted to levels comparable across countries based on data on 2005 PPP‐adjusted real per capita GDP from the World Bank's World Development Indicators. 29 " The data from 1870 to 1945 at five‐year intervals are unpublished estimates to be described in Barro and Lee(2015). 30 " One concern is that the comparison between the long‐term and short‐term results depends on sample selection with respect to which countries have long‐term data. However, the post‐1960 results in Table 1, column 3, do not differ greatly if the sample comprises only the 28 countries included in the long‐term sample in Table 5. The estimated convergence rate for this 28‐country sample observed since 1960 (266 total observations) is −0.0198 (SE = 0.0040). 31 " Acemoglu et al. (2008, Table 7) argue that lagged per capita GDP lacks significant explanatory power for the Polity indicator over long samples for 25 countries starting in 1875. 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I appreciate excellent research assistance from David Hsu and Ricardo Perez‐Truglia and comments from Philippe Aghion, Abhijit Banerjee, Xavier Gabaix, Elhanan Helpman, Chad Jones, Peter Klenow, Jong‐Wha Lee, Andrei Shleifer, Ugo Troiano, Jose Ursúa and Romain Wacziarg. I appreciate support of this research from the National Science Foundation. © 2015 Royal Economic Society

Yu,, Miaojie

doi: 10.1111/ecoj.12127pmid: N/A

Abstract This article explores how reductions in tariffs on imported inputs and final goods affect the productivity of large Chinese trading firms, with the special tariff treatment that processing firms receive on imported inputs. Firm‐level input and output tariffs are constructed. Both types of tariff reductions have positive impacts on productivity that are weaker as firms' share of processing imports grows. The impact of input tariff reductions on productivity improvement, overall, is weaker than that of output tariff reductions, although the opposite is true for non‐processing firms only. Both tariff reductions are found to contribute at least 14.5% to economy‐wide productivity growth. The effect of trade liberalisation on firm productivity is one of the most important topics in empirical trade research. Initially, trade economists primarily focused on the effect of cutting tariffs on final goods. At present, research interest has shifted to exploration of the effect of tariff reductions on imported intermediate inputs, which is usually greater than the effect on final goods (Amiti and Konings, 2007; Goldberg et al., 2010; Topalova and Khandelwal, 2011). Amiti and Konings (2007) analyse Indonesian firm‐level data and find that firms' gains from reduction of input tariffs are at least twice as much as those from reduction of output tariffs. Furthermore, Topalova and Khandelwal (2011) find that Indian firms' gains from input tariff reduction could be ten times greater than those from output tariff reduction in several industries. They forcefully argue that the primary reason for this result is that access to better intermediate inputs through the reduction of input tariffs is more important than the pro‐competitive effect of the reduction of output tariffs. Different from such findings, the present article shows that reducing output tariffs has had a greater effect on productivity improvement than reducing input tariffs for large Chinese trading firms in the new century. A 10 percentage point fall in output (input) tariffs leads to a productivity gain of 9.2 (5.1)%. The positive impact of both types of tariff reductions on productivity improvement is weaker as the firm's share of processing imports grows. Such results are primarily attributable to the special tariff treatment afforded to imported inputs by processing firms as opposed to non‐processing firms in China. Processing imports, which account for half of total imports in China, have zero tariffs. Further tariff reductions on imported intermediate inputs have no impact on firms that entirely engage in processing trade but still have some impact on firms that engage in both processing and non‐processing trade. As the firm's processing share grows, input tariff reductions have a smaller impact on productivity gains. Similarly, as firms' processing share increases, the share of domestic sales decreases accordingly; and the pro‐competition effects from the reductions in output tariffs are hence weaker. The current article contributes to the literature in at least three important ways. First, it enriches the understanding of the economic growth of China, the second largest economy and the largest exporter of goods in the world. It is widely believed that China's huge foreign trade volume, a 10% of world trade, is a fundamental cause of the country's rapid economic growth. However, this conjecture is rarely supported by using Chinese micro firm‐level data.1 This study aims to fill in this gap. Using highly disaggregated transaction‐level customs data and firm‐level production data from 2000–6, the article thoroughly explores the nexus between foreign trade and firm productivity. Second, processing trade is an important type of trade in many developing countries, such as Indonesia, Mexico and Vietnam. Processing trade is the process by which a domestic firm initially obtains raw materials or intermediate inputs from abroad and, after local processing, exports the value‐added final goods (Feenstra and Hanson, 2005). Governments typically encourage processing trade by offering tariff reductions or even exemptions on the processing of intermediate goods. Although there are some studies on trade reform in both developed and developing countries,2 the interaction between trade reform and processing trade is rarely explored. Hence, understanding the productivity gains from trade reform under the special tariff treatments afforded to processing trade is essential. Last but not least, aside from adopting the widely accepted method of measuring tariffs at the sector level, I take a step forward to measure both output tariffs and input tariffs at the firm level. Perhaps because of data restrictions, previous studies have usually measured tariffs at the industrial level by using input–output tables, as in Amiti and Konings (2007), or by measuring effective tariff protection as in Topalova and Khandelwal (2011). However, such a convenient approach might face a possible pitfall because input–output tables mix up both imported intermediate inputs and domestic intermediate inputs that are not directly relevant to tariff reductions. Using input–output tables may not accurately measure the level of trade protection faced by firms. Thanks to the rich information covered by both Chinese firm‐level production data and transaction‐level trade data, I am able to construct novel measures of firm‐specific input and output tariffs to estimate the effect of trade reforms on firm productivity. To my knowledge, this is the first attempt to measure tariffs at the firm level in the literature, although it is worthwhile to stress that my estimation results remain robust when using conventional industry‐level measures of tariffs. I also carefully control for two sets of endogeneity issues of firm‐level tariffs and firms' self‐selection to processing activities. Several endogeneity problems plague the firm‐level input and output tariffs. The first one results from tariff measures themselves. Because a firm may import multiple products, it is useful to construct an import‐based weight to reflect the importance of products for the firm. However, imports and tariffs are negatively correlated. In the extreme case, imports and their associated import shares are zero for prohibitive tariffs. As a result, the measure of input tariffs faces a downward bias. To address this endogeneity problem, throughout all the estimation, firm‐level tariffs are constructed using time‐invariant weights based on the firm's imports in the first year it appears in the sample. The second endogeneity problem relates to a possible reverse causality of tariffs with respect to productivity. Tariffs may be granted in response to domestic special interest groups, the pressure of which could be significant in countries such as India (Topalova and Khandelwal, 2011) or low in countries such as Indonesia (Amiti and Konings, 2007). Given that China acceded to the WTO in 2001, domestic pressure might not have played a key role during 2000–6. However, for the sake of completeness, an (IV) approach is adopted to control for possible reverse causality. Another set of endogeneity issues is of firms' self‐selection to processing activities. Observing that some Chinese firms are involved in both processing and ordinary trade, whereas others are only involved in one type of trade, I measure the processing variable in two ways. First, I use a processing indicator to identify whether a firm engages in processing trade. If a firm imports any products for processing purposes, as revealed in the customs data, such a firm is defined as a processing firm. However, the firm's processing share is endogenous. A firm would first decide whether to engage in processing trade and, if so, the extent to which it will engage in processing imports. To address such self‐selection behaviour, I rely on a type‐2 Tobit model. In the first‐step probit estimates, I find that low‐productivity firms self‐select to engage in processing trade, possibly to enjoy the free duty on imported intermediate inputs. After obtaining the firm's fitted extent of processing imports from the second‐step Heckman estimates, I use it as a measure of the processing indicator in the main estimates of the effects of tariffs on firm productivity to control for the endogeneity of the firm's processing decision. All else being constant, a high degree of engagement in processing trade is shown to reduce firm productivity. To explore the relationship between firm productivity and output and input tariffs, I follow the standard procedure to investigate the nexus in two steps. First, the firm's total factor productivity (TFP) is measured based on a production function using the methodology of Olley and Pakes (1996), with a number of necessary modifications and extensions to fit the Chinese context. As processing firms and non‐processing firms could use different technologies to produce products even within an industry, I estimate firm TFP for processing firms and non‐processing firms separately within an industry. I also take the firm's learning from processing trade into account (De Loecker, 2013). Although the augmented Olley–Pakes approach is capable of controlling for the possible simultaneity bias and selection bias caused by regular OLS estimates, it relies on the important assumption that capital is more actively responsive to unobserved productivity. However, China is a labour‐abundant country and hence has relatively low labour costs. In the face of a productivity shock, Chinese firms usually adjust their labour input to re‐optimise production behaviour (Blomström and Kokko, 1996). Therefore, I adopt three alternative approaches to measure firm TFP: (i) labour productivity; (ii) the Levinsohn–Petrin (2003) TFP; and (iii) the Blundell and Bond (1998) system‐GMM TFP. Given that the system‐GMM TFP has an additional advantage in controlling for the role of lagged firm productivity to avoid possible serial correlation in the TFP estimation (Fernandes, 2007), I use it as the main measure of firm TFP. It is also important to understand the mechanisms through which firm productivity improves in response to trade reforms. Inspired by previous studies, such as Amiti and Konings (2007), Goldberg et al. (2010) and Bustos (2011), the impact of input tariffs on productivity is straightforward, as lower tariffs induce a larger variety of inputs. By contrast, the impact of output tariffs on productivity could work directly by pressuring firms to be more productive, and/or indirectly by weeding out less‐productive firms. This article finds that the pro‐competition effect is mostly through the channels that pressure firms to be more productive, which is in line with the findings of Horn et al. (1995). Several possible channels – such as import scope and research and development (R&D) – are also discussed. Unlike Amiti and Konings (2007), my data set includes information that allows the firm's product scope (in export markets) to be directly measured as in Goldberg et al. (2010). In addition, similar to Bustos (2011), the analysis takes into consideration information on R&D expenses. Finally, as economy‐wide productivity is an essential measure of a country's welfare, my final step is to add firm productivity to economy‐wide productivity by using Domar's (1961) weight, which corrects for possible aggregation bias due to the ignorance of vertical integration in an open economy. In brief, I find that both output and input tariff reductions contribute at least 14.5% to economy‐wide productivity growth during the sample period. The remainder of the article is organised as follows. Section 1 introduces the special tariff treatment on Chinese processing trade. Section 2 describes the unique data used in the analysis. Section 3 discusses key variables and the econometric method. Section 4 presents the empirical evidence. Finally, Section 5 concludes. 1. Special Tariff Treatment on Processing Trade Processing trade in China began in the early 1980s. As an important means of trade liberalisation, the government encourages Chinese firms to import all or part of the raw materials and intermediate inputs, and re‐export final value‐added goods after local processing or assembly. As of 2012, the General Administration of Customs reports 16 specific types of processing trade in China.3 Among these types of trade, two are the most important, namely, processing with assembly and processing with inputs.4 Both types of processing trade are duty‐free but they are characterised by an important difference. For processing with assembly, a domestic Chinese firm obtains raw materials and parts from its foreign trading partners without any payment. However, after local processing, the firm has to sell its products to the same foreign trading partner by charging an assembly fee. By contrast, for processing with inputs, a domestic Chinese firm pays for raw materials from a foreign seller. After local processing, the Chinese firm can then sell its final goods to other foreign countries. Figure 1 shows that, compared with ordinary imports, processing imports in China accounted for just a small proportion of total imports in the early 1980s. However, China's processing imports dramatically increased in the early 1990s and began to dominate ordinary imports in 1992, when China officially announced the adoption of a market economy. Going forward, processing imports accounted for more than 50% of the country's total imports. Interestingly, processing imports with assembly were more popular in the 1980s because most Chinese firms lacked the capital needed to import. Since the 1990s, processing imports with inputs have been more prevalent. This trend can be seen clearly in Figure 2: within processing imports, the ratio of processing with assembly over processing with inputs declined from 0.41 in 2000 to 0.32 in 2006. The primary objective of the current article is to determine how a firm's TFP reacts to output and input tariff reductions in the presence of special tariff treatments on processing trade. Therefore, understanding whether a firm engages in processing activities is important. All Chinese firms are classified into four types, namely, non‐importing firms and three types of importing firms: ordinary importers, hybrid processing importers and pure processing importers. As shown in Figure 3, non‐importing firms do not have any imports; all raw materials and intermediate inputs are locally acquired. However, non‐importing firms can sell their final goods domestically and internationally (as shown by arrow (1)). Fig. 1. Open in new tabDownload slide China's Processing Imports Versus Ordinary Imports Fig. 1. Open in new tabDownload slide China's Processing Imports Versus Ordinary Imports Fig. 2. Open in new tabDownload slide China's Processing Imports: Assembly Versus Inputs Sources. Customs trade data (2000–6), author's own compilation. Fig. 2. Open in new tabDownload slide China's Processing Imports: Assembly Versus Inputs Sources. Customs trade data (2000–6), author's own compilation. Fig. 3. Open in new tabDownload slide Four Types of Chinese Firms Note. Dotted lines denote firms' processing imports/exports; solid lines represent firms' non‐processing imports/exports. Fig. 3. Open in new tabDownload slide Four Types of Chinese Firms Note. Dotted lines denote firms' processing imports/exports; solid lines represent firms' non‐processing imports/exports. Among the three types of importers, ordinary importers are firms that do not use any processing of imported intermediate inputs, although they import non‐processing intermediate inputs and could sell their final goods in both domestic and foreign markets (arrow (2)).5 In sharp contrast, pure processing importers are firms engaged only in processing activities, shown by the dotted lines in the Figure. Pure processing importers purchase 100% of their raw materials and intermediate inputs abroad and re‐export their final value‐added goods (arrow (5)). Such firms clearly enjoy the privilege of duty‐free imports. Finally, and perhaps the most interesting type of firm, hybrid processing importers engage in both ordinary imports (arrow (3)) and processing imports (arrow (4)). Such firms enjoy free duties for their processing imports, but still pay duties for ordinary imports. Here it is important to stress that the processing trade of both hybrid and pure processing importers could include any processing type, such as assembly and processing with inputs. 2. Data To investigate the impact of trade liberalisation on firm productivity, I rely on the following three disaggregated, large panel data sets: tariff data, firm‐level production data and product‐level trade data. Tariff data can be accessed directly from the WTO and the trade analysis and information system (TRAINS).6 China's tariff data are available at the Harmonised System (HS) six‐digit disaggregated level for 2000–6. Given that the product‐level trade data are at the HS eight‐digit level, the product‐level trade data are aggregated to the HS six‐digit level to correspond with the tariff data. As I am interested in measuring the average effect of trade liberalisation on firm productivity, I use the ad valorem duty at the six‐digit level to measure trade liberalisation. 2.1. Firm‐level Production Data The sample is derived from a rich firm‐level panel data set that covers between 162,885 firms (in 2000) and 301,961 firms (in 2006). The data are collected and maintained by China's National Bureau of Statistics (NBS) in an annual survey of manufacturing enterprises. Complete information on the three major accounting statements (i.e. balance sheet, profit and loss account, and cash flow statement) is available. In brief, the data set covers two types of manufacturing firms – all state‐owned enterprises (SOEs) and non‐SOEs whose annual sales exceed RMB 5 million ($770,000).7 The data set includes more than 100 financial variables listed in the main accounting statements of these firms. Although the data set contains rich information, some samples are still noisy and are therefore misleading, largely because of misreporting by some firms.8 Following Cai and Liu (2009), I clean the sample and omit outliers by using the following criteria. First, observations with missing key financial variables (such as total assets, net value of fixed assets, sales and gross value of the firm's output productivity) are excluded. Second, I drop firms with fewer than eight workers as they fall under a different legal regime, as mentioned in Brandt et al. (2012). Following Feenstra et al. (2013a), I delete observations according to the basic rules of the Generally Accepted Accounting Principles (GAAP) if any of the following are true: (i) liquid assets are greater than total assets; (ii) total fixed assets are greater than total assets; (iii) the net value of fixed assets is greater than total assets, (iv) the firm's identification number is missing; or (v) an invalid established time exists (e.g. the opening month is later than December or earlier than January). After applying such a stringent filter to guarantee the quality of the production data, the filtered firm data are reduced by about 50% in each year, as shown in columns (3) and (4) of Appendix Table A1. Note that, in China's customs data set, some Chinese firms do not have their own production activity but only export goods collected from other domestic firms or import goods from abroad and then sell them to other domestic companies (Ahn et al., 2010).9 To ensure the preciseness of the estimates, I exclude such trading companies from the sample in all the estimates. In particular, firms with names including any Chinese characters for Trading Company or Importing and Exporting Company are excluded from the sample.10 2.2. Product‐level Trade Data The extremely disaggregated product‐level trade transaction data are obtained from China's General Administration of Customs. It records a variety of information for each trading firm's product list, including trading price, quantity and value at the HS eight‐digit level. More importantly, this rich data set not only includes both import and export data but also breaks down the data into several specific types of processing trade, such as processing with assembly and processing with inputs. Table 1 reports a simple statistical summary for Chinese product‐level trade data by shipment and year for 2000–6. Overall, when focusing on the highly disaggregated HS eight‐digit level, approximately 35% of the 18,599,507 transaction‐level observations are ordinary trade and 65% refer to processing trade. Similar proportions are obtained when measuring by trade volume: around 43% of trade volume comprises ordinary trade. Processing with inputs accounts for around 30%, whereas processing with assembly only is around 10%. The remaining 17% represents other types of processing trade, aside from assembly and processing with inputs. Table 1 Chinese Transaction‐level Trade Data by Shipment and Year Imports by shipment . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Percentage of number of observations (HS eight‐digit) Ordinary imports 2.57 3.54 3.77 5.17 6.04 6.80 7.30 35.19 Processing imports with assembly 2.46 2.72 2.37 2.59 2.77 2.79 2.77 18.47 Processing imports with inputs 3.90 4.14 3.57 4.67 5.33 5.74 5.61 32.95 Other types of processing imports 1.42 1.55 1.70 1.71 2.03 2.24 2.77 13.40 Total 10.34 11.95 11.41 14.13 16.16 17.57 18.44 100 Percentage of import value Ordinary imports 3.12 3.87 3.71 5.87 7.74 8.86 10.46 43.64 Processing imports with assembly 0.87 0.98 0.98 1.22 1.68 2.11 2.31 10.16 Processing imports with inputs 2.02 2.21 2.39 3.87 5.24 6.52 7.15 29.40 Other types of processing imports 1.01 1.24 1.43 1.93 2.85 3.35 4.99 16.80 Total 7.02 8.30 8.52 12.89 17.51 20.85 24.91 100 Imports by shipment . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Percentage of number of observations (HS eight‐digit) Ordinary imports 2.57 3.54 3.77 5.17 6.04 6.80 7.30 35.19 Processing imports with assembly 2.46 2.72 2.37 2.59 2.77 2.79 2.77 18.47 Processing imports with inputs 3.90 4.14 3.57 4.67 5.33 5.74 5.61 32.95 Other types of processing imports 1.42 1.55 1.70 1.71 2.03 2.24 2.77 13.40 Total 10.34 11.95 11.41 14.13 16.16 17.57 18.44 100 Percentage of import value Ordinary imports 3.12 3.87 3.71 5.87 7.74 8.86 10.46 43.64 Processing imports with assembly 0.87 0.98 0.98 1.22 1.68 2.11 2.31 10.16 Processing imports with inputs 2.02 2.21 2.39 3.87 5.24 6.52 7.15 29.40 Other types of processing imports 1.01 1.24 1.43 1.93 2.85 3.35 4.99 16.80 Total 7.02 8.30 8.52 12.89 17.51 20.85 24.91 100 Open in new tab Table 1 Chinese Transaction‐level Trade Data by Shipment and Year Imports by shipment . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Percentage of number of observations (HS eight‐digit) Ordinary imports 2.57 3.54 3.77 5.17 6.04 6.80 7.30 35.19 Processing imports with assembly 2.46 2.72 2.37 2.59 2.77 2.79 2.77 18.47 Processing imports with inputs 3.90 4.14 3.57 4.67 5.33 5.74 5.61 32.95 Other types of processing imports 1.42 1.55 1.70 1.71 2.03 2.24 2.77 13.40 Total 10.34 11.95 11.41 14.13 16.16 17.57 18.44 100 Percentage of import value Ordinary imports 3.12 3.87 3.71 5.87 7.74 8.86 10.46 43.64 Processing imports with assembly 0.87 0.98 0.98 1.22 1.68 2.11 2.31 10.16 Processing imports with inputs 2.02 2.21 2.39 3.87 5.24 6.52 7.15 29.40 Other types of processing imports 1.01 1.24 1.43 1.93 2.85 3.35 4.99 16.80 Total 7.02 8.30 8.52 12.89 17.51 20.85 24.91 100 Imports by shipment . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Percentage of number of observations (HS eight‐digit) Ordinary imports 2.57 3.54 3.77 5.17 6.04 6.80 7.30 35.19 Processing imports with assembly 2.46 2.72 2.37 2.59 2.77 2.79 2.77 18.47 Processing imports with inputs 3.90 4.14 3.57 4.67 5.33 5.74 5.61 32.95 Other types of processing imports 1.42 1.55 1.70 1.71 2.03 2.24 2.77 13.40 Total 10.34 11.95 11.41 14.13 16.16 17.57 18.44 100 Percentage of import value Ordinary imports 3.12 3.87 3.71 5.87 7.74 8.86 10.46 43.64 Processing imports with assembly 0.87 0.98 0.98 1.22 1.68 2.11 2.31 10.16 Processing imports with inputs 2.02 2.21 2.39 3.87 5.24 6.52 7.15 29.40 Other types of processing imports 1.01 1.24 1.43 1.93 2.85 3.35 4.99 16.80 Total 7.02 8.30 8.52 12.89 17.51 20.85 24.91 100 Open in new tab 2.3. Merged Data Set Firm‐level production data are crucial in measuring TFP, whereas product‐level trade transaction data are non‐substitutable in identifying a processing firm. However, researchers face some technical challenges in merging the two data sets. Although the data sets share a common variable (i.e. the firm's identification number), the coding system in each data set is completely different.11 Hence, the firm's identification number cannot serve as a bridge to match the two data sets. To address this challenge, following Yu and Tian (2012), I use two methods to match the two data sets by using other common variables. First, I match the two data sets by using each firm's Chinese name and year. That is, if a firm has an exact Chinese name in both data sets in a particular year, it should be the same firm.12 As described carefully in Appendix A, I obtain 83,679 matched firms in total by using the raw production data set and the number is reduced to 69,623 in total by using the more accurate filtered production data set as described above. To increase the number of qualified matching firms as much as possible, I then use another matching technique to serve as a supplement. Namely, I rely on two other common variables to identify the firms: postal code and the last seven digits of the firm's phone number. The rationale is that firms should have a unique phone number within a postal district. Although this method seems straightforward, there are subtle technical and practical difficulties.13 The detailed merging procedures are explained in Appendix A. After merging both product‐level trade data and firm‐level production data, I finally obtain 76,823 common trading firms, including both importers and exporters.14 Briefly, the merged data set accounts for around 40% of the filtered full‐sample, firm‐level production data set in terms of the number of exporters, and around 53% in terms of export value. By way of comparison, my matching success rate is highly comparable to that in other studies that use the same data sets, such as Ge et al. (2011) and Wang and Yu (2012). How successful is the matching using this technique? Table 2 first compares the merged data and the full‐sample customs trade data sets. Of the total 56,459 importing firms in the merged data, ordinary importers account for 38.1% whereas processing importers account for 61.9%. These numbers are close to their counterparts from the full‐sample customs data – 27.3% for ordinary importers and 72.7% for processing importers – as shown in the last column of Table 2.15 The proportions of hybrid processing importers and pure processing importers by year in both the merged data and the full‐sample data sets are also reported in the bottom two rows of Table 2. Given that the firm‐level production data set is crucial for the construction of the regressand (i.e. firm TFP), Table 3 shows how much of total sales and total employment are accounted for by the merged data set each year during 2000–6. In particular, the proportion of exports in the merged sample over exports in the full‐sample production data varies from 50% to around 58% during the sample period, suggesting that some firms enter and exit in the merged sample that is used for the estimation. The merged data set includes both exporters and importers.16 Moreover, Table 4 compares the differences between the merged data set and the full‐sample firm‐level data set. The merged sample has clearly higher means of sales, exports and number of employees than those in the full‐sample firm‐level data set. These findings suggest that the merged sample is skewed towards large firms. Thus, my findings are valid for large Chinese trading firms. Table 2 Merged Importers by Firm Type . Merged sample . . Percentage . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Full sample . Total importers 8.8 9.9 10.6 12.4 19.4 18.0 21.0 100.0 100.0 Ordinary importers 2.4 3.0 3.7 5.0 7.5 7.3 9.1 38.1 27.3 Processing importers 6.4 6.9 6.9 7.4 12.0 10.7 11.8 61.9 72.7 Hybrid processing importers 3.0 3.2 3.5 3.9 5.8 5.3 6.0 30.7 53.0 Pure processing importers 3.4 3.6 3.4 3.5 6.2 5.4 5.9 31.2 19.7 . Merged sample . . Percentage . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Full sample . Total importers 8.8 9.9 10.6 12.4 19.4 18.0 21.0 100.0 100.0 Ordinary importers 2.4 3.0 3.7 5.0 7.5 7.3 9.1 38.1 27.3 Processing importers 6.4 6.9 6.9 7.4 12.0 10.7 11.8 61.9 72.7 Hybrid processing importers 3.0 3.2 3.5 3.9 5.8 5.3 6.0 30.7 53.0 Pure processing importers 3.4 3.6 3.4 3.5 6.2 5.4 5.9 31.2 19.7 Notes There are 56,459 importers in total in the matched data whereas 217,372 firm importers are included in the full‐sample trade data. Open in new tab Table 2 Merged Importers by Firm Type . Merged sample . . Percentage . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Full sample . Total importers 8.8 9.9 10.6 12.4 19.4 18.0 21.0 100.0 100.0 Ordinary importers 2.4 3.0 3.7 5.0 7.5 7.3 9.1 38.1 27.3 Processing importers 6.4 6.9 6.9 7.4 12.0 10.7 11.8 61.9 72.7 Hybrid processing importers 3.0 3.2 3.5 3.9 5.8 5.3 6.0 30.7 53.0 Pure processing importers 3.4 3.6 3.4 3.5 6.2 5.4 5.9 31.2 19.7 . Merged sample . . Percentage . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Total . Full sample . Total importers 8.8 9.9 10.6 12.4 19.4 18.0 21.0 100.0 100.0 Ordinary importers 2.4 3.0 3.7 5.0 7.5 7.3 9.1 38.1 27.3 Processing importers 6.4 6.9 6.9 7.4 12.0 10.7 11.8 61.9 72.7 Hybrid processing importers 3.0 3.2 3.5 3.9 5.8 5.3 6.0 30.7 53.0 Pure processing importers 3.4 3.6 3.4 3.5 6.2 5.4 5.9 31.2 19.7 Notes There are 56,459 importers in total in the matched data whereas 217,372 firm importers are included in the full‐sample trade data. Open in new tab Table 3 Firm‐level Production Information in Merged Versus Full‐sample Data by Year Types of firms (%) . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Average . Sales 23.7 24.0 23.8 24.6 27.8 25.8 28.3 25.5 Exports 51.9 50.1 52.9 50.0 55.2 51.6 57.9 52.8 Number of employees 20.2 20.9 21.6 23.0 26.5 25.5 28.7 23.8 Types of firms (%) . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Average . Sales 23.7 24.0 23.8 24.6 27.8 25.8 28.3 25.5 Exports 51.9 50.1 52.9 50.0 55.2 51.6 57.9 52.8 Number of employees 20.2 20.9 21.6 23.0 26.5 25.5 28.7 23.8 Notes The values in this panel are the proportions that were obtained by dividing sales/exports/number of employees in the matched data by their counterparts in the full‐sample data, respectively. The last column reports the year‐average percentage over 2000–6. Open in new tab Table 3 Firm‐level Production Information in Merged Versus Full‐sample Data by Year Types of firms (%) . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Average . Sales 23.7 24.0 23.8 24.6 27.8 25.8 28.3 25.5 Exports 51.9 50.1 52.9 50.0 55.2 51.6 57.9 52.8 Number of employees 20.2 20.9 21.6 23.0 26.5 25.5 28.7 23.8 Types of firms (%) . 2000 . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Average . Sales 23.7 24.0 23.8 24.6 27.8 25.8 28.3 25.5 Exports 51.9 50.1 52.9 50.0 55.2 51.6 57.9 52.8 Number of employees 20.2 20.9 21.6 23.0 26.5 25.5 28.7 23.8 Notes The values in this panel are the proportions that were obtained by dividing sales/exports/number of employees in the matched data by their counterparts in the full‐sample data, respectively. The last column reports the year‐average percentage over 2000–6. Open in new tab Table 4 Comparison of the Merged Data Set and the Full‐sample Production Data Set . Merged data . Full‐sample data . Variables . Mean . Min. . Max. . Mean . Min. . Max. . Sales (RMB 1,000) 150,053 5000 1.57e+08 85,065 5000 1.57e+08 Exports (RMB 1,000) 53,308 0 1.52e+08 16,544 0 1.52e+08 Number of employees 478 8 157,213 274 8 165,878 . Merged data . Full‐sample data . Variables . Mean . Min. . Max. . Mean . Min. . Max. . Sales (RMB 1,000) 150,053 5000 1.57e+08 85,065 5000 1.57e+08 Exports (RMB 1,000) 53,308 0 1.52e+08 16,544 0 1.52e+08 Number of employees 478 8 157,213 274 8 165,878 Open in new tab Table 4 Comparison of the Merged Data Set and the Full‐sample Production Data Set . Merged data . Full‐sample data . Variables . Mean . Min. . Max. . Mean . Min. . Max. . Sales (RMB 1,000) 150,053 5000 1.57e+08 85,065 5000 1.57e+08 Exports (RMB 1,000) 53,308 0 1.52e+08 16,544 0 1.52e+08 Number of employees 478 8 157,213 274 8 165,878 . Merged data . Full‐sample data . Variables . Mean . Min. . Max. . Mean . Min. . Max. . Sales (RMB 1,000) 150,053 5000 1.57e+08 85,065 5000 1.57e+08 Exports (RMB 1,000) 53,308 0 1.52e+08 16,544 0 1.52e+08 Number of employees 478 8 157,213 274 8 165,878 Open in new tab 3. Measures and Empirics In this Section, I first introduce the measures of the three key variables: firm TFP, firm‐specific output tariffs and firm‐specific input tariffs. For comparison, I also introduce the measure of industry‐specific output and input tariffs. Finally, I discuss my empirical investigation of the effect of tariff reductions on productivity. 3.1. TFP Measures I use the augmented Olley and Pakes (1996) approach to construct measures of Chinese firm‐level TFP following Amiti and Konings (2007). Assuming a Cobb–Douglas production function, the usual estimation equation is as follows: lnYitj=β0j+βmjlnMitj+βkjlnKitj+βljlnLitj+ϵit,(1) where Yitj ⁠, Mitj ⁠, Kitj and Litj refer to firm i's output, materials, capital and labour in industry j in year t, respectively. Traditionally, TFP is measured by the estimated Solow residual which is the difference between the true data on output and the fitted value using the OLS approach. However, the OLS approach suffers from two problems: simultaneity bias and selection bias. At least some shocks to TFP changes could be observed by the firm early enough for it to change its input decisions to maximise profit. Thus, firm TFP could have a reverse endogeneity on firm input choices. Moreover, firms with low productivity that have collapsed and exited the market are excluded from the data set, indicating that the samples used for the regression are not randomly selected, which, in turn, results in estimation bias. Olley and Pakes (1996) successfully provide a semi‐parametric approach to address those two biases. Subsequently, numerous studies, such as those by De Loecker (2011, 2013) and De Loecker et al. (2012), among others, have modified and tailored their approaches to calculating TFP. In the present article, I adopt the Olley–Pakes approach to estimate and calculate a firm's TFP with some extensions. Appendix B provides the detailed estimation procedure. First and foremost, I estimate the production function for processing and non‐processing firms separately in each industry. The idea is that different industries may use different technologies; hence, firm TFP (denoted TFPOP1) is estimated separately for each industry. Equally important, even within an industry, processing firms (especially those firms engaged in processing with assembly) may use completely different technologies than non‐processing firms, given that processing firms with assembly receive only imported material passively without making any profit‐maximising input choices (Feenstra and Hanson, 2005). For the non‐processing firm TFP estimates, since a non‐processing importing firm may or may not export its final goods, I also include an export dummy to allow different TFP realisation between exporting non‐processing firms and non‐exporting non‐processing firms. By the same token, I include an import dummy in the control function to allow different TFP realisation between non‐processing importers and non‐processing non‐importers (but exporters). Note that two such dummies are not necessary for processing firms as, by definition, processing firms must import inputs and sell their products abroad. Possibly, firms could learn by processing imports. If productivity gains from processing imports occur simultaneously with investment, TFPOP1 may have a bias on the estimated capital coefficient. Thus, ignorance of controlling for the effect of the previous period's processing activity on firm productivity may cause another bias of measured productivity. Inspired by De Loecker (2013), as an alternative approach to estimate TFP (denoted by TFPOP2), I consider another control function in which both processing and non‐processing firms are pooled together. More importantly, a processing dummy (i.e. a dummy that takes the value one if a firm has any processing imports and zero otherwise) is also incorporated in the control function (see Appendix B for details). This is done because processing imports may affect firm productivity and, accordingly, the TFP trajectory of a processing firm is endogenously different compared with the trajectory of a non‐processing firm. Second, I use deflated prices at the industry level to measure TFP. The measured TFP is expected to capture the firm's true technical efficiency only. However, here the measured TFP is also likely to pick up differences in price, price‐cost markups and even input usage across firms (De Loecker, 2011; De Loecker and Warzynski, 2012). Admittedly, an ideal way to remove price differences across firms would be to adopt firm‐specific price deflators (Foster et al., 2007). However, as in many other studies, such price data are unavailable.17 Following De Loecker et al. (2012), I use the industrial price to deflate the firm's output.18 Turning to the issue of price‐cost markups, as stressed by Bernard et al. (2003), once the price‐cost markup is positively associated with true efficiency, even revenue‐based productivity can work well to capture the true efficiency, as is done with physical efficiency. Third, I take China's WTO accession in 2001 into account, as such a positive demand shock would push Chinese firms to expand their economic scales, which, in turn, would exaggerate the simultaneous bias of their measured TFP. In particular, a WTO dummy (i.e. equal to one after 2001 and zero otherwise) is included in the estimation of the capital coefficient, as discussed in Appendix B. Fourth, the prevalence of SOEs also affects firm productivity. SOEs in China are usually accompanied by state intervention and do not necessarily make profit‐maximising choices (Hsieh and Klenow, 2009). Therefore, it is important to construct an SOE indicator and add it to the control function in the first‐step Olley–Pakes estimates.19 Finally, it is necessary to construct a real investment variable when using the Olley and Pakes (1996) approach. I adopt the perpetual inventory method as the law of motion for real capital and real investment. Nominal and real capital stocks are constructed as in Brandt et al. (2012). Rather than assigning an arbitrary number for the depreciation ratio, I use the exact firm's real depreciation provided by the Chinese firm‐level data set. Appendix Table B1 presents the estimated coefficients for the production function and the associated log of TFP by industry for processing firms and non‐processing firms, respectively. The implied scale elasticities are quite close to constant returns‐to‐scale elasticities for both processing firms and non‐processing firms within each industry. The augmented Olley–Pakes approach assumes that capital responds to the unobserved productivity shock with a Markov process, whereas other input factors respond without any dynamic effects. However, labour may also be correlated with an unobserved productivity shock. As highlighted by Ackerberg et al. (2006), it is unlikely that there is enough variation left to identify the labour coefficient by using the Olley–Pakes approach. This consideration may fit China's case more closely, given that the country is labour abundant. When facing an unobserved productivity shock, firms might re‐optimise their production behaviour by adjusting their labour rather than their capital. I use the Blundell and Bond (1998) system‐GMM approach to capture the dynamic effects of other input factors. By assuming that the unobserved productivity shock depends on a firm's previous period?s realisations, the system‐GMM approach models TFP as affected by all types of inputs in both current and past realisations. In particular, this model has the following dynamic representation: lnyitj=γ0j+γ1jlnLitj+γ2jlnLi,t−1j+(γ3jlnLitj+γ4jlnLi,t−1j)PEit+γ5jlnKitj+γ6jlnKi,t−1j+(γ7jlnKitj+γ8jlnKi,t−1j)PEit+γ9jlnMitj+γ10jlnMi,t−1j+(γ11jlnMitj+γ12jlnMi,t−1j)PEit+γ13jlnyi,t−1j+γ14jlnyi,t−1jPEit+γ15PEit+ςi+ζt+ωit,(2) where ςi is firm i's fixed effect, ζt is the year‐specific fixed effect, and PEit is a processing indicator that takes the value one if a firm has any processing imports and zero otherwise. The idiosyncratic term ωit is serially uncorrelated if no measurement error exists.20 Consistent estimates of the coefficients in the model can be obtained by using a system‐GMM approach. The idea is that labour and material inputs are not taken as exogenously given but are instead allowed to change over time as capital grows. Appendix Table C1 presents the estimated coefficients for system‐GMM firm TFP by industry.21 Overall, the estimated log TFP increases 0.17 log points (from 2.28 in 2001 to 2.45 in 2006), registering a 2.62% annual growth rate, which is very close to the findings in Brandt et al. (2012). 3.2. Firm‐Specific Tariffs A firm could produce multiple products and, thus, its productivity could be affected by multiple tariff lines. Hence, it is important to properly measure the input tariff level faced by firms. As mentioned above, processing imports are duty‐free in China. Given that a firm could engage in both processing imports (P) and non‐processing imports (O), I construct a firm‐specific input tariff index (FITit) as follows: FITit=∑k∈Omi,initial_yeark∑k∈Mmi,initial_yearkτtk,(3) where mi,initial_yeark is firm i's imports of product k in the first year the firm appears in the sample. Note that O ∪ P = M where M is the set of the firm's total imports. The set of processing imports does not appear in (3) because processing imports, again, are duty‐free. The firm's input tariffs are constructed by using time‐invariant weights to avoid the well‐known endogeneity of weighted tariffs: imports are negatively associated with tariffs. For products with prohibitive tariffs, their imports and the associated import share would be zero. Accordingly, if the import weight is measured in the current period, the measure of firm tariffs would face a downward bias. Therefore, following Topalova and Khandelwal (2011), I measure the import weight for each product using data for the firm's first year in the sample. Turning to the construction of firm‐level output tariffs, product‐level domestic sales would be an ideal proxy for capturing the role of each product within a firm. However, such data are unavailable. Hence, I rely on an index to circumvent this data restriction. As a more productive firm is not only capable of selling its products domestically, but also internationally (Melitz, 2003), a product would, in general, be sold domestically if it is sold abroad. Assuming a product is sold domestically and internationally in the same proportions, I consider a following weighted output tariff index (FOTit) for firm i in year t: FOTit=∑kXi,initial_yeark∑kXi,initial_yearkτtk,(4) where τtk is the ad valorem tariff of product k in year t. The ratio in the parentheses is the value weight of product k, measured by the firm's exports of product k in its initial year in the sample, Xi,initial_yeark ⁠, over the firm's total exports in the initial year.22 Inspired by Topalova and Khandelwal (2011), exports for each product are fixed at the initial period to avoid possible reverse causality in firm productivity with respect to measured output tariffs. This measure suffers from two important caveats. First, a firm may sell a product at home but not abroad (i.e. it is a pure domestic firm), which could be fairly reasonable as recent studies show that multi‐product firms often sell different products at home and abroad (Bernard et al. 2011; Arkolakis and Muendler, 2012). In this case, the export weight for such a product in (4) is zero and the firm's output tariff measure fails to capture any pro‐competition effects. This argument also holds for pure exporting firms that sell their products abroad only (around 12.2% of firms are pure exporters in my matched data). To ensure that my main estimation results are not biased by such firms, I drop pure domestic firms and pure exporting firms from the sample in all regressions. Second, the exported and domestic shares of a product are assumed to be equal. Note that this is a strong assumption indeed as the product composition of exports may be very different from the composition of domestic sales. This is especially true for China, which holds an important position in global supply chains (GSCs) and produces some intermediates that cannot be used in the domestic production sector.23 Because of data restrictions, I am not able to check this out directly. However, as this problem would bias the measure of firm output tariffs differently depending on the industry and depending on the intensity of the sector of processing firms, I run further regressions by distinguishing more integrated industries from less integrated industries and by separating the sample by the intensity of the sector in processing firms. As shown in the text later, all such robustness checks suggest that my main results are still valid even considering such within‐firm differences in product composition. Columns (1)–(4) in Table 5 report firm‐specific input and output tariffs computed using (3) and (4), respectively. The average firm‐specific output tariffs were cut in half from around 15.6% in 2000 to 7.4% in 2006, and their standard deviation also dropped by around 50% over the same period. Firm‐specific input tariffs are much lower than output tariffs. Input tariffs also exhibit a sharp declining trend during the sample period. Table 5 China's Output Tariffs and Input Tariffs by Year . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . . Mean . SD . Mean . SD . Mean . SD . Mean . SD . Year . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 15.57 12.03 2.54 4.90 21.43 8.78 3.00 3.63 2001 12.39 9.40 2.37 5.06 17.77 6.07 2.98 3.78 2002 9.63 8.22 1.68 3.53 14.28 6.05 1.41 1.66 2003 8.82 7.51 1.94 3.70 12.46 5.21 0.41 0.27 2004 7.59 7.08 1.87 3.59 11.27 4.60 0.36 0.25 2005 7.00 6.78 1.71 3.53 10.49 4.46 0.34 0.21 2006 7.46 6.46 2.18 3.72 10.27 4.20 0.35 0.18 All years 8.29 7.65 1.98 3.82 11.88 5.63 0.69 0.15 . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . . Mean . SD . Mean . SD . Mean . SD . Mean . SD . Year . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 15.57 12.03 2.54 4.90 21.43 8.78 3.00 3.63 2001 12.39 9.40 2.37 5.06 17.77 6.07 2.98 3.78 2002 9.63 8.22 1.68 3.53 14.28 6.05 1.41 1.66 2003 8.82 7.51 1.94 3.70 12.46 5.21 0.41 0.27 2004 7.59 7.08 1.87 3.59 11.27 4.60 0.36 0.25 2005 7.00 6.78 1.71 3.53 10.49 4.46 0.34 0.21 2006 7.46 6.46 2.18 3.72 10.27 4.20 0.35 0.18 All years 8.29 7.65 1.98 3.82 11.88 5.63 0.69 0.15 Notes Columns (1)–(4) report the mean and standard deviation of firm output tariffs and firm input tariffs with initial time‐invariant weights as described in (4) and (3), in the text. Columns (5) and (6) report the mean and standard deviation of industry‐level output tariffs and columns (7)–(8) report the mean and standard deviation of industry‐level input tariffs that are constructed using the 2002 input?output table for China. Open in new tab Table 5 China's Output Tariffs and Input Tariffs by Year . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . . Mean . SD . Mean . SD . Mean . SD . Mean . SD . Year . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 15.57 12.03 2.54 4.90 21.43 8.78 3.00 3.63 2001 12.39 9.40 2.37 5.06 17.77 6.07 2.98 3.78 2002 9.63 8.22 1.68 3.53 14.28 6.05 1.41 1.66 2003 8.82 7.51 1.94 3.70 12.46 5.21 0.41 0.27 2004 7.59 7.08 1.87 3.59 11.27 4.60 0.36 0.25 2005 7.00 6.78 1.71 3.53 10.49 4.46 0.34 0.21 2006 7.46 6.46 2.18 3.72 10.27 4.20 0.35 0.18 All years 8.29 7.65 1.98 3.82 11.88 5.63 0.69 0.15 . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . . Mean . SD . Mean . SD . Mean . SD . Mean . SD . Year . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 15.57 12.03 2.54 4.90 21.43 8.78 3.00 3.63 2001 12.39 9.40 2.37 5.06 17.77 6.07 2.98 3.78 2002 9.63 8.22 1.68 3.53 14.28 6.05 1.41 1.66 2003 8.82 7.51 1.94 3.70 12.46 5.21 0.41 0.27 2004 7.59 7.08 1.87 3.59 11.27 4.60 0.36 0.25 2005 7.00 6.78 1.71 3.53 10.49 4.46 0.34 0.21 2006 7.46 6.46 2.18 3.72 10.27 4.20 0.35 0.18 All years 8.29 7.65 1.98 3.82 11.88 5.63 0.69 0.15 Notes Columns (1)–(4) report the mean and standard deviation of firm output tariffs and firm input tariffs with initial time‐invariant weights as described in (4) and (3), in the text. Columns (5) and (6) report the mean and standard deviation of industry‐level output tariffs and columns (7)–(8) report the mean and standard deviation of industry‐level input tariffs that are constructed using the 2002 input?output table for China. Open in new tab 3.3. Industry‐specific Tariffs Similar to Amiti and Konings (2007), the sector output tariffs at the two‐digit Chinese industry classification (CIC) level are obtained by taking a simple average of the HS six‐digit codes within each two‐digit CIC industry code.24 The industry‐level input tariff index is measured by IITft=∑ninputnf2002∑ninputnf2002τnt,(5) where IITft denotes the industry‐level input tariffs facing firms in industry f in year t. τnt is the import tariff of input n in year t. The weight in parentheses is measured as the cost share of input n in the production of industry f, for which data can be obtained from by China's input–output table for 2002.25 As shown in columns (5)–(8) in Table 5, the information in these columns is in line with that obtained by using the firm‐level tariffs in columns (1)–(4): both output and input tariffs dramatically fell over the sample period. Similar patterns can be found from their standard deviations. Firm‐specific output tariffs seem to be lower than industrial output tariffs. In sharp contrast, firm‐specific input tariffs are higher than industry‐specific input tariffs. One possible reason for the under‐measurement of industrial input tariffs is that the inclusion of non‐importing firms in intermediate input industries biases the industrial input weight in (5) which does not show up in the corresponding firm‐specific input tariffs.26 The simple correlations reported in Table 6 confirm this point: industry‐specific input tariffs are only weakly correlated to firm‐specific input tariffs (|corr.| = 0.06), whereas industry‐specific output tariffs are strongly correlated to firm‐specific output tariffs, as expected (|corr.| = 0.48). Table 6 Simple Correlations of China's Output Tariffs and Input Tariffs . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . Firm output tariffs 1.000 Firm input tariffs 0.092 1.000 Industry output tariffs 0.477 −0.073 1.000 Industry input tariffs 0.328 −0.062 0.578 1.000 . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . Firm output tariffs 1.000 Firm input tariffs 0.092 1.000 Industry output tariffs 0.477 −0.073 1.000 Industry input tariffs 0.328 −0.062 0.578 1.000 Open in new tab Table 6 Simple Correlations of China's Output Tariffs and Input Tariffs . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . Firm output tariffs 1.000 Firm input tariffs 0.092 1.000 Industry output tariffs 0.477 −0.073 1.000 Industry input tariffs 0.328 −0.062 0.578 1.000 . Firm output tariffs . Firm input tariffs . Industry output tariffs . Industry input tariffs . Firm output tariffs 1.000 Firm input tariffs 0.092 1.000 Industry output tariffs 0.477 −0.073 1.000 Industry input tariffs 0.328 −0.062 0.578 1.000 Open in new tab 3.4. Empirical Specification To investigate the effects of input and output tariff reductions on firm productivity, I consider the following empirical framework: lnTFPit=β0+β1FOTit+β2FOTit×PEit+β3FITit+β4FITit×PEit+β5PEit+θXit+ϖi+ηt+μit,(6) where lnTFPit is the logarithm of firm i's measured TFP in industry j in year t, whereas FITit and FOTit denote firm‐level input tariffs and output tariffs as measured in (3) and (4), respectively. The augmented Olley–Pakes TFP is adopted for the baseline estimates, but the system‐GMM TFP is adopted as the main measure, given that it enjoys rich, measured flexibility. PEit is a processing indicator that equals one if firms import any processing products in year t, and zero otherwise. An interaction term between the firm's output (input) tariff and the processing indicator is also included to capture a possible heterogeneous effect of output (input) tariff reductions on firm productivity between processing and ordinary firms. In addition, β5 in (6) measures other possible gains from processing trade not caused by trade liberalisation. Xit denotes other firm characteristics, such as type of ownership (i.e. SOEs or multinational firms). SOEs are traditionally believed to have relatively low economic efficiency and, hence, low productivity (Hsieh and Klenow, 2009). By contrast, multinational firms have higher productivity in part because of international technology spillovers (Keller and Yeaple, 2009) or fewer financial constraints (Manova et al., 2009). Therefore, I construct two indicators to measure the roles of SOEs and multinational firms. In particular, a firm is classified as a foreign firm if it has any investments from other countries (regimes). A large proportion of the inflow of foreign investment comes from Hong Kong/Macao/Taiwan, so these investments are considered in the construction of such an indicator.27 As a result, 77% of trading firms are classified as multinational affiliates.28 Similarly, I construct an indicator for SOEs, which is one if a firm has any investment from the government, and zero otherwise. Finally, the error term is divided into three components: (i) firm‐specific fixed effects ϖi to control for time‐invariant but unobservable factors such as managerial ability; (ii) year‐specific fixed effects ηt to control for firm‐invariant factors such as an appreciation of the renminbi (RMB); and (iii) an idiosyncratic effect μit with normal distribution μit∼N(0,σi2) to control for other unspecified factors. However, the empirical specification above faces an identification challenge. The processing indicator in (6) is a relatively crude measure of processing activity, which may overestimate the role of processing firms. For example, if a firm has only a very small proportion of processing imports over total imports, it is still classified as a processing firm, yet its primary operation remains in ordinary trade. To overcome this challenge, I consider a continuous measure of the extent to which a firm is engaged in processing trade to replace the processing indicator, and the extent of processing engagement (Pextit) is measured through firm i's total processing imports over total imports in year t. In particular, I consider the following specification for my main estimation: lnTFPit=β0+β1FOTit+β2FOTit×Pextit+β3FITit+β4FITit×Pextit+β5Pextit+θXit+ϖi+ηt+μit.(7) Yet, a new identification challenge arises from the coefficients of the variable Pextit itself and its interaction terms: β2, β4 and β5. These coefficients differ across industries as different industries use different technologies (Pavcnik, 2002). More importantly, even within an industry, the decision to engage in processing trade is endogenous to firms. Previous works, such as Dai et al. (2012), find that less‐productive firms self‐select to engage in processing trade. If so, a firm's extent of processing engagement is also endogenous as firms with a high extent of processing engagement may be less productive. That is, β2, β4 and β5 vary across firms. My estimating equation thus has random coefficients that are correlated with the endogenous extent of processing engagement, so it is a correlated random coefficients (CRC) model (Wooldridge, 2008). Heckman and Vytlacil (1998) recommend replacing the endogenous variable in a CRC model – or the extent of processing engagement in my case – with its predicted value.29 In the next Section, I estimate the extent of processing engagement with a Heckman procedure, or type‐2 Tobit model, using the exogenous variables Zit which is be specified in the next Section. In particular, I have Pextit=E(Pextit|Zit)+ϵit,withE(ϵit|Zit)=0.(8) By substituting (8) into (7), I obtain: lnTFPit=β0+β1FOTit+β2FOTit×E(Pextit|Zit)+β3FITit+β4FITit×E(Pextit|Zit)+β5E(Pextit|Zit)+θXit+ϖi+ηt+εit,(9) where the error term is εit = (β2FOTit + β4FITit + β5)εit + μit. 30 All the terms appearing within this error have zero expected value conditional on Zit, so that εit is conditionally uncorrelated with these exogenous variables and they can be used for estimation. Finally, as suggested by Wooldridge (2008), a correction to the standard errors must be made to reflect the use of estimated regressors in (9), which I implement by bootstrapping. 4. Estimation Results 4.1. Baseline Results As described above, the merged data set is skewed towards large trading firms, which are the main focus of the present article. Still, it is worthwhile checking whether the relatively high attrition rate of the merged data set affects the estimation results. Hence, my estimation begins with a comparison between the full‐sample data set and the merged data set. I start off the estimation in Table 7 by using conventional industry‐level tariffs, as introduced in subsection 11. Columns (1) and (2) first run regressions using full‐sample firm data. As processing information is not included in the full‐sample firm data, it is ignored in the estimation. As firms in different industries would adopt different technologies, it would be inappropriate to combine firms across all industries without controlling for industrial differences (Pavcnik, 2002). Therefore, I control for industry‐level fixed effects at the two‐digit CIC level in the estimates in column (1). It turns out that both industrial output tariffs and input tariffs are negatively and statistically significantly correlated with firm productivity, which is consistent with the findings of many other studies. Column (2) takes a step forward to control for firm‐specific fixed effects and year‐specific fixed effects. The coefficient of industry output tariffs is still negative and significant. Strikingly enough, the coefficient of industry input tariffs is positive. However, this is not a worry as the coefficient is statistically insignificant. One possible reason for such an unanticipated finding is the inclusion of non‐importing firms that appeared in the full‐sample firm data set but did not directly benefit from reductions in tariffs on the imported intermediate inputs. Table 7 Benchmark Estimates for Comparisons Regressand: ln TFPijtOP . Full‐sample data set . Merged data set . (1) . (2) . (3) . (4) . Industry output tariffs −0.563** −0.264*** −0.601*** −0.154* (−2.77) (−8.42) (−5.09) (−1.91) Industry input tariffs −2.54** 0.133 −1.46*** −1.45*** (−4.97) (0.93) (−4.08) (−3.53) Industry‐specific fixed effects Yes No Yes No Firm‐specific fixed effects No Yes No Yes Year‐specific fixed effects No Yes No Yes Observations 315,416 315,416 82,570 82,570 Prob. > F 0.000 0.000 0.000 0.000 R2 0.21 0.13 0.34 0.02 Regressand: ln TFPijtOP . Full‐sample data set . Merged data set . (1) . (2) . (3) . (4) . Industry output tariffs −0.563** −0.264*** −0.601*** −0.154* (−2.77) (−8.42) (−5.09) (−1.91) Industry input tariffs −2.54** 0.133 −1.46*** −1.45*** (−4.97) (0.93) (−4.08) (−3.53) Industry‐specific fixed effects Yes No Yes No Firm‐specific fixed effects No Yes No Yes Year‐specific fixed effects No Yes No Yes Observations 315,416 315,416 82,570 82,570 Prob. > F 0.000 0.000 0.000 0.000 R2 0.21 0.13 0.34 0.02 Notes t‐values are in parentheses. Significant at *10%, **5% and ***1%. Regressions in columns (1) and (2) use the entire sample for Chinese firms (2000–6), whereas those in columns (3) and (4) use the matched sample for Chinese trading firms (2000–6). Regressions in columns (1) and (3) are clustered at the two‐digit Chinese industry level. Industry input tariffs are calculated by using the 2002 time‐invariant input–output matrix for China as described in (5) in the text. Regressions in columns (1) and (3) are clustered at the one‐digit industry level. Open in new tab Table 7 Benchmark Estimates for Comparisons Regressand: ln TFPijtOP . Full‐sample data set . Merged data set . (1) . (2) . (3) . (4) . Industry output tariffs −0.563** −0.264*** −0.601*** −0.154* (−2.77) (−8.42) (−5.09) (−1.91) Industry input tariffs −2.54** 0.133 −1.46*** −1.45*** (−4.97) (0.93) (−4.08) (−3.53) Industry‐specific fixed effects Yes No Yes No Firm‐specific fixed effects No Yes No Yes Year‐specific fixed effects No Yes No Yes Observations 315,416 315,416 82,570 82,570 Prob. > F 0.000 0.000 0.000 0.000 R2 0.21 0.13 0.34 0.02 Regressand: ln TFPijtOP . Full‐sample data set . Merged data set . (1) . (2) . (3) . (4) . Industry output tariffs −0.563** −0.264*** −0.601*** −0.154* (−2.77) (−8.42) (−5.09) (−1.91) Industry input tariffs −2.54** 0.133 −1.46*** −1.45*** (−4.97) (0.93) (−4.08) (−3.53) Industry‐specific fixed effects Yes No Yes No Firm‐specific fixed effects No Yes No Yes Year‐specific fixed effects No Yes No Yes Observations 315,416 315,416 82,570 82,570 Prob. > F 0.000 0.000 0.000 0.000 R2 0.21 0.13 0.34 0.02 Notes t‐values are in parentheses. Significant at *10%, **5% and ***1%. Regressions in columns (1) and (2) use the entire sample for Chinese firms (2000–6), whereas those in columns (3) and (4) use the matched sample for Chinese trading firms (2000–6). Regressions in columns (1) and (3) are clustered at the two‐digit Chinese industry level. Industry input tariffs are calculated by using the 2002 time‐invariant input–output matrix for China as described in (5) in the text. Regressions in columns (1) and (3) are clustered at the one‐digit industry level. Open in new tab The rest of the regressions reported in Table 7 use the merged data set, which only includes large trading firms. For a close comparison with columns (1) and (2), the estimates in column (3) control for industry‐level fixed effects, whereas those in column (4) control for firm‐specific and year‐specific fixed effects. The coefficients of both industry output tariffs and input tariffs are found to be negative and significant.31 I include the processing indicator (i.e. one if a firm has any processing imports and zero otherwise) in the first three columns of Table 8, given that processing information is available in the merged data set. To check whether the estimation results are sensitive to different TFP measures, column (1) uses TFPOP1 in which the productivities of processing firms and non‐processing firms are estimated using different control functions, whereas column (2) uses TFPOP2 in which productivities of processing firms and non‐processing firms are jointly estimated as the regressand. In addition, columns (1) and (2) abstract from the interaction term between output (input) tariffs and the processing indicator. After controlling for firm‐specific and year‐specific fixed effects, both industry output tariffs and industry input tariffs are negatively correlated with firm productivity. Their coefficients are statistically significant. Meanwhile, the coefficient of the processing indicator is negative and significant, indicating that processing firms have low productivity. However, the Olley–Pakes TFP measure that is used in columns (1) and (2) of Table 8 still suffers from three possible pitfalls. First, the Olley–Pakes approach does not allow output to exhibit any serial correlation, which is likely. Second, it assumes that firms will mostly adjust their capital usage when facing an exogenous shock. However, this may not be the case for China, given that Chinese firms are able to access relatively cheap labour. Finally, there are many missing values for investment in the Chinese firm data, which are essential for computing the Olley–Pakes TFP.32 By way of comparison, the system‐GMM TFP measure is better at overcoming such pitfalls: It has enough flexibility to allow for possible serial autocorrelation and to allow firms to adjust all inputs including not only capital, but also labour and materials. In addition, the computation of system‐GMM TFP no longer relies on investment as a proxy variable. I therefore use the system‐GMM TFP as the main measure of firm productivity from column (3) of Table 8 to the rest estimates in the article. To examine the possibly heterogenous impact of tariff reductions on firm productivity, column (3) of Table 8 includes interaction terms for the processing indicator and industry output and input tariffs. The coefficients of output tariffs and input tariffs themselves and their interaction with the processing indicator are still statistically significant. However, the processing indicator exhibits an erratic sign, although it is insignificant. I suspect this is because the processing indicator is a relatively crude measure of processing activity, which may overestimate the role of processing firms. For example, if a firm has only a very small proportion of processing imports over total imports, it is still classified as a processing firm, yet its primary operation remains in ordinary trade. I then consider a continuous measure of the extent to which a firm is engaged in processing trade to replace the processing indicator in the rest of Table 8; the extent of processing engagement is measured by the firm's total processing imports over total imports each year. Table 8 Preliminary Estimates Tariffs measure: . Industry tariffs . Firm tariffs . Processing measure: . Processing dummy . Extent of processing imports . . lnTFPijtOP1 . lnTFPijtOP2 . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . (6) . Output tariffs −0.161** −0.715*** −1.010*** −1.074*** −1.069*** −0.315*** (−1.98) (−12.53) (−25.17) (−11.20) (−9.92) (−4.61) Output tariffs × processing variable −0.099* −0.614*** −0.604*** −0.234*** (−1.79) (−4.92) (−4.21) (−2.69) Input tariffs −1.468*** −1.332*** −0.656*** −1.667*** −1.379** −0.572*** (−3.57) (−5.19) (−5.13) (−2.90) (−2.26) (−5.37) Input tariffs × processing variable 0.561*** 2.233*** 2.251*** 2.409*** (3.26) (3.56) (3.33) (8.01) Processing variable −0.010* −0.011** 0.001 −0.097*** −0.077*** −0.180*** (−1.76) (−2.53) (0.14) (−6.66) (−4.62) (−17.54) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Firm‐specific fixed effects Yes Yes Yes Yes Yes Yes Pure domestic firms Yes Yes Yes Yes No No Pure exporting firms Yes Yes Yes Yes No No Observations 82,558 82,314 97,299 35,172 24,457 27,679 R2 0.02 0.01 0.03 0.12 0.12 0.09 Tariffs measure: . Industry tariffs . Firm tariffs . Processing measure: . Processing dummy . Extent of processing imports . . lnTFPijtOP1 . lnTFPijtOP2 . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . (6) . Output tariffs −0.161** −0.715*** −1.010*** −1.074*** −1.069*** −0.315*** (−1.98) (−12.53) (−25.17) (−11.20) (−9.92) (−4.61) Output tariffs × processing variable −0.099* −0.614*** −0.604*** −0.234*** (−1.79) (−4.92) (−4.21) (−2.69) Input tariffs −1.468*** −1.332*** −0.656*** −1.667*** −1.379** −0.572*** (−3.57) (−5.19) (−5.13) (−2.90) (−2.26) (−5.37) Input tariffs × processing variable 0.561*** 2.233*** 2.251*** 2.409*** (3.26) (3.56) (3.33) (8.01) Processing variable −0.010* −0.011** 0.001 −0.097*** −0.077*** −0.180*** (−1.76) (−2.53) (0.14) (−6.66) (−4.62) (−17.54) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Firm‐specific fixed effects Yes Yes Yes Yes Yes Yes Pure domestic firms Yes Yes Yes Yes No No Pure exporting firms Yes Yes Yes Yes No No Observations 82,558 82,314 97,299 35,172 24,457 27,679 R2 0.02 0.01 0.03 0.12 0.12 0.09 Notes Robust t‐values are in parentheses. Significant at *10%, **5% and ***1%. Regressions in columns (1)–(5) use industry‐level output tariffs and input tariffs, which are calculated using the 2002 time‐invariant input–output matrix for China as described in (5) in the text. Regressions in column (6) use firm‐specific output tariffs and input tariffs which are computed using the time‐invariant weight in the initial period that the firm first appears in the data set. Columns (1)–(3) use a processing dummy (one if a firm has any processing imports and zero otherwise), whereas columns (4)–(6) use the extent of processing imports as a proxy for the processing variable. Regressands in columns (1)–(2) are Olley–Pakes TFP with different first‐step control functions as introduced in Appendix B, whereas those in columns (3)–(6) are system‐GMM TFP. Open in new tab Table 8 Preliminary Estimates Tariffs measure: . Industry tariffs . Firm tariffs . Processing measure: . Processing dummy . Extent of processing imports . . lnTFPijtOP1 . lnTFPijtOP2 . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . (6) . Output tariffs −0.161** −0.715*** −1.010*** −1.074*** −1.069*** −0.315*** (−1.98) (−12.53) (−25.17) (−11.20) (−9.92) (−4.61) Output tariffs × processing variable −0.099* −0.614*** −0.604*** −0.234*** (−1.79) (−4.92) (−4.21) (−2.69) Input tariffs −1.468*** −1.332*** −0.656*** −1.667*** −1.379** −0.572*** (−3.57) (−5.19) (−5.13) (−2.90) (−2.26) (−5.37) Input tariffs × processing variable 0.561*** 2.233*** 2.251*** 2.409*** (3.26) (3.56) (3.33) (8.01) Processing variable −0.010* −0.011** 0.001 −0.097*** −0.077*** −0.180*** (−1.76) (−2.53) (0.14) (−6.66) (−4.62) (−17.54) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Firm‐specific fixed effects Yes Yes Yes Yes Yes Yes Pure domestic firms Yes Yes Yes Yes No No Pure exporting firms Yes Yes Yes Yes No No Observations 82,558 82,314 97,299 35,172 24,457 27,679 R2 0.02 0.01 0.03 0.12 0.12 0.09 Tariffs measure: . Industry tariffs . Firm tariffs . Processing measure: . Processing dummy . Extent of processing imports . . lnTFPijtOP1 . lnTFPijtOP2 . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . (6) . Output tariffs −0.161** −0.715*** −1.010*** −1.074*** −1.069*** −0.315*** (−1.98) (−12.53) (−25.17) (−11.20) (−9.92) (−4.61) Output tariffs × processing variable −0.099* −0.614*** −0.604*** −0.234*** (−1.79) (−4.92) (−4.21) (−2.69) Input tariffs −1.468*** −1.332*** −0.656*** −1.667*** −1.379** −0.572*** (−3.57) (−5.19) (−5.13) (−2.90) (−2.26) (−5.37) Input tariffs × processing variable 0.561*** 2.233*** 2.251*** 2.409*** (3.26) (3.56) (3.33) (8.01) Processing variable −0.010* −0.011** 0.001 −0.097*** −0.077*** −0.180*** (−1.76) (−2.53) (0.14) (−6.66) (−4.62) (−17.54) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Firm‐specific fixed effects Yes Yes Yes Yes Yes Yes Pure domestic firms Yes Yes Yes Yes No No Pure exporting firms Yes Yes Yes Yes No No Observations 82,558 82,314 97,299 35,172 24,457 27,679 R2 0.02 0.01 0.03 0.12 0.12 0.09 Notes Robust t‐values are in parentheses. Significant at *10%, **5% and ***1%. Regressions in columns (1)–(5) use industry‐level output tariffs and input tariffs, which are calculated using the 2002 time‐invariant input–output matrix for China as described in (5) in the text. Regressions in column (6) use firm‐specific output tariffs and input tariffs which are computed using the time‐invariant weight in the initial period that the firm first appears in the data set. Columns (1)–(3) use a processing dummy (one if a firm has any processing imports and zero otherwise), whereas columns (4)–(6) use the extent of processing imports as a proxy for the processing variable. Regressands in columns (1)–(2) are Olley–Pakes TFP with different first‐step control functions as introduced in Appendix B, whereas those in columns (3)–(6) are system‐GMM TFP. Open in new tab Column (4) of Table 8 gives the results of a regression of system‐GMM firm TFP on industry‐level input and output tariffs. The coefficients of the output and input tariffs are still negative and statistically significant. The variable for the extent of processing imports turns out to be negative and significant. As one of the novel measures of the present article is firm‐specific output and input tariffs, I now turn to compare the estimation results using industry‐level tariffs and firm‐level tariffs. Because firm‐specific output tariffs, as introduced in (4), cannot apply to pure domestic firms or pure exporting firms, I drop such firms in column (5) with measures of industry‐level output and input tariffs and in column (6) with measures of firm‐specific output and input tariffs for comparison. The coefficients of output (input) tariffs in columns (5) and (6) are all negative and statistically significant. In terms of economic magnitudes, the differences in the coefficients of output (input) tariffs between the two columns are sizable. When moving from the industry‐level measure of output tariffs in column (5) to the firm‐specific measure of input tariffs in column (6), the coefficient is reduced from −1.07 to −0.32. Likewise, the point estimate of the input tariffs is reduced more than half moving from the measure of industrial input tariffs to the measure of firm‐specific input tariffs. Such sizable differences indicate the pitfalls of using industry‐level measures of tariffs. First, output tariff reductions for some products in an industry are not directly relevant to a firm in the same industry if the firm never produces such products. Thus, the pro‐competitive effects would be overestimated if output tariffs were measured at the industry level. By the same token, the cost‐saving effects of cutting input tariffs are also overstated with the industry measure of input tariffs. Second, compared with output tariffs, the estimation bias for input tariffs could be more severe as the industry measure of input tariffs is also contaminated by the use of an input–output matrix, which also mixed up both imported intermediate inputs and domestic intermediate inputs that are not directly relevant to the cut in tariffs. Finally, ignorance of the ‘free‐duty’ phenomenon for processing imports generates an additional measurement error in industrial input tariffs for Chinese firms. To avoid such possible estimation bias, I use a firm‐specific measure of tariffs in the rest of the article. 4.2. Self‐selection to Processing Columns (4) and (5) of Table 8 use the extent of processing imports and its interaction with output and input tariffs, but the processing imports variable is endogenous. As shown in column (1) of Table 8, processing firms are associated with low productivity. Thus, it is interesting to compare the TFP trajectories of processing firms with those of non‐processing firms. As shown in the last column of Table 9, processing firms, overall, are less productive than non‐processing firms. Interestingly, the productivity difference between processing and non‐processing firms roughly decreases over the years, suggesting that a catching‐up process of processing firms may take place.33 Such comparisons are straightforward. However, they bear a cost because processing firms may be very different from non‐processing firms in terms of size. To overcome such a pitfall, as suggested by Imbens (2004), I perform the nearest‐neighbour matching between the treatment group (i.e. processing firms) and the control group (i.e. non‐processing firms) by choosing the number of firm employees and firm sales as covariates. Each processing firm would find its most similar non‐processing firm. Table 9 reports both the estimates for average treatment for the treated (ATT) and average treatment for the control (ATC). For instance, the coefficient of ATT for all processing firms is 0.037 and highly statistically significant, suggesting that, overall, productivity for processing firms is lower than that for similar non‐processing firms. Table 9 TFP Trajectories of Processing Versus Non‐processing Firms by Year Firm productivity (⁠ lnTFPijtGMM ⁠) . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Overall . Non‐processing firms 2.458 2.465 2.518 2.544 2.585 2.625 2.576 Processing firms 2.416 2.432 2.462 2.539 2.575 2.629 2.551 Difference 0.042*** 0.033*** 0.056*** 0.005 0.010* −0.003 0.025*** (2.90) (2.57) (4.98) (0.64) (1.74) (−0.58) (7.63) Comparisons using nearest‐neighbour matching Average treatment on the treated 0.040*** 0.032*** 0.014 0.034*** 0.032*** 0.051*** 0.031*** (3.64) (3.08) (1.30) (5.08) (5.88) (9.24) (10.13) Average treatment on the control 0.031*** 0.018*** 0.004 0.037*** 0.027*** 0.041*** 0.027*** (2.60) (2.18) (0.46) (4.92) (5.57) (7.86) (9.60) Firm productivity (⁠ lnTFPijtGMM ⁠) . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Overall . Non‐processing firms 2.458 2.465 2.518 2.544 2.585 2.625 2.576 Processing firms 2.416 2.432 2.462 2.539 2.575 2.629 2.551 Difference 0.042*** 0.033*** 0.056*** 0.005 0.010* −0.003 0.025*** (2.90) (2.57) (4.98) (0.64) (1.74) (−0.58) (7.63) Comparisons using nearest‐neighbour matching Average treatment on the treated 0.040*** 0.032*** 0.014 0.034*** 0.032*** 0.051*** 0.031*** (3.64) (3.08) (1.30) (5.08) (5.88) (9.24) (10.13) Average treatment on the control 0.031*** 0.018*** 0.004 0.037*** 0.027*** 0.041*** 0.027*** (2.60) (2.18) (0.46) (4.92) (5.57) (7.86) (9.60) Notes t‐values corrected for clustering at the firm level are in parentheses. Significant at *10%, **5% and ***1%. Estimates for both average treatment on the treated (i.e. processing firms) and average treatment on the control (i.e. non‐processing firms) are obtained by using the nearest‐neighbour matching approach in which firm size and firm sales are chosen as covariates. Open in new tab Table 9 TFP Trajectories of Processing Versus Non‐processing Firms by Year Firm productivity (⁠ lnTFPijtGMM ⁠) . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Overall . Non‐processing firms 2.458 2.465 2.518 2.544 2.585 2.625 2.576 Processing firms 2.416 2.432 2.462 2.539 2.575 2.629 2.551 Difference 0.042*** 0.033*** 0.056*** 0.005 0.010* −0.003 0.025*** (2.90) (2.57) (4.98) (0.64) (1.74) (−0.58) (7.63) Comparisons using nearest‐neighbour matching Average treatment on the treated 0.040*** 0.032*** 0.014 0.034*** 0.032*** 0.051*** 0.031*** (3.64) (3.08) (1.30) (5.08) (5.88) (9.24) (10.13) Average treatment on the control 0.031*** 0.018*** 0.004 0.037*** 0.027*** 0.041*** 0.027*** (2.60) (2.18) (0.46) (4.92) (5.57) (7.86) (9.60) Firm productivity (⁠ lnTFPijtGMM ⁠) . 2001 . 2002 . 2003 . 2004 . 2005 . 2006 . Overall . Non‐processing firms 2.458 2.465 2.518 2.544 2.585 2.625 2.576 Processing firms 2.416 2.432 2.462 2.539 2.575 2.629 2.551 Difference 0.042*** 0.033*** 0.056*** 0.005 0.010* −0.003 0.025*** (2.90) (2.57) (4.98) (0.64) (1.74) (−0.58) (7.63) Comparisons using nearest‐neighbour matching Average treatment on the treated 0.040*** 0.032*** 0.014 0.034*** 0.032*** 0.051*** 0.031*** (3.64) (3.08) (1.30) (5.08) (5.88) (9.24) (10.13) Average treatment on the control 0.031*** 0.018*** 0.004 0.037*** 0.027*** 0.041*** 0.027*** (2.60) (2.18) (0.46) (4.92) (5.57) (7.86) (9.60) Notes t‐values corrected for clustering at the firm level are in parentheses. Significant at *10%, **5% and ***1%. Estimates for both average treatment on the treated (i.e. processing firms) and average treatment on the control (i.e. non‐processing firms) are obtained by using the nearest‐neighbour matching approach in which firm size and firm sales are chosen as covariates. Open in new tab The estimates in Table 9 hint that low‐productivity firms may self‐select to engage in processing trade. To control for this, I introduce a type‐2 Tobit model or, equivalently, a bivariate sample selection model (Cameron and Trivedi, 2005). The type‐2 Tobit specification includes: (i) a processing participation equation, Processingit=0ifVit<01ifVit≥0,(10) where Vit denotes a latent variable faced by firm i; and (ii) an ‘outcome’ equation whereby the firm's extent of processing imports is modelled as a linear function of other variables. In particular, I estimate the following selection equation using a probit model: Pr(Processingit=1)=Pr(Vit≥0)=Φ(α0+α1lnTFPit−1+α2SOEit−1+α3FIEit−1+α4lnLit−1+α5Tenureit−1+λj+ςt),(11) where Φ(.) is the cumulative density function of the normal distribution. In addition to the logarithm of the firm's TFP, a firm's decision to engage in processing trade is also affected by other factors, such as its ownership (whether it is an SOE or a multinational firm) and size (measured by the logarithm of the number of employees). Note that the bivariate sample selection estimation require an excluded variable that affects the firm's processing decision but does not appear in the extent of processing equation (Cameron and Trivedi, 2005). Here the firm's age (Tenureit−1) serves this purpose, as previous studies have found that a firm's export probability is higher for older firms (Amiti and Davis, 2011). By contrast, my sample also reveals that the simple correlation between a firm's extent of processing imports and the firm's age is close to nil (−0.04), suggesting that the firm's age can be excluded in the second‐step Heckman estimates.34 All regressors in the type‐2 Tobit selection model are of a one‐period lag as it usually takes time for such factors to affect a firm's processing choice. Finally, I include the three‐digit CIC industrial dummies, λj, and year dummies, ςt, to control for other unspecified factors. Table 10 The Heckman Two‐step Estimates of Bivariate Selection Model Heckman two‐step: . 1st step . 2nd step . Regressand: . Processing indicator . Extent of processing . One‐period lag of log TFP (⁠ lnTFPijtGMM −0.126*** (−7.23) −0.176*** (−15.17) One‐period lag of log Labour 0.152*** (25.55) 0.031*** (3.23) One‐period lag of SOEs indicator −0.160*** (−2.82) −0.039 (−1.47) One‐period lag of foreign indicator 0.978*** (68.97) 0.299*** (5.05) One‐period lag of firm tenure 0.004*** (5.02) – Inverse Mills ratio – 0.172** (2.10) Year‐specific fixed effects Yes Yes Industry‐specific fixed effects Yes Yes Observations 58,629 21,232 Heckman two‐step: . 1st step . 2nd step . Regressand: . Processing indicator . Extent of processing . One‐period lag of log TFP (⁠ lnTFPijtGMM −0.126*** (−7.23) −0.176*** (−15.17) One‐period lag of log Labour 0.152*** (25.55) 0.031*** (3.23) One‐period lag of SOEs indicator −0.160*** (−2.82) −0.039 (−1.47) One‐period lag of foreign indicator 0.978*** (68.97) 0.299*** (5.05) One‐period lag of firm tenure 0.004*** (5.02) – Inverse Mills ratio – 0.172** (2.10) Year‐specific fixed effects Yes Yes Industry‐specific fixed effects Yes Yes Observations 58,629 21,232 Notes t‐values corrected for clustering at the firm level are in parentheses. Significant at *10%, **5% and ***1%. The sample selection model is presented in (10) and (11) in the text. The regressand in the first‐step is the firm's processing dummy, whereas that in the second step is the firm's extent of processing imports. Firm‐level system‐GMM TFP is adopted as a measure of firm productivity. Firm tenure is used as an exclusion variable that appeared in the first step but not the second step. The three‐digit Chinese industry‐specific fixed effects and year‐specific fixed effects are also included in the estimation. Open in new tab Table 10 The Heckman Two‐step Estimates of Bivariate Selection Model Heckman two‐step: . 1st step . 2nd step . Regressand: . Processing indicator . Extent of processing . One‐period lag of log TFP (⁠ lnTFPijtGMM −0.126*** (−7.23) −0.176*** (−15.17) One‐period lag of log Labour 0.152*** (25.55) 0.031*** (3.23) One‐period lag of SOEs indicator −0.160*** (−2.82) −0.039 (−1.47) One‐period lag of foreign indicator 0.978*** (68.97) 0.299*** (5.05) One‐period lag of firm tenure 0.004*** (5.02) – Inverse Mills ratio – 0.172** (2.10) Year‐specific fixed effects Yes Yes Industry‐specific fixed effects Yes Yes Observations 58,629 21,232 Heckman two‐step: . 1st step . 2nd step . Regressand: . Processing indicator . Extent of processing . One‐period lag of log TFP (⁠ lnTFPijtGMM −0.126*** (−7.23) −0.176*** (−15.17) One‐period lag of log Labour 0.152*** (25.55) 0.031*** (3.23) One‐period lag of SOEs indicator −0.160*** (−2.82) −0.039 (−1.47) One‐period lag of foreign indicator 0.978*** (68.97) 0.299*** (5.05) One‐period lag of firm tenure 0.004*** (5.02) – Inverse Mills ratio – 0.172** (2.10) Year‐specific fixed effects Yes Yes Industry‐specific fixed effects Yes Yes Observations 58,629 21,232 Notes t‐values corrected for clustering at the firm level are in parentheses. Significant at *10%, **5% and ***1%. The sample selection model is presented in (10) and (11) in the text. The regressand in the first‐step is the firm's processing dummy, whereas that in the second step is the firm's extent of processing imports. Firm‐level system‐GMM TFP is adopted as a measure of firm productivity. Firm tenure is used as an exclusion variable that appeared in the first step but not the second step. The three‐digit Chinese industry‐specific fixed effects and year‐specific fixed effects are also included in the estimation. Open in new tab Table 10 reports the estimation results for the type‐2 Tobit selection model. From the first‐step probit estimates (11), low‐productivity firms are more likely to engage in processing trade. Similarly, large and foreign firms are more likely to engage in processing trade. However, SOEs are less likely to become processing firms. Finally, as predicted, firms that were established earlier are more likely to engage in processing trade. I then include the computed inverse Mills ratio obtained in the first‐step probit estimates in the second‐step Heckman estimation as an additional regressor. It turns out that the estimated coefficients have exactly identical signs as obtained in the first‐step estimates. Thus, after controlling for the endogenous selection of processing, I obtain the fitted value of the firm's extent of processing, which is used to replace the firm's actual extent of processing in the rest of estimates, as discussed above. 4.3. Endogeneity Issues The specifications in Tables 7 and 8 face three possible endogeneity problems. The first one relates to the measure of firm input tariffs, because imports and tariffs are strongly correlated. This problem is essentially solved by using measures of tariffs based on time‐invariant weights. The second relates to the possible reverse causality between firm productivity and exports. As the firm's productivity improves, its exports may grow faster for some products than for others. The disproportional growth in exports of some products would challenge the validity of a time‐variant measure of firm output tariffs. To avoid this possibility, measures of tariffs based on time‐invariant weights, as in (4), have been used in all specifications. However, there is still another possible reverse causality problem. Although tariff reductions are regulated by the GATT/WTO agreements, they are still, to some extent, endogenous because firms in low‐productivity sectors would lobby the government for protection (Grossman and Helpman, 1994), that is, to maintain related internationally negotiated tariffs at a relatively high level. I control for such reverse causality by using an IV approach. Identifying a good instrument for tariffs is challenging. Inspired by Amiti and Konings (2007), here I construct a one‐year lag of firm‐specific output tariffs and input tariffs as instruments.35 The economic rationale is as follows. The government generally has difficulty in removing the high protection status quo from an industry with high tariffs, possibly because of domestic pressure from special interest groups. Hence, compared with other sectors, industries with high tariffs one year ago would still be expected to have relatively high tariffs at present. Column (1) of Table 11 presents 2SLS fixed‐effects estimates using the previous tariffs with time‐invariant weights as instruments.36 After controlling for reverse causality, reductions in both firm input tariffs and firm output tariffs lead to firm productivity growth. As noted before, the measure of firm output tariffs may suffer from a pitfall because of the assumption of equal shares between domestic sales and exports for each product produced, as the product composition of exports may be different from that of domestic sales by the sector integration of GSCs and by the intensity of the sectors in processing firms. To address this concern, besides dropping pure domestic firms and pure exporters from the sample, I run two sets of auxiliary regressions. First, all industries are classified into two groups (more integrated and less integrated) according to their ‘production depth’ of engaging (GSCs) which is measured by the value‐added ratio to gross industrial output (OECD, 2010). By taking the mean of such ratios across two‐digit level industries as a cut‐off, columns (2) and (3) regress the impact of tariff reductions on firm productivity by the extent of GSCs integrating. Second, columns (4) and (5) run regressions for sectors with high (low) intensity of the sectors in processing firms, respectively, in which the intensity is measured by share of number of processing firms over number of total firms in each industry and the mean of the ratios across industries is taken as the cut‐off. In all cases, the coefficients of output and input tariffs are significant and in line with my previous findings. Table 11 IV Estimates with Measure of System‐GMM TFP Regressand: lnTFPijtGMM . All sample . GSCs integrated . Processing intensity . . Less . More . Low . High . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.319*** −0.825*** −1.962*** −1.657** −1.941*** (−4.60) (−2.13) (−3.66) (−3.98) (−4.47) Firm output tariffs × fitted extent of processing 0.817* 0.802 1.184 1.321 1.765*** (1.72) (1.18) (1.41) (1.53) (2.67) Firm input tariffs −1.712*** −2.821*** −1.519*** −1.883** −3.447** (−3.46) (−3.57) (−2.76) (−3.50) (−2.32) Firm input tariffs × fitted extent of processing 2.460*** 2.497* 2.818** 3.478** 3.546* (2.54) (1.75) (2.71) (2.65) (1.72) Fitted extent of processing −0.740*** −1.005*** −0.778*** −0.944*** −0.833*** (−17.66) (−15.99) (−10.28) (−12.28) (−11.95) Kleibergen–Paap rank LM χ2 statistic 87.75† 428.5† 961.9† 883.6† 639.1† Kleibergen–Paap rank Wald F statistic 95.94† 112.0† 257.1† 234.2† 171.3† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 22,812 8,374 14,438 13,633 9,179 R2 0.17 0.18 0.16 0.16 0.23 First‐stage regressions IV1: Firm output tariffs with a lag 0.004*** 0.005*** 0.003*** 0.003*** 0.004*** (12.03) (9.91) (9.38) (8.40) (4.19) IV2: Firm output tariffs with a lag × fitted extent of processing 0.004*** 0.004*** 0.004*** 0.005*** 0.004*** (19.15) (12.67) (5.92) (11.72) (7.69) IV3: Firm input tariffs with a lag 0.005*** 0.004*** 0.005*** 0.005*** 0.005*** (8.89) (19.62) (4.22) (7.95) (3.82) IV4: Firm input tariffs with a lag × fitted extent of processing 0.008*** 0.008*** 0.008*** 0.007*** 0.010*** (14.31) (9.02) (7.85) (10.33) (9.01) Regressand: lnTFPijtGMM . All sample . GSCs integrated . Processing intensity . . Less . More . Low . High . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.319*** −0.825*** −1.962*** −1.657** −1.941*** (−4.60) (−2.13) (−3.66) (−3.98) (−4.47) Firm output tariffs × fitted extent of processing 0.817* 0.802 1.184 1.321 1.765*** (1.72) (1.18) (1.41) (1.53) (2.67) Firm input tariffs −1.712*** −2.821*** −1.519*** −1.883** −3.447** (−3.46) (−3.57) (−2.76) (−3.50) (−2.32) Firm input tariffs × fitted extent of processing 2.460*** 2.497* 2.818** 3.478** 3.546* (2.54) (1.75) (2.71) (2.65) (1.72) Fitted extent of processing −0.740*** −1.005*** −0.778*** −0.944*** −0.833*** (−17.66) (−15.99) (−10.28) (−12.28) (−11.95) Kleibergen–Paap rank LM χ2 statistic 87.75† 428.5† 961.9† 883.6† 639.1† Kleibergen–Paap rank Wald F statistic 95.94† 112.0† 257.1† 234.2† 171.3† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 22,812 8,374 14,438 13,633 9,179 R2 0.17 0.18 0.16 0.16 0.23 First‐stage regressions IV1: Firm output tariffs with a lag 0.004*** 0.005*** 0.003*** 0.003*** 0.004*** (12.03) (9.91) (9.38) (8.40) (4.19) IV2: Firm output tariffs with a lag × fitted extent of processing 0.004*** 0.004*** 0.004*** 0.005*** 0.004*** (19.15) (12.67) (5.92) (11.72) (7.69) IV3: Firm input tariffs with a lag 0.005*** 0.004*** 0.005*** 0.005*** 0.005*** (8.89) (19.62) (4.22) (7.95) (3.82) IV4: Firm input tariffs with a lag × fitted extent of processing 0.008*** 0.008*** 0.008*** 0.007*** 0.010*** (14.31) (9.02) (7.85) (10.33) (9.01) Notes t‐values in parentheses are obtained using bootstrapped standard errors. Significant at *10%, **5% and ***1%. Column (1) includes the entire sample in the regression. Columns (2) and (3) include sectors that are less (more) integrated in global supply chains (GSCs), respectively, using the industrial average ratio of value‐added to gross industrial output as cut‐offs. Columns (4) and (5) include the sectors with low (high) intensity of the sectors in processing firms, respectively, in which the intensity is measured by share of number of processing firms over number of total firms in each industry. †(‡) indicates significance of the p‐value at the 1 (5)% level. In the first‐stage regressions, IV1 reports the coefficient of the firm output tariffs with initial time‐invariant weight and one‐period lag of tariffs, using firm output tariffs with initial time‐invariant weight and current tariffs as the regressand. IV2 reports the coefficient of the interaction between fitted extent of processing obtained from the second‐step Heckman estimates in Table 8 and firm output tariffs with initial time‐invariant weight and one‐period lag of tariffs, using the interaction between fitted extent to processing and current tariffs as the regressand. Similarly, IV3 reports the coefficient of the firm input tariffs with initial time‐invariant weight and one‐period lag of tariffs using firm input tariffs with initial time‐invariant weight and current tariffs as the regressand. IV4 reports the coefficient of the interaction between fitted extent of processing and firm input tariffs with initial time‐invariant weight and one‐period lag of tariffs, using the interaction between fitted extent of processing and firm output tariffs with initial time‐invariant weight and current tariffs as the regressand. Pure domestic firms and pure exporters are dropped from the sample in all estimates. Open in new tab Table 11 IV Estimates with Measure of System‐GMM TFP Regressand: lnTFPijtGMM . All sample . GSCs integrated . Processing intensity . . Less . More . Low . High . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.319*** −0.825*** −1.962*** −1.657** −1.941*** (−4.60) (−2.13) (−3.66) (−3.98) (−4.47) Firm output tariffs × fitted extent of processing 0.817* 0.802 1.184 1.321 1.765*** (1.72) (1.18) (1.41) (1.53) (2.67) Firm input tariffs −1.712*** −2.821*** −1.519*** −1.883** −3.447** (−3.46) (−3.57) (−2.76) (−3.50) (−2.32) Firm input tariffs × fitted extent of processing 2.460*** 2.497* 2.818** 3.478** 3.546* (2.54) (1.75) (2.71) (2.65) (1.72) Fitted extent of processing −0.740*** −1.005*** −0.778*** −0.944*** −0.833*** (−17.66) (−15.99) (−10.28) (−12.28) (−11.95) Kleibergen–Paap rank LM χ2 statistic 87.75† 428.5† 961.9† 883.6† 639.1† Kleibergen–Paap rank Wald F statistic 95.94† 112.0† 257.1† 234.2† 171.3† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 22,812 8,374 14,438 13,633 9,179 R2 0.17 0.18 0.16 0.16 0.23 First‐stage regressions IV1: Firm output tariffs with a lag 0.004*** 0.005*** 0.003*** 0.003*** 0.004*** (12.03) (9.91) (9.38) (8.40) (4.19) IV2: Firm output tariffs with a lag × fitted extent of processing 0.004*** 0.004*** 0.004*** 0.005*** 0.004*** (19.15) (12.67) (5.92) (11.72) (7.69) IV3: Firm input tariffs with a lag 0.005*** 0.004*** 0.005*** 0.005*** 0.005*** (8.89) (19.62) (4.22) (7.95) (3.82) IV4: Firm input tariffs with a lag × fitted extent of processing 0.008*** 0.008*** 0.008*** 0.007*** 0.010*** (14.31) (9.02) (7.85) (10.33) (9.01) Regressand: lnTFPijtGMM . All sample . GSCs integrated . Processing intensity . . Less . More . Low . High . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.319*** −0.825*** −1.962*** −1.657** −1.941*** (−4.60) (−2.13) (−3.66) (−3.98) (−4.47) Firm output tariffs × fitted extent of processing 0.817* 0.802 1.184 1.321 1.765*** (1.72) (1.18) (1.41) (1.53) (2.67) Firm input tariffs −1.712*** −2.821*** −1.519*** −1.883** −3.447** (−3.46) (−3.57) (−2.76) (−3.50) (−2.32) Firm input tariffs × fitted extent of processing 2.460*** 2.497* 2.818** 3.478** 3.546* (2.54) (1.75) (2.71) (2.65) (1.72) Fitted extent of processing −0.740*** −1.005*** −0.778*** −0.944*** −0.833*** (−17.66) (−15.99) (−10.28) (−12.28) (−11.95) Kleibergen–Paap rank LM χ2 statistic 87.75† 428.5† 961.9† 883.6† 639.1† Kleibergen–Paap rank Wald F statistic 95.94† 112.0† 257.1† 234.2† 171.3† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 22,812 8,374 14,438 13,633 9,179 R2 0.17 0.18 0.16 0.16 0.23 First‐stage regressions IV1: Firm output tariffs with a lag 0.004*** 0.005*** 0.003*** 0.003*** 0.004*** (12.03) (9.91) (9.38) (8.40) (4.19) IV2: Firm output tariffs with a lag × fitted extent of processing 0.004*** 0.004*** 0.004*** 0.005*** 0.004*** (19.15) (12.67) (5.92) (11.72) (7.69) IV3: Firm input tariffs with a lag 0.005*** 0.004*** 0.005*** 0.005*** 0.005*** (8.89) (19.62) (4.22) (7.95) (3.82) IV4: Firm input tariffs with a lag × fitted extent of processing 0.008*** 0.008*** 0.008*** 0.007*** 0.010*** (14.31) (9.02) (7.85) (10.33) (9.01) Notes t‐values in parentheses are obtained using bootstrapped standard errors. Significant at *10%, **5% and ***1%. Column (1) includes the entire sample in the regression. Columns (2) and (3) include sectors that are less (more) integrated in global supply chains (GSCs), respectively, using the industrial average ratio of value‐added to gross industrial output as cut‐offs. Columns (4) and (5) include the sectors with low (high) intensity of the sectors in processing firms, respectively, in which the intensity is measured by share of number of processing firms over number of total firms in each industry. †(‡) indicates significance of the p‐value at the 1 (5)% level. In the first‐stage regressions, IV1 reports the coefficient of the firm output tariffs with initial time‐invariant weight and one‐period lag of tariffs, using firm output tariffs with initial time‐invariant weight and current tariffs as the regressand. IV2 reports the coefficient of the interaction between fitted extent of processing obtained from the second‐step Heckman estimates in Table 8 and firm output tariffs with initial time‐invariant weight and one‐period lag of tariffs, using the interaction between fitted extent to processing and current tariffs as the regressand. Similarly, IV3 reports the coefficient of the firm input tariffs with initial time‐invariant weight and one‐period lag of tariffs using firm input tariffs with initial time‐invariant weight and current tariffs as the regressand. IV4 reports the coefficient of the interaction between fitted extent of processing and firm input tariffs with initial time‐invariant weight and one‐period lag of tariffs, using the interaction between fitted extent of processing and firm output tariffs with initial time‐invariant weight and current tariffs as the regressand. Pure domestic firms and pure exporters are dropped from the sample in all estimates. Open in new tab Several tests were performed to verify the quality of the instruments. First, I use the Kleibergen–Paap LM χ2 statistic to check whether the excluded instruments are correlated with the endogenous regressors. As shown in Table 11, the null hypothesis that the model is under‐identified is rejected at the 1% significance level. Second, the Kleibergen–Paap (2006) F‐statistics provide strong evidence for rejecting the null hypothesis that the first stage is weakly identified at a highly significant level.37 Finally, the first‐stage estimates reported in the lower module of Table 11 offer strong evidence to justify such instruments. In particular, all the t‐values of the instruments are significant. Finally, standard errors are corrected for the use of the estimated regressors by bootstrapping.38 4.4. Further Robustness Checks of 2SLS Estimates It is also worthwhile checking whether the effects of firm‐level input and output tariffs on firm productivity pick up only the role of firm size, given that large firms usually have high productivity, or whether the effects are sensitive to the inclusion of the firm's type of ownership. I therefore include an SOE indicator, a foreign indicator, and the log of labour (i.e. a measure of firm size) in all the 2SLS estimates in Table 12. Because measured TFP may also pick up the difference in prices and price‐cost mark‐ups across firms, column (1) of Table 12 performs the 2SLS estimates using the logarithm of the firm's labour productivity as the regressand. As the log of firm labour is already used as the denominator of the regressand, it is no longer appropriate to include it as a control variable for firm size in the regression. I instead use the log of the firm's capital‐labour ratio as a proxy. To check further whether my main findings are sensitive to the measure of firm TFP and the empirical specifications, column (2) also uses the Levinsohn–Petrin (2003) TFP as the regressand while controlling for other variables as in column (1). Column (3) still uses the system‐GMM as the regressand but includes the above‐mentioned controlling variables. Overall, the main findings of the estimates in these columns are highly consistent with those in Table 11: the impact of input tariff reductions on productivity improvement, overall, is weaker than that of output tariff reductions. The firm's gains from tariff reductions are diminishing as the firm's processing imports share increases. Thus far, the effect of China's import tariff reductions on firm efficiency has been carefully investigated. However, although China has substantially reduced its import tariffs in the new century, Chinese exporters have also enjoyed large tariff reductions in their export destinations. Access to large foreign markets could possibly create incentives for productivity upgrading, especially if such investments require substantial fixed costs. Thus, controlling for tariff reductions in China's export destinations is also worthwhile to obtain a precise estimate of the effect of import tariff reductions on firm TFP. Table 12 More Robust IV Estimates . ln LPijt . lnTFPijtLevP . lnTFPijtGMM . Weighted lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.980*** −1.217** −1.100*** −1.096*** −1.159*** (−3.49) (−2.02) (−4.51) (−4.62) (−4.47) Firm output tariffs × fitted extent of processing 2.260** −0.106 0.677 0.675 0.812** (2.03) (−0.08) (1.63) (1.47) (1.96) Firm input tariffs −3.866** −5.069*** −1.380*** −1.378*** −1.589*** (−2.30) (−2.69) (−2.66) (−2.47) (−2.57) Firm input tariffs × fitted extent of processing 8.610*** 10.309*** 2.448** 2.435** 2.664** (2.36) (2.59) (2.12) (2.09) (2.06) Fitted extent of processing −2.737*** −2.901*** −1.251*** −1.251*** −1.311*** (−22.42) (−23.00) (−26.78) (−23.61) (−27.83) SOEs indicator −0.619*** −0.369*** −0.187*** −0.187*** −0.188*** (−11.60) (−5.15) (−7.71) (−7.51) (−7.81) Foreign ownership indicator 0.493*** 0.475*** 0.220*** 0.220*** 0.229*** (19.38) (24.15) (27.24) (32.40) (28.84) Firm size 0.325*** 0.559*** 0.068*** 0.068*** 0.072*** (51.51) (81.26) (34.23) (29.81) (24.59) Firm external tariffs 0.001 0.001 (1.09) (1.22) Kleibergen–Paap rank LM χ2 statistic 106.5† 92.00† 105.4† 105.4† 105.5† Kleibergen–Paap rank Wald F statistic 54.98† 47.78† 55.18† 55.10† 55.10† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 19,296 15,759 19,283 19,283 19,283 R2 0.40 0.53 0.30 0.30 0.65 . ln LPijt . lnTFPijtLevP . lnTFPijtGMM . Weighted lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.980*** −1.217** −1.100*** −1.096*** −1.159*** (−3.49) (−2.02) (−4.51) (−4.62) (−4.47) Firm output tariffs × fitted extent of processing 2.260** −0.106 0.677 0.675 0.812** (2.03) (−0.08) (1.63) (1.47) (1.96) Firm input tariffs −3.866** −5.069*** −1.380*** −1.378*** −1.589*** (−2.30) (−2.69) (−2.66) (−2.47) (−2.57) Firm input tariffs × fitted extent of processing 8.610*** 10.309*** 2.448** 2.435** 2.664** (2.36) (2.59) (2.12) (2.09) (2.06) Fitted extent of processing −2.737*** −2.901*** −1.251*** −1.251*** −1.311*** (−22.42) (−23.00) (−26.78) (−23.61) (−27.83) SOEs indicator −0.619*** −0.369*** −0.187*** −0.187*** −0.188*** (−11.60) (−5.15) (−7.71) (−7.51) (−7.81) Foreign ownership indicator 0.493*** 0.475*** 0.220*** 0.220*** 0.229*** (19.38) (24.15) (27.24) (32.40) (28.84) Firm size 0.325*** 0.559*** 0.068*** 0.068*** 0.072*** (51.51) (81.26) (34.23) (29.81) (24.59) Firm external tariffs 0.001 0.001 (1.09) (1.22) Kleibergen–Paap rank LM χ2 statistic 106.5† 92.00† 105.4† 105.4† 105.5† Kleibergen–Paap rank Wald F statistic 54.98† 47.78† 55.18† 55.10† 55.10† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 19,296 15,759 19,283 19,283 19,283 R2 0.40 0.53 0.30 0.30 0.65 Notes t‐values in parentheses are obtained using bootstrapped standard errors. Significant at *10%, **5% and ***1%. †indicates significance of p‐value at the 1% level. The regressand is log of value‐added labour productivity (ln LPijt) in column (1) and Levinsohn–Petrin (2003) TFP (⁠ lnTFPijtLevP ⁠) in column (2), and conventional measure of system‐GMM TFP (⁠ lnTFPijtGMM ⁠) in columns (3) and (4). The regressand in column (5) is weighted system‐GMM TFP which is calculated by multiplying lnTFPijtGMM with their relative standard deviations across firms within an industry at the two‐digit level. In all IV estimates, I control for year‐specific fixed effects and time‐invariant two‐digit level Chinese industry fixed‐effects. Firm size in columns (2)–(5) is proxied by log of firm labour, whereas in column (1) it is proxied by firm's capital‐labour ratio. All instruments used are the same as those in Table 9. Pure domestic firms and pure exporters are dropped from the sample. Open in new tab Table 12 More Robust IV Estimates . ln LPijt . lnTFPijtLevP . lnTFPijtGMM . Weighted lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.980*** −1.217** −1.100*** −1.096*** −1.159*** (−3.49) (−2.02) (−4.51) (−4.62) (−4.47) Firm output tariffs × fitted extent of processing 2.260** −0.106 0.677 0.675 0.812** (2.03) (−0.08) (1.63) (1.47) (1.96) Firm input tariffs −3.866** −5.069*** −1.380*** −1.378*** −1.589*** (−2.30) (−2.69) (−2.66) (−2.47) (−2.57) Firm input tariffs × fitted extent of processing 8.610*** 10.309*** 2.448** 2.435** 2.664** (2.36) (2.59) (2.12) (2.09) (2.06) Fitted extent of processing −2.737*** −2.901*** −1.251*** −1.251*** −1.311*** (−22.42) (−23.00) (−26.78) (−23.61) (−27.83) SOEs indicator −0.619*** −0.369*** −0.187*** −0.187*** −0.188*** (−11.60) (−5.15) (−7.71) (−7.51) (−7.81) Foreign ownership indicator 0.493*** 0.475*** 0.220*** 0.220*** 0.229*** (19.38) (24.15) (27.24) (32.40) (28.84) Firm size 0.325*** 0.559*** 0.068*** 0.068*** 0.072*** (51.51) (81.26) (34.23) (29.81) (24.59) Firm external tariffs 0.001 0.001 (1.09) (1.22) Kleibergen–Paap rank LM χ2 statistic 106.5† 92.00† 105.4† 105.4† 105.5† Kleibergen–Paap rank Wald F statistic 54.98† 47.78† 55.18† 55.10† 55.10† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 19,296 15,759 19,283 19,283 19,283 R2 0.40 0.53 0.30 0.30 0.65 . ln LPijt . lnTFPijtLevP . lnTFPijtGMM . Weighted lnTFPijtGMM . Regressand: . (1) . (2) . (3) . (4) . (5) . Firm output tariffs −1.980*** −1.217** −1.100*** −1.096*** −1.159*** (−3.49) (−2.02) (−4.51) (−4.62) (−4.47) Firm output tariffs × fitted extent of processing 2.260** −0.106 0.677 0.675 0.812** (2.03) (−0.08) (1.63) (1.47) (1.96) Firm input tariffs −3.866** −5.069*** −1.380*** −1.378*** −1.589*** (−2.30) (−2.69) (−2.66) (−2.47) (−2.57) Firm input tariffs × fitted extent of processing 8.610*** 10.309*** 2.448** 2.435** 2.664** (2.36) (2.59) (2.12) (2.09) (2.06) Fitted extent of processing −2.737*** −2.901*** −1.251*** −1.251*** −1.311*** (−22.42) (−23.00) (−26.78) (−23.61) (−27.83) SOEs indicator −0.619*** −0.369*** −0.187*** −0.187*** −0.188*** (−11.60) (−5.15) (−7.71) (−7.51) (−7.81) Foreign ownership indicator 0.493*** 0.475*** 0.220*** 0.220*** 0.229*** (19.38) (24.15) (27.24) (32.40) (28.84) Firm size 0.325*** 0.559*** 0.068*** 0.068*** 0.072*** (51.51) (81.26) (34.23) (29.81) (24.59) Firm external tariffs 0.001 0.001 (1.09) (1.22) Kleibergen–Paap rank LM χ2 statistic 106.5† 92.00† 105.4† 105.4† 105.5† Kleibergen–Paap rank Wald F statistic 54.98† 47.78† 55.18† 55.10† 55.10† Year‐specific fixed effects Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Observations 19,296 15,759 19,283 19,283 19,283 R2 0.40 0.53 0.30 0.30 0.65 Notes t‐values in parentheses are obtained using bootstrapped standard errors. Significant at *10%, **5% and ***1%. †indicates significance of p‐value at the 1% level. The regressand is log of value‐added labour productivity (ln LPijt) in column (1) and Levinsohn–Petrin (2003) TFP (⁠ lnTFPijtLevP ⁠) in column (2), and conventional measure of system‐GMM TFP (⁠ lnTFPijtGMM ⁠) in columns (3) and (4). The regressand in column (5) is weighted system‐GMM TFP which is calculated by multiplying lnTFPijtGMM with their relative standard deviations across firms within an industry at the two‐digit level. In all IV estimates, I control for year‐specific fixed effects and time‐invariant two‐digit level Chinese industry fixed‐effects. Firm size in columns (2)–(5) is proxied by log of firm labour, whereas in column (1) it is proxied by firm's capital‐labour ratio. All instruments used are the same as those in Table 9. Pure domestic firms and pure exporters are dropped from the sample. Open in new tab To measure tariff reductions in a firm's export destination markets, I construct an index of firm‐specific external tariffs (FETit) as follows:39 FETit=∑kXitk∑kXitk∑cXiktc∑cXiktcτktc,(12) where τktc is product k's ad valorem tariff imposed by export destination country c in year t. A firm may export multiple types of products to multiple countries. The ratio in the second set of parentheses in (12), Xiktc/∑cXiktc ⁠, measures the export ratio of product k produced by firm i but consumed in country c, yielding a weighted external tariff across Chinese firms' export destinations. Similarly, the first term in parentheses in (12), Xitk/∑kXitk ⁠, measures the proportion of product k's exports over firm i's total exports. The mean of the firm‐specific external tariff is only 0.9%, which is significantly lower than its counterpart for firm‐specific import tariffs on final goods (8.3%). This makes good economic sense. The most important export destinations for Chinese firms are developed countries, such as the US and the countries of the EU, which usually set substantially lower import tariffs on exporters from developing countries like China. Column (4) of Table 12 presents the estimation results including a variable for the firm's external tariffs in the regressions. The coefficient of firm external tariffs is statistically insignificant. One possible reason for this is that Chinese firms had already entered foreign markets before 2000. Thus, tariff reductions in Chinese firms' export destinations have no statistically significant effect in reducing the fixed costs of exports. Still, the regressand used in all the estimation is a measure of TFP, estimated in various ways. As the observations are estimated but not observed, it is worthwhile controlling for the fact that some observations are estimated more precisely than the others. Therefore, I compute the standard deviation of system‐GMM TFP both across firms within an industry and across all firms and divide its sector average by the total average to multiply the firm's system‐GMM TFP as the regressand in the last column of Table 12.40 I obtain similar results as before: the effect of firm tariffs on productivity declines as the firm's processing imports grow. The overall impact of output tariff reductions is stronger than that of input tariff reductions. Finally, the great flexibility of the system‐GMM estimation method indeed provides a unique opportunity to obtain the effects of tariff reduction on firm productivity using a one‐step approach. That is, the coefficients of both input coefficients for the production function and tariffs are obtained simultaneously. I hence experiment with this in Appendix Table C2, as additional robustness checks.41 4.5. Discussion of Channels The article has presented rich evidence that both output and input tariff reductions boost firm productivity. However, we still have little understanding about the channels through which these effects occur. The impact of input tariffs on productivity is relatively direct, as lower tariffs induce access to a larger variety of imported intermediate inputs (Helpern et al., 2010).42 Reductions in output tariffs are found to have a pro‐competitive effect. However, it is less clear whether such a pro‐competitive effect is realised through improvement in the efficiency of firms that are present in the market, or through weeding out the less‐productive firms from the market. To test these two possible channels, I first include an always‐present firms indicator (i.e. it equals one if the firm is present in all years during 2000–6 and otherwise zero) in column (1) of Table 13. The always‐present indicator has a positive and significant sign, suggesting that always‐present firms are more productive. To check whether low‐productivity firms collapse and exit from the market, column (2) includes an exit indicator that takes the value one if firms exit from the market in the next year and zero otherwise. The insignificant sign of the exiting dummy suggests that exiters do not have a significant productivity difference compared with non‐exiting firms. This finding is different from the predictions in Melitz (2003). Amiti and Konings (2007) argue that tariff reductions could result in firms switching their scope from low to high‐productivity products. However, they do not have information on firm scope because of Indonesian data restrictions. Thus, they use a switching dummy as a compromise. However, my merged data set includes information on exporters' scope. Many Chinese firms export multiple products, with the maximum reaching 745 export products. The logarithm of the firm's export scope is included in column (3) of Table 13, and its coefficient is positive and significant, suggesting that firms exporting more products have higher productivity. In column (4), the log of the firm's scope is then interacted with firm‐specific input and output tariffs. The interaction of output tariffs and log scope is found to be significant, whereas that of input tariffs and log scope is insignificant, indicating that at least a few gains from output tariff reductions are attributable to product switching, as also found by Amiti and Konings (2007) with their more limited data. However, this channel is not important for input tariff reductions. Last but not least, firms' productivity gains from trade reform may also result from the channel of investing in new technologies (Bustos, 2011). Firms with higher R&D expenses are expected to have higher productivity. This conjecture is verified in column (5) of Table 13 by including a variable for the firm's log R&D. In the last column, the logarithm of R&D is also interacted with the firm‐specific input and output tariffs. Interestingly, the interaction coefficients of the output and input tariffs and R&D are insignificant, showing that the gains from both output and input tariff reductions do not result from investing in new technologies. One reason is the limited firm R&D data in my sample: around 80% of the observations do not contain valid R&D expenses,43 thus the effect of R&D is under‐estimated for firms to realise gains from tariff reductions. Table 13 IV Estimates for Channels . Firm's selection . Multi‐product firms . R&D expenses . Regressand: ln TFPijtGMM . (1) . (2) . (3) . (4) . (5) . (6) . Firm output tariffs −1.081*** −1.086*** −0.838*** −0.468 −1.119*** −1.628 (−4.10) (−3.44) (−3.51) (−1.54) (−2.16) (−1.17) Firm output tariffs × fitted extent of processing 0.934** 0.934* 1.026** 1.139*** 0.421 0.785 (2.03) (1.82) (2.30) (2.38) (0.37) (0.51) Firm output tariffs × log of firm's scope −0.263*** (−3.45) Firm output tariffs × log of firm's R&D 0.061 (0.40) Firm input tariffs −1.671*** −1.672*** −1.267*** −1.199*** −2.060* −0.899 (−4.10) (−2.88) (−3.36) (−3.31) (−1.73) (−0.52) Firm input tariffs × fitted extent of processing 3.557*** 3.575*** 4.065*** 3.486*** 4.711 3.889 (4.07) (2.94) (4.33) (4.29) (1.53) (1.35) Firm input tariffs × log of firm's scope 0.224 (1.08) Firm input tariffs × log of firm's R&D −0.150 (−0.73) Fitted extent of processing −1.500*** −1.501*** −1.467*** −1.461*** −1.471*** −1.476*** (−40.41) (−29.71) (−35.02) (−32.76) (−10.87) (−9.16) SOEs indicator −0.249*** −0.238*** −0.216*** −0.217*** −0.245*** −0.244*** (−12.94) (−12.89) (−9.87) (−8.38) (−8.56) (−6.80) Foreign ownership indicator 0.281*** 0.282*** 0.228*** 0.229*** 0.310*** 0.309*** (40.84) (39.83) (28.95) (29.88) (18.64) (19.74) Log of labour 0.079*** 0.079*** 0.061*** 0.061*** 0.078*** 0.078*** (34.27) (31.90) (35.40) (27.95) (14.85) (13.77) Log of capital‐labour ratio 0.033*** 0.033*** 0.021*** 0.019*** 0.044*** 0.045*** (18.21) (15.31) (8.82) (8.68) (6.97) (8.74) Firm exits next year 0.009 (0.92) Always‐present firm indicator 0.013* (1.89) Log of firm's scope 0.042*** 0.059*** (19.57) (8.16) Log of R&D 0.028*** 0.028*** (11.33) (2.18) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Yes Observations 19,190 19,190 19,190 19,190 3,331 3,331 R2 0.38 0.38 0.40 0.40 0.47 0.47 . Firm's selection . Multi‐product firms . R&D expenses . Regressand: ln TFPijtGMM . (1) . (2) . (3) . (4) . (5) . (6) . Firm output tariffs −1.081*** −1.086*** −0.838*** −0.468 −1.119*** −1.628 (−4.10) (−3.44) (−3.51) (−1.54) (−2.16) (−1.17) Firm output tariffs × fitted extent of processing 0.934** 0.934* 1.026** 1.139*** 0.421 0.785 (2.03) (1.82) (2.30) (2.38) (0.37) (0.51) Firm output tariffs × log of firm's scope −0.263*** (−3.45) Firm output tariffs × log of firm's R&D 0.061 (0.40) Firm input tariffs −1.671*** −1.672*** −1.267*** −1.199*** −2.060* −0.899 (−4.10) (−2.88) (−3.36) (−3.31) (−1.73) (−0.52) Firm input tariffs × fitted extent of processing 3.557*** 3.575*** 4.065*** 3.486*** 4.711 3.889 (4.07) (2.94) (4.33) (4.29) (1.53) (1.35) Firm input tariffs × log of firm's scope 0.224 (1.08) Firm input tariffs × log of firm's R&D −0.150 (−0.73) Fitted extent of processing −1.500*** −1.501*** −1.467*** −1.461*** −1.471*** −1.476*** (−40.41) (−29.71) (−35.02) (−32.76) (−10.87) (−9.16) SOEs indicator −0.249*** −0.238*** −0.216*** −0.217*** −0.245*** −0.244*** (−12.94) (−12.89) (−9.87) (−8.38) (−8.56) (−6.80) Foreign ownership indicator 0.281*** 0.282*** 0.228*** 0.229*** 0.310*** 0.309*** (40.84) (39.83) (28.95) (29.88) (18.64) (19.74) Log of labour 0.079*** 0.079*** 0.061*** 0.061*** 0.078*** 0.078*** (34.27) (31.90) (35.40) (27.95) (14.85) (13.77) Log of capital‐labour ratio 0.033*** 0.033*** 0.021*** 0.019*** 0.044*** 0.045*** (18.21) (15.31) (8.82) (8.68) (6.97) (8.74) Firm exits next year 0.009 (0.92) Always‐present firm indicator 0.013* (1.89) Log of firm's scope 0.042*** 0.059*** (19.57) (8.16) Log of R&D 0.028*** 0.028*** (11.33) (2.18) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Yes Observations 19,190 19,190 19,190 19,190 3,331 3,331 R2 0.38 0.38 0.40 0.40 0.47 0.47 Notes t‐values in parentheses are obtained using bootstrapped standard errors. Significant at *10%, **5% and ***1%. The two‐digit Chinese industry‐specific fixed effects are included in the estimation. Open in new tab Table 13 IV Estimates for Channels . Firm's selection . Multi‐product firms . R&D expenses . Regressand: ln TFPijtGMM . (1) . (2) . (3) . (4) . (5) . (6) . Firm output tariffs −1.081*** −1.086*** −0.838*** −0.468 −1.119*** −1.628 (−4.10) (−3.44) (−3.51) (−1.54) (−2.16) (−1.17) Firm output tariffs × fitted extent of processing 0.934** 0.934* 1.026** 1.139*** 0.421 0.785 (2.03) (1.82) (2.30) (2.38) (0.37) (0.51) Firm output tariffs × log of firm's scope −0.263*** (−3.45) Firm output tariffs × log of firm's R&D 0.061 (0.40) Firm input tariffs −1.671*** −1.672*** −1.267*** −1.199*** −2.060* −0.899 (−4.10) (−2.88) (−3.36) (−3.31) (−1.73) (−0.52) Firm input tariffs × fitted extent of processing 3.557*** 3.575*** 4.065*** 3.486*** 4.711 3.889 (4.07) (2.94) (4.33) (4.29) (1.53) (1.35) Firm input tariffs × log of firm's scope 0.224 (1.08) Firm input tariffs × log of firm's R&D −0.150 (−0.73) Fitted extent of processing −1.500*** −1.501*** −1.467*** −1.461*** −1.471*** −1.476*** (−40.41) (−29.71) (−35.02) (−32.76) (−10.87) (−9.16) SOEs indicator −0.249*** −0.238*** −0.216*** −0.217*** −0.245*** −0.244*** (−12.94) (−12.89) (−9.87) (−8.38) (−8.56) (−6.80) Foreign ownership indicator 0.281*** 0.282*** 0.228*** 0.229*** 0.310*** 0.309*** (40.84) (39.83) (28.95) (29.88) (18.64) (19.74) Log of labour 0.079*** 0.079*** 0.061*** 0.061*** 0.078*** 0.078*** (34.27) (31.90) (35.40) (27.95) (14.85) (13.77) Log of capital‐labour ratio 0.033*** 0.033*** 0.021*** 0.019*** 0.044*** 0.045*** (18.21) (15.31) (8.82) (8.68) (6.97) (8.74) Firm exits next year 0.009 (0.92) Always‐present firm indicator 0.013* (1.89) Log of firm's scope 0.042*** 0.059*** (19.57) (8.16) Log of R&D 0.028*** 0.028*** (11.33) (2.18) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Yes Observations 19,190 19,190 19,190 19,190 3,331 3,331 R2 0.38 0.38 0.40 0.40 0.47 0.47 . Firm's selection . Multi‐product firms . R&D expenses . Regressand: ln TFPijtGMM . (1) . (2) . (3) . (4) . (5) . (6) . Firm output tariffs −1.081*** −1.086*** −0.838*** −0.468 −1.119*** −1.628 (−4.10) (−3.44) (−3.51) (−1.54) (−2.16) (−1.17) Firm output tariffs × fitted extent of processing 0.934** 0.934* 1.026** 1.139*** 0.421 0.785 (2.03) (1.82) (2.30) (2.38) (0.37) (0.51) Firm output tariffs × log of firm's scope −0.263*** (−3.45) Firm output tariffs × log of firm's R&D 0.061 (0.40) Firm input tariffs −1.671*** −1.672*** −1.267*** −1.199*** −2.060* −0.899 (−4.10) (−2.88) (−3.36) (−3.31) (−1.73) (−0.52) Firm input tariffs × fitted extent of processing 3.557*** 3.575*** 4.065*** 3.486*** 4.711 3.889 (4.07) (2.94) (4.33) (4.29) (1.53) (1.35) Firm input tariffs × log of firm's scope 0.224 (1.08) Firm input tariffs × log of firm's R&D −0.150 (−0.73) Fitted extent of processing −1.500*** −1.501*** −1.467*** −1.461*** −1.471*** −1.476*** (−40.41) (−29.71) (−35.02) (−32.76) (−10.87) (−9.16) SOEs indicator −0.249*** −0.238*** −0.216*** −0.217*** −0.245*** −0.244*** (−12.94) (−12.89) (−9.87) (−8.38) (−8.56) (−6.80) Foreign ownership indicator 0.281*** 0.282*** 0.228*** 0.229*** 0.310*** 0.309*** (40.84) (39.83) (28.95) (29.88) (18.64) (19.74) Log of labour 0.079*** 0.079*** 0.061*** 0.061*** 0.078*** 0.078*** (34.27) (31.90) (35.40) (27.95) (14.85) (13.77) Log of capital‐labour ratio 0.033*** 0.033*** 0.021*** 0.019*** 0.044*** 0.045*** (18.21) (15.31) (8.82) (8.68) (6.97) (8.74) Firm exits next year 0.009 (0.92) Always‐present firm indicator 0.013* (1.89) Log of firm's scope 0.042*** 0.059*** (19.57) (8.16) Log of R&D 0.028*** 0.028*** (11.33) (2.18) Year‐specific fixed effects Yes Yes Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Yes Yes Observations 19,190 19,190 19,190 19,190 3,331 3,331 R2 0.38 0.38 0.40 0.40 0.47 0.47 Notes t‐values in parentheses are obtained using bootstrapped standard errors. Significant at *10%, **5% and ***1%. The two‐digit Chinese industry‐specific fixed effects are included in the estimation. Open in new tab 4.6. Economic Magnitudes and Welfare Contributions This subsection discusses the economic magnitudes of tariff reductions. As shown in the IV estimates in column (1) of Table 11, the regressand is in logarithms whereas the regressors are in levels. Thus, the estimated key coefficients can be interpreted as semi‐elasticities. With tariffs as natural numbers used in the regressions (e.g. the mean of firm output tariffs is 0.083, as reported in Table 5), the own coefficient of the firm output (input) tariffs is −1.32 (−1.71). Measuring tariffs in percentage points (so the mean of firm output tariffs in the sample is 8.3 percentage points), such coefficients are changed to −0.0132 (−0.0171), implying that a 10 percentage point fall in output tariffs for non‐processing firms leads to a 0.132(0.171) increase in log TFP, or equivalently, a productivity gain of 13.2 (17.1)%.44 Equally important, the firm's productivity gains from cutting input and output tariffs become smaller as the firm's processing imports share grows. On average, the impact of the output tariff reductions on productivity improvement is −0.013 + 0.008 × 0.49 = −0.0092, given that the mean of the fitted extent of processing is 0.49, implying that a 10 percentage point fall in output tariffs leads to a productivity gain of 9.2%. Analogously, the average impact of a reduction in input tariffs is −0.017 + 0.025 × 0.49 = −0.0051, indicating that a 10 percentage point fall in input tariffs leads to a productivity gain of 5.1%, almost 56% as high as the gains from reducing output tariffs.45 Average firm output tariffs were cut 8.2 percentage points (from 15.6% in 2000 to 7.4% in 2006), which thus predicts 0.009 × 8.2 = 7.4% productivity gain and contributes 44.4% of the 0.17 log point increase in firm productivity covered in the sample. By the same token, the average firm input tariffs were cut 0.36 percentage points (from 2.54% in 2000 to 2.18% in 2006), which thus predict 0.005 × 0.36 = 0.18% productivity gain and contributes 1.1% of the 0.17 log point increase in log of TFP. Adding these numbers, tariff reductions, overall, contribute around 45.5% to productivity growth for the firms covered in the sample. As economy‐wide productivity growth is one of the best measures of a country's standard of living, my final step is to offer a more intuitive economic interpretation for the contribution of tariff reductions to China's aggregated productivity growth. The adding‐up of firm productivity to economy‐wide productivity is non‐trivial as, because of the presence of vertical integration, intermediate inputs across firms (sectors) contribute to aggregated productivity by allowing productivity gains in successive firms (sectors) to augment one another (OECD, 2001).46 As initiated by Domar (1961) and later elaborated by Hulten (1978) and Feenstra et al. (2013b), the economy‐wide TFP can be aggregated by using the ‘Domar weight’ which is defined by each firm's gross output relative to economy‐wide absorption (i.e. total gross output minus trade surplus). I then calculate the aggregated TFP using Domar weights for each year. It turns out that aggregated log of TFP increases around 0.53 log points (from 0.56 in 2000 to 1.09 in 2006).47 As described before, both output and input tariff reductions, on average, lead to productivity gains of 7.54% + 0.11% = 0.076, and thus contribute to 14.5% of the 0.53 log point increase in economy‐wide log productivity. A final remark is that the calculation here presumes that tariff cuts have no impact on firm productivity beyond the sample. As tariff reductions still, in reality, have beneficial ripple effects beyond the set of firms in the sample, the calculated contribution to the whole economy should be interpreted as a lower‐bound number. 5. Concluding Remarks To explore how reductions in tariffs on imported inputs and final goods affect firm productivity, the article has exploited the special tariff treatment afforded to imported inputs by processing firms as opposed to non‐processing firms in China. As a popular trade pattern in a large number of developing countries, including China, processing trade plays an important role in the realisation of productivity gains. Overall, I find that the impact of output tariff reduction is greater than that of input tariff reduction for large Chinese trading firms. More interestingly, the positive impact of reduction in input (output) tariffs on firm productivity is weaker as firms' processing import share grows. This article is one of the first to explore the role of processing trade in Chinese firms' productivity gains. The rich data set enables the determination of whether a firm engages in processing trade and the examination of the effect of the firm's extent of processing trade engagement on productivity. With such information, firm‐level input and output tariffs were also constructed, as one of the first attempts in the literature, which, in turn, enriches the understanding of the economic effects of China's special tariff reforms in processing trade. Appendix A. Matching Production and Trade Data Sets My discussion on matching the two data sets (i.e. firm‐level production data and firm‐customs data) here draws heavily from Yu and Tian (2012). As mentioned in the text, I go through two steps to merge transaction‐level trade data with firm‐level production data. In the first step, I match the two data sets by firm name and year. The year variable is necessarily an auxiliary identifier because some firms could have different names across years and newcomers could possibly take their original names. Using the raw (i.e. unfiltered) production data set, I come up with 83,679 merged firms; this number is further reduced to 69,623 with the more accurately filtered production data set. In the second step, I use another matching technique as a supplement. In particular, I adopt two other common variables to identify firms: postal code and the last seven digits of a firm's phone number. The rationale is that firms should have different and unique phone numbers within a postal district. Although this method seems straightforward, subtle technical and practical difficulties still exist. For instance, the production‐level trade data set includes both area codes and a hyphen in the phone numbers, whereas the firm‐level production data set does not. Therefore, I use the last seven digits of the phone number to serve as the proxy for firm identification for two reasons. First, in 2000–6, some large Chinese cities (e.g. Shantou in Guangdong province) added one more digit at the start of their seven‐digit phone numbers. Therefore, using the last seven digits of the number will not confuse firm identification. Second, in the original data set, phone numbers are defined as a string of characters with the phone postal code; however, it is inappropriate to de‐string such characters to numerals because a hyphen is used to connect the postal code and phone number. Using the last seven‐digit sub‐string neatly solves this problem. A firm might not include information on its name in either the trade or the production data set. Similarly, a firm could lose its phone and/or postal code information. To be sure that the merged data set can cover as many common firms as possible, I then include observations in the matched data set if a firm occurs in either the name‐adopted matched data set or the phone‐and‐post‐adopted matched data set. As shown in Appendix Table A1, column (1) reports the number of observations of HS eight‐digit monthly transaction‐level trade data from China's General Administration of Customs by year. As shown at the bottom of column (1), there are more than 118 million monthly trade transactions conducted by 286,819 firms during the seven years, as shown in column (2). Meanwhile, if no further data cleaning and stringent filter criteria are adopted as introduced in the text, column (3) shows that there are 615,591 large manufacturing firms in China. However, after stringent filtering according to GAAP requirements, around 70% of them survive – number of the filtered firms is 438,165 as seen at the bottom of column (4). Accordingly, column (5) reports the number of matched firms using exactly identical company names in both trade data set and raw production data set. By contrast, column (6) reports number of matched firms using exactly identical company names in both the trade data set and the filtered production data set, which results in 69,623 matched firms. Column (7) reports the number of matched firms using exactly identical company names and exactly identical postal codes and phone numbers in both the trade and raw production data sets. The number of merged firms increases to 91,299. By way of comparison, my matching performance is highly comparable with that of other similar studies. For example, Ge et al. (2011) use the same data sets and similar matching techniques and end up with 86,336 merged firms. Finally, if I match the more stringent filtered production data set with the firm‐level data set using exactly identical company names and postal–phone code numbers but drop firms whose customs‐reported exports are higher than NBS‐reported firm sales, I end up with 76,823 firms in total, as shown in the last column of Appendix Table A1. I use these firms to run the regressions because they are the most reliable firms that can pass various stringent filtering processes in the firm production data. After merging both the product‐level trade data and the firm‐level production data, the 76,823 common trading firms account for approximately 27% of the 286,819 firms in the product‐level trade data set and approximately 17% of the 438,146 valid firms in the firm‐level production data set (11% of the valid firms are exporters, whereas 6% of them are importers). Given that only 27% of firms are exporters in the firm‐level production data set (Feenstra et al., 2013b), the merged data set hence accounts for around 40% of the filtered full‐sample firm‐level production data set in terms of number of exporters, and around 53% of exports in terms of export value. Table A1 Matched Statistics – Number of Firms Year . Trade data . Production data . Matched data . Transactions . Firms . Raw firms . Filtered firms . w/Raw firms . w/Filtered firms . w/Raw firms . w/Filtered firms . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 10,586,696 80,232 162,883 83,628 18,580 12,842 21,425 15,748 2001 12,667,685 87,404 169,031 100,100 21,583 15,645 24,959 19,091 2002 14,032,675 95,579 181,557 110,530 24,696 18,140 28,759 22,291 2003 18,069,404 113,147 196,222 129,508 28,898 21,837 33,901 26,930 2004 21,402,355 134,895 277,004 199,927 44,338 35,007 49,891 40,711 2005 24,889,639 136,604 271,835 198,302 44,387 34,958 49,891 40,387 2006 16,685,377 197,806 301,960 224,854 53,748 42,833 49,680 47,591 All years 118,333,831 286,819 615,951 438,165 83,679 69,623 91,299 76,823 Year . Trade data . Production data . Matched data . Transactions . Firms . Raw firms . Filtered firms . w/Raw firms . w/Filtered firms . w/Raw firms . w/Filtered firms . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 10,586,696 80,232 162,883 83,628 18,580 12,842 21,425 15,748 2001 12,667,685 87,404 169,031 100,100 21,583 15,645 24,959 19,091 2002 14,032,675 95,579 181,557 110,530 24,696 18,140 28,759 22,291 2003 18,069,404 113,147 196,222 129,508 28,898 21,837 33,901 26,930 2004 21,402,355 134,895 277,004 199,927 44,338 35,007 49,891 40,711 2005 24,889,639 136,604 271,835 198,302 44,387 34,958 49,891 40,387 2006 16,685,377 197,806 301,960 224,854 53,748 42,833 49,680 47,591 All years 118,333,831 286,819 615,951 438,165 83,679 69,623 91,299 76,823 Notes Column (1) reports number of observations of HS eight‐digit monthly transaction‐level trade data from China's General Administration of Customs by year. Column (2) reports number of firms covered in the transaction‐level trade data by year. Column (3) reports number of firms covered in the firm‐level production data set compiled by China's National Bureau of Statistics without any filter and cleaning. By contrast, column (4) presents number of firms covered in the firm‐level production data set with careful filtering according to GAAP requirements. Accordingly, column (5) reports number of matched firms using exactly identical company names in both the trade data set and the raw production data set. By contrast, column (6) reports number of matched firms using exactly identical company names in both the trade data set and the filtered production data set. Column (7) reports number of matched firms using exactly identical company names and exactly identical postal codes and phone numbers in both the trade data set and the raw production data set. By contrast, column (8) reports number of matched firms using exactly identical company names and exactly identical postal codes and phone numbers in both the trade data set and the filtered production data set. Open in new tab Table A1 Matched Statistics – Number of Firms Year . Trade data . Production data . Matched data . Transactions . Firms . Raw firms . Filtered firms . w/Raw firms . w/Filtered firms . w/Raw firms . w/Filtered firms . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 10,586,696 80,232 162,883 83,628 18,580 12,842 21,425 15,748 2001 12,667,685 87,404 169,031 100,100 21,583 15,645 24,959 19,091 2002 14,032,675 95,579 181,557 110,530 24,696 18,140 28,759 22,291 2003 18,069,404 113,147 196,222 129,508 28,898 21,837 33,901 26,930 2004 21,402,355 134,895 277,004 199,927 44,338 35,007 49,891 40,711 2005 24,889,639 136,604 271,835 198,302 44,387 34,958 49,891 40,387 2006 16,685,377 197,806 301,960 224,854 53,748 42,833 49,680 47,591 All years 118,333,831 286,819 615,951 438,165 83,679 69,623 91,299 76,823 Year . Trade data . Production data . Matched data . Transactions . Firms . Raw firms . Filtered firms . w/Raw firms . w/Filtered firms . w/Raw firms . w/Filtered firms . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . 2000 10,586,696 80,232 162,883 83,628 18,580 12,842 21,425 15,748 2001 12,667,685 87,404 169,031 100,100 21,583 15,645 24,959 19,091 2002 14,032,675 95,579 181,557 110,530 24,696 18,140 28,759 22,291 2003 18,069,404 113,147 196,222 129,508 28,898 21,837 33,901 26,930 2004 21,402,355 134,895 277,004 199,927 44,338 35,007 49,891 40,711 2005 24,889,639 136,604 271,835 198,302 44,387 34,958 49,891 40,387 2006 16,685,377 197,806 301,960 224,854 53,748 42,833 49,680 47,591 All years 118,333,831 286,819 615,951 438,165 83,679 69,623 91,299 76,823 Notes Column (1) reports number of observations of HS eight‐digit monthly transaction‐level trade data from China's General Administration of Customs by year. Column (2) reports number of firms covered in the transaction‐level trade data by year. Column (3) reports number of firms covered in the firm‐level production data set compiled by China's National Bureau of Statistics without any filter and cleaning. By contrast, column (4) presents number of firms covered in the firm‐level production data set with careful filtering according to GAAP requirements. Accordingly, column (5) reports number of matched firms using exactly identical company names in both the trade data set and the raw production data set. By contrast, column (6) reports number of matched firms using exactly identical company names in both the trade data set and the filtered production data set. Column (7) reports number of matched firms using exactly identical company names and exactly identical postal codes and phone numbers in both the trade data set and the raw production data set. By contrast, column (8) reports number of matched firms using exactly identical company names and exactly identical postal codes and phone numbers in both the trade data set and the filtered production data set. Open in new tab Appendix B. The Augmented Olley–Pakes TFP Measures In this Appendix, I estimate the measured Olley–Pakes TFP by taking the role of processing trade into account. In the article, the Olley–Pakes TFP is estimated in three ways: (i) TFPOP which is used in the full‐sample estimates in columns (1) and (2) in Table 7; (ii) TFPOP1 which separates processing firms and non‐processing firms into two groups and uses different control function approaches, as discussed below, and is used in columns (3) and (4) in Table 7 and column (1) in Table 8; and (iii) TFPOP2 which pools processing firms and non‐processing firms together for estimation and is used in column (2) in Table 8. It is important to stress that different versions of Olley–Pakes TFP do not qualitatively change my estimation results. By assuming that the expectation of future realisation of the unobserved productivity shock, υit, relies on its contemporaneous value, firm i's investment is modelled as an increasing function of both unobserved productivity and log capital, kit ≡ ln Kit. Following previous works, such as Amiti and Konings (2007), the Olley–Pakes approach was revised by adding other control variables as extra arguments of the investment function as follows: Iit=I~(lnKit,υit,FXit,WTOt,SOEit),(B.1) where FXit is a dummy to measure whether firm i exports in year t as firm's export decision may affect firm investment. As my firm data set is from 2000 to 2006, I include a WTO dummy (i.e. one for a year after 2001 and zero for before) in the investment function. Finally, given the importance of state intervention, SOEs would have different decision behaviour than non‐SOEs. I therefore include an SOE dummy in the investment function as well. Therefore, the inverse function of (B.1) is υit=I~−1(lnKit,Iit,FXit,WTOt,SOEit) ⁠. The unobserved productivity also depends on log capital and other arguments. The estimation specification (M.1) in the text can now be written as follows: lnYit=β0+βmlnMit+βllnLit+g(lnKit,Iit,FXit,WTOt,SOEit)+ϵit,(B.2) where g(·) is defined as βklnKit+I~−1(lnKit,Iit,FXit,WTOt,SOEit) ⁠. Following Olley and Pakes (1996), fourth‐order polynomials in log‐capital, log‐investment, firm's export dummy and import dummy are used to approximate g(·).48 With this specification, the coefficient of labour βl and that of materials βm can be estimated as the first‐step procedure. The three different versions of Olley–Pakes TFP use different control functions. The control function of TFPOP which is used in the full‐sample estimates cannot control for the firm's import status, as the full‐sample production data set does not report import status. However, the import dummy is incorporated in the other two approaches (TFPOP1 and TFPOP2) when using a matched sample to estimate. The difference between TFPOP1 and TFPOP2 is whether processing firms are separated from non‐processing firms. B.1. TFPOP Used in the Full‐sample Data Set In the full‐sample data set, information on the firm's import status and processing status is unavailable. I hence adopt the following functional form: g(lnKit,Iit,FXit,WTOt,SOEit)=(α0+α1WTOt+α2FXit+α3SOEit)∑h=04∑q=04δhq(lnKit)hIitq.(B.3) In the first step, I obtain estimates of β^m and β^l for non‐processing (ordinary) firms. I then calculate the residual Rit which is defined as Rit≡lnYit−β^mlnMit−β^llnLit ⁠. The next step is to obtain an unbiased estimated coefficient of βk. To correct the selection bias as mentioned above, Amiti and Konings (2007) suggest estimating the probability of a survival indicator on a high‐order polynomial in log‐capital and log‐investment. One can then accurately estimate the following specification: Rit=βklnKit+I~−1(gi,t−1−βklnKi,t−1,p^ri,t−1)+ϵit,(B.4) where p^ri denotes the fitted value for the probability of the firm 's exit in the next year. As the specific ‘true’ functional form of the inverse function I~−1(·) is unknown, it is appropriate to use fourth‐order polynomials in gi,t−1 and lnKi,t−1 to approximate it. In addition, (B.4) also requires the estimated coefficients of the log‐capital in the first and second terms to be identical. Therefore, non‐linear least squares seems to be the most desirable econometric technique. Finally, the Olley–Pakes type of TFP for ordinary firm i in industry j is obtained once the estimated coefficient β^k is obtained: lnTFPijtOP=lnYit−β^mlnMit−β^klnKit−β^llnLit.(B.5) B.2. TFPOP1 with Separate Estimates for Processing and Non‐processing Firms By contrast, the control functions used in TFPOP1 for processing firms and non‐processing firms are different. If a firm is engaged in any processing imports, it is defined as a processing firm; otherwise it is defined as a non‐processing (ordinary) firm. I first separate all firms in the sample into two groups – non‐processing (ordinary) firms and processing firms. The control function for non‐processing firms in the first‐step estimates takes the following form: gord(lnKit,Iit,FXit,IMit,WTOt,SOEit)=(θ0+θ1WTOt+θ2FXit+θ3IMit+θ4SOEit)∑h=04∑q=04δhqord(lnKit)hIitq,(B.6) where IMit denotes the import dummy that takes the value one if firm i in year t is an importer, and zero otherwise. The estimates in the second step are identical to the corresponding estimates in the first approach TFPOP. The Olley–Pakes type of TFP for ordinary firm i in industry j is obtained once the estimated coefficient β^kord is obtained: lnTFPijtord=lnYit−β^mordlnMit−β^kordlnKit−β^lordlnLit.(B.7) The estimates for processing firms have two important differences from those for ordinary firms. First, the coefficients of all inputs are allowed to be different because processing firms could use different technologies from ordinary firms. Second, because processing firms, by definition, are both importers and exporters, I do not need to introduce the export dummy or the import dummy in their investment function or the fourth‐order polynomials. That is, the polynomials for processing firms are as follows: gproc(lnKit,Iit,WTOt,SOEit)=(γ0+γ1WTOt+γ2SOEit)∑h=04∑q=04δhqproc(lnKit)hIitq.(B.8) The rest of the procedures for processing firm TFP are the same as their counterparts for non‐processing firms. The Olley–Pakes type of TFP for processing firm i in industry j is obtained as follows: lnTFPijtproc=lnYit−β^mproclnMit−β^kproclnKit−β^lproclnLit.(B.9) I hence obtain two different sets of TFP for ordinary firms and processing firms. Their estimated input coefficients and measured TFP are shown in Appendix Table B1. The series of TFPOP1 is obtained by stacking them together. B.3. TFPOP2 with Learning from Processing Following De Loecker (2013), I now allow firms to learn from processing trade. Therefore, the export dummy is endogenously correlated with firm investment. To obtain TFPOP2, the difference from standard Olley–Pakes estimates is the first‐step estimation. I first insert the processing dummy, PEit, into the investment function as follows: Iit=I~(lnKit,υit,FXit,IMit,WTOt,SOEit,PEit).(B.10) Therefore, the inverse function of (B.10) is υit=I~−1(lnKit,Iit,FXit,IMit,WTOt,SOEit,PEit) ⁠. To capture the possible learning effects from processing, the export decision was presumed to be made prior to the realisation of firm productivity. Hence, the productivity processing function g(·) is defined as βk ln Kit + υit+1 where the productivity realisation υit+1 uses the following polynomial specification as in De Loecker (2013): υit+1=∑s=04∑m=04βsmPEitsυitm+ζit+1(B.11) with E(ζit+1PEit) = 0. Note that firm innovation ζit+1 thus is different from the standard Olley–Pakes step where ζit+1 = υit+1 − υit. Compared with other dummies, such as the exporting dummy, the processing dummy is not only used in the second‐step estimates, but also in the first‐step estimates. Similarly, the inverse investment function can be characterised as the following control function: υit=(λ0+λ1WTOt+λ2FXit+λ3IMit+λ4PEit+λ5SOEit)∑h=04∑q=04δhq(lnKit)hIitq. Table B1 Estimates of Olley–Pakes TFP by Processing and Ordinary Firms Separately Chinese industry . Ordinary firms . Processing . Labour . Materials . Capital . Labour . Materials . Capital . 13 0.242 0.875 0.052 0.116 0.884 0.066 14 0.023 0.926 0.050 0.037 0.925 0.074 15 0.185 0.508 0.268 0.243 0.505 0.088 17 0.017 0.884 0.059 0.089 0.834 0.041 18 0.054 0.858 0.076 0.177 0.669 0.142 19 0.126 0.895 0.023 0.118 0.808 0.000 20 0.126 0.895 0.023 0.044 0.913 0.003 21 0.055 0.917 0.042 0.101 0.873 0.103 22 0.111 0.907 0.008 0.027 0.896 0.063 23 0.023 0.821 0.039 0.105 0.836 0.025 24 0.068 0.764 0.123 0.104 0.863 0.036 26 0.086 0.795 0.063 0.007 0.927 0.024 27 0.108 0.862 0.040 0.038 0.860 0.038 28 0.116 0.789 0.033 0.016 0.837 0.041 29 0.061 0.569 0.174 0.073 0.938 0.032 30 0.118 0.633 0.182 0.125 0.696 0.114 31 0.073 0.851 0.047 0.050 0.870 0.035 32 0.046 0.976 0.051 0.038 0.961 0.010 33 0.053 0.815 0.080 0.055 0.850 0.076 34 0.041 0.867 0.048 0.044 0.883 0.026 35 0.065 0.875 0.024 0.032 0.917 0.026 36 0.090 0.823 0.076 0.038 0.869 0.111 37 0.058 0.888 0.047 0.054 0.924 0.029 39 0.013 0.830 0.103 0.102 0.826 0.000 40 0.071 0.831 0.072 0.086 0.878 0.086 41 0.081 0.906 0.015 0.139 0.567 0.168 42 0.055 0.917 0.045 0.142 0.818 0.094 Chinese industry . Ordinary firms . Processing . Labour . Materials . Capital . Labour . Materials . Capital . 13 0.242 0.875 0.052 0.116 0.884 0.066 14 0.023 0.926 0.050 0.037 0.925 0.074 15 0.185 0.508 0.268 0.243 0.505 0.088 17 0.017 0.884 0.059 0.089 0.834 0.041 18 0.054 0.858 0.076 0.177 0.669 0.142 19 0.126 0.895 0.023 0.118 0.808 0.000 20 0.126 0.895 0.023 0.044 0.913 0.003 21 0.055 0.917 0.042 0.101 0.873 0.103 22 0.111 0.907 0.008 0.027 0.896 0.063 23 0.023 0.821 0.039 0.105 0.836 0.025 24 0.068 0.764 0.123 0.104 0.863 0.036 26 0.086 0.795 0.063 0.007 0.927 0.024 27 0.108 0.862 0.040 0.038 0.860 0.038 28 0.116 0.789 0.033 0.016 0.837 0.041 29 0.061 0.569 0.174 0.073 0.938 0.032 30 0.118 0.633 0.182 0.125 0.696 0.114 31 0.073 0.851 0.047 0.050 0.870 0.035 32 0.046 0.976 0.051 0.038 0.961 0.010 33 0.053 0.815 0.080 0.055 0.850 0.076 34 0.041 0.867 0.048 0.044 0.883 0.026 35 0.065 0.875 0.024 0.032 0.917 0.026 36 0.090 0.823 0.076 0.038 0.869 0.111 37 0.058 0.888 0.047 0.054 0.924 0.029 39 0.013 0.830 0.103 0.102 0.826 0.000 40 0.071 0.831 0.072 0.086 0.878 0.086 41 0.081 0.906 0.015 0.139 0.567 0.168 42 0.055 0.917 0.045 0.142 0.818 0.094 Notes This Table reports the estimates of log of Olley–Pakes TFP (lnTFPOP1) by separating ordinary firms and processing firms. The Chinese industries and associated codes are classified as follows: processing of foods (13), manufacture of foods (14), beverages (15), textiles (17), apparel (18), leather (19), timber (20), furniture (21), paper (22), printing (23), articles for cultures and sports (24), petroleum (25), raw chemicals (26), medicines (27), chemical fibres (28), rubber (29), plastics (30), non‐metallic minerals (31), smelting of ferrous metals (32), smelting of non‐ferrous metals (33), metal (34), general machinery (35), special machinery (36), transport equipment (37), electrical machinery (39), communication equipment (40), measuring instruments (41) and manufacture of artwork (42). I do not report the standard errors for each estimated coefficient to save space, although they are available upon request. Open in new tab Table B1 Estimates of Olley–Pakes TFP by Processing and Ordinary Firms Separately Chinese industry . Ordinary firms . Processing . Labour . Materials . Capital . Labour . Materials . Capital . 13 0.242 0.875 0.052 0.116 0.884 0.066 14 0.023 0.926 0.050 0.037 0.925 0.074 15 0.185 0.508 0.268 0.243 0.505 0.088 17 0.017 0.884 0.059 0.089 0.834 0.041 18 0.054 0.858 0.076 0.177 0.669 0.142 19 0.126 0.895 0.023 0.118 0.808 0.000 20 0.126 0.895 0.023 0.044 0.913 0.003 21 0.055 0.917 0.042 0.101 0.873 0.103 22 0.111 0.907 0.008 0.027 0.896 0.063 23 0.023 0.821 0.039 0.105 0.836 0.025 24 0.068 0.764 0.123 0.104 0.863 0.036 26 0.086 0.795 0.063 0.007 0.927 0.024 27 0.108 0.862 0.040 0.038 0.860 0.038 28 0.116 0.789 0.033 0.016 0.837 0.041 29 0.061 0.569 0.174 0.073 0.938 0.032 30 0.118 0.633 0.182 0.125 0.696 0.114 31 0.073 0.851 0.047 0.050 0.870 0.035 32 0.046 0.976 0.051 0.038 0.961 0.010 33 0.053 0.815 0.080 0.055 0.850 0.076 34 0.041 0.867 0.048 0.044 0.883 0.026 35 0.065 0.875 0.024 0.032 0.917 0.026 36 0.090 0.823 0.076 0.038 0.869 0.111 37 0.058 0.888 0.047 0.054 0.924 0.029 39 0.013 0.830 0.103 0.102 0.826 0.000 40 0.071 0.831 0.072 0.086 0.878 0.086 41 0.081 0.906 0.015 0.139 0.567 0.168 42 0.055 0.917 0.045 0.142 0.818 0.094 Chinese industry . Ordinary firms . Processing . Labour . Materials . Capital . Labour . Materials . Capital . 13 0.242 0.875 0.052 0.116 0.884 0.066 14 0.023 0.926 0.050 0.037 0.925 0.074 15 0.185 0.508 0.268 0.243 0.505 0.088 17 0.017 0.884 0.059 0.089 0.834 0.041 18 0.054 0.858 0.076 0.177 0.669 0.142 19 0.126 0.895 0.023 0.118 0.808 0.000 20 0.126 0.895 0.023 0.044 0.913 0.003 21 0.055 0.917 0.042 0.101 0.873 0.103 22 0.111 0.907 0.008 0.027 0.896 0.063 23 0.023 0.821 0.039 0.105 0.836 0.025 24 0.068 0.764 0.123 0.104 0.863 0.036 26 0.086 0.795 0.063 0.007 0.927 0.024 27 0.108 0.862 0.040 0.038 0.860 0.038 28 0.116 0.789 0.033 0.016 0.837 0.041 29 0.061 0.569 0.174 0.073 0.938 0.032 30 0.118 0.633 0.182 0.125 0.696 0.114 31 0.073 0.851 0.047 0.050 0.870 0.035 32 0.046 0.976 0.051 0.038 0.961 0.010 33 0.053 0.815 0.080 0.055 0.850 0.076 34 0.041 0.867 0.048 0.044 0.883 0.026 35 0.065 0.875 0.024 0.032 0.917 0.026 36 0.090 0.823 0.076 0.038 0.869 0.111 37 0.058 0.888 0.047 0.054 0.924 0.029 39 0.013 0.830 0.103 0.102 0.826 0.000 40 0.071 0.831 0.072 0.086 0.878 0.086 41 0.081 0.906 0.015 0.139 0.567 0.168 42 0.055 0.917 0.045 0.142 0.818 0.094 Notes This Table reports the estimates of log of Olley–Pakes TFP (lnTFPOP1) by separating ordinary firms and processing firms. The Chinese industries and associated codes are classified as follows: processing of foods (13), manufacture of foods (14), beverages (15), textiles (17), apparel (18), leather (19), timber (20), furniture (21), paper (22), printing (23), articles for cultures and sports (24), petroleum (25), raw chemicals (26), medicines (27), chemical fibres (28), rubber (29), plastics (30), non‐metallic minerals (31), smelting of ferrous metals (32), smelting of non‐ferrous metals (33), metal (34), general machinery (35), special machinery (36), transport equipment (37), electrical machinery (39), communication equipment (40), measuring instruments (41) and manufacture of artwork (42). I do not report the standard errors for each estimated coefficient to save space, although they are available upon request. Open in new tab The second‐step estimates are standard as above. After obtaining the coefficients of capital, labour and materials, the TFPOP2 is calculated as follows: lnTFPijtOP2=lnYit−β^mlnMit−β^klnKit−β^llnLit.(B.12) Appendix C. Derivation of Domar‐aggregation Productivity This Appendix interprets how to add firm productivity to economy‐wide aggregate productivity using Domar's (1961) weight under an open‐economy set‐up. The Appendix draws heavily from OECD (2001) and Feenstra et al.(2013a). The challenging part of the aggregation comes from the fact that domestic intermediate inputs used by firms do not show up in the economy‐wide production possibility frontier (PPF), as they represent intra‐industry flows that are absorbed in a process of vertical integration. To concretise this idea, consider the following PPF: T(FA,N,IM,π)=0,(C.1) where FA denotes China's final absorption (or equivalently, final demand), N denotes all domestic primary inputs such as capital and labour, IM is imported intermediate inputs and π is aggregate TFP. By assuming inputs are homogenous of degree zero in FA, N, IM and π and perfectly competitive markets, the productivity change can be traced as follows: dlnπdt=dlnFAdt−PNNPFAFAdlnNdt−PIMIMPFAFAdlnIMdt,(C.2) where (PNN)/(PFAFA) is the share of primary inputs in total final absorption and (PIM IM)/(PFAFA) is the share of imported intermediate inputs in total final absorption. Both terms sum to unity because of zero profit in a perfectly competitive set‐up. To link the aggregate economy with firm‐level economic activities, each term in (C.2) can be decomposed as follows: dlnFAdt=∑iPiFAiPFAFAdlnFAidtdlnNdt=∑iPNiNiPNNdlnNidtdlnIMdt=∑iPIMiIMiPIMIMdlnIMidt.(C.3) That is, aggregated final demand (aggregated primary inputs, aggregated imported intermediate inputs) can be written as a weighted average of firms' demand (primary inputs, imported intermediate inputs). By inserting (C.3) back into (C.2), I obtain: dlnπdt=∑iPiFAiPFAFAdlnFAidt−PNNPFAFA∑iPNiNiPNNdlnNidt−PIMIMPFAFA∑iPIMiIMiPIMIMdlnIMidt.(C.4) Turning to measures of firm productivity, consider the following production function, which is homogenous of degree one: Yi=πif(Ni,Mi,IMi),(C.5) where Yi, Ni, Mi, and IMi denote firm i's output, primary inputs, domestic intermediate inputs and imported intermediate inputs, respectively. πi is the Hicks‐neutral TFP. Total differentiate (C.5) to obtain the following equation: dlnπidt=dlnYidt−PNiNiPiYidlnNidt−PMiMiPiYidlnMidt−PIMiIMiPiYidlnIMidt.(C.6) Note that each firm gets zero profit as the market structure is perfect competition, which implies: PiYi=PNiNi+PMiMi+PIMiIMi.(C.7) Thus, the input shares in the last three terms in (C.6) sum to unity. Meanwhile, the firm's total demand (i.e. demand for intermediate goods and final goods) is equal to its production value (i.e. supply): PiYi=∑kPiYki+PiFAi, where prices for intermediate demand use and for final use are assumed to be equal for simplicity and Yki denotes firm i's deliveries of its product to firm k. Totally differentiate the above equation to obtain: dlnFAidt=PiYiPiFAidlnYidt−∑kPiYkiPiYidlnYkidt.(C.8) By inserting (C.8) into (C.4), I obtain: dlnπdt=∑iPiYiPFAFAdlnYidt−∑kPiYkiPiYidlnYkidt−PNiNiPiYidlnNidt−PIMiIMiPiYidlnIMidt.(C.9) Table C1 Estimates of System‐GMM Firm TFP by Industry . Estimated coefficients . TFP . SD . Weighted TFD . Tests (p‐value) . Labour . Materials . Capital . . . TFP . AR(1) . AR(2) . Hansen . Chinese industry . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . 13 0.094 0.718 0.010 2.575 0.387 2.884 0.000 0.987 0.443 14 0.089 0.828 0.003 2.528 0.380 2.776 0.000 0.396 0.603 15 0.077 0.677 0.152 2.677 0.465 3.599 0.063 0.724 1.00 17 0.065 0.748 0.002 2.523 0.298 2.175 0.007 0.389 0.569 18 0.068 0.724 0.020 2.447 0.326 2.303 0.000 0.317 0.834 19 0.050 0.868 0.029 2.488 0.323 2.320 0.015 0.858 0.676 20 0.015 0.844 0.010 2.851 0.412 3.398 0.011 0.510 0.548 21 0.114 0.795 0.001 2.650 0.309 2.367 0.000 0.051 0.808 22 0.151 0.655 0.011 2.705 0.338 2.644 0.424 0.570 1.00 23 0.178 0.474 0.051 2.618 0.341 2.578 0.036 0.059 0.846 24 0.098 0.609 0.058 2.485 0.281 2.018 0.030 0.411 0.990 25 0.017 0.700 0.173 2.865 0.498 4.127 0.156 0.744 1.00 26 0.142 0.701 0.034 2.669 0.353 2.721 0.000 0.868 0.222 27 0.014 0.748 0.054 2.764 0.350 2.797 0.008 0.988 0.712 28 0.052 0.812 0.088 2.674 0.326 2.520 0.082 0.280 1.00 29 0.165 0.633 0.025 2.593 0.348 2.606 0.015 0.691 0.899 30 0.128 0.865 0.022 2.690 0.335 2.605 0.000 0.303 0.371 31 0.105 0.769 0.019 2.626 0.343 2.600 0.000 0.936 0.034 32 0.001 0.876 0.001 2.864 0.388 3.212 0.060 0.233 0.909 33 0.068 0.805 0.057 2.592 0.386 2.888 0.914 0.682 0.896 34 0.022 0.840 0.021 2.480 0.318 2.279 0.009 0.161 0.788 35 0.108 0.782 0.003 2.527 0.313 2.286 0.000 0.473 0.726 36 0.091 0.719 0.089 2.604 0.356 2.681 0.000 0.845 0.537 37 0.103 0.813 0.034 2.637 0.359 2.737 0.090 0.893 0.393 39 0.309 0.628 0.101 2.503 0.394 2.847 0.049 – 0.743 40 0.158 0.729 0.021 2.833 0.451 3.692 0.013 – 0.368 41 0.061 0.889 0.012 2.682 0.465 3.603 0.028 0.281 0.767 42 0.088 0.667 0.012 2.450 0.295 2.090 0.041 – 0.564 . Estimated coefficients . TFP . SD . Weighted TFD . Tests (p‐value) . Labour . Materials . Capital . . . TFP . AR(1) . AR(2) . Hansen . Chinese industry . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . 13 0.094 0.718 0.010 2.575 0.387 2.884 0.000 0.987 0.443 14 0.089 0.828 0.003 2.528 0.380 2.776 0.000 0.396 0.603 15 0.077 0.677 0.152 2.677 0.465 3.599 0.063 0.724 1.00 17 0.065 0.748 0.002 2.523 0.298 2.175 0.007 0.389 0.569 18 0.068 0.724 0.020 2.447 0.326 2.303 0.000 0.317 0.834 19 0.050 0.868 0.029 2.488 0.323 2.320 0.015 0.858 0.676 20 0.015 0.844 0.010 2.851 0.412 3.398 0.011 0.510 0.548 21 0.114 0.795 0.001 2.650 0.309 2.367 0.000 0.051 0.808 22 0.151 0.655 0.011 2.705 0.338 2.644 0.424 0.570 1.00 23 0.178 0.474 0.051 2.618 0.341 2.578 0.036 0.059 0.846 24 0.098 0.609 0.058 2.485 0.281 2.018 0.030 0.411 0.990 25 0.017 0.700 0.173 2.865 0.498 4.127 0.156 0.744 1.00 26 0.142 0.701 0.034 2.669 0.353 2.721 0.000 0.868 0.222 27 0.014 0.748 0.054 2.764 0.350 2.797 0.008 0.988 0.712 28 0.052 0.812 0.088 2.674 0.326 2.520 0.082 0.280 1.00 29 0.165 0.633 0.025 2.593 0.348 2.606 0.015 0.691 0.899 30 0.128 0.865 0.022 2.690 0.335 2.605 0.000 0.303 0.371 31 0.105 0.769 0.019 2.626 0.343 2.600 0.000 0.936 0.034 32 0.001 0.876 0.001 2.864 0.388 3.212 0.060 0.233 0.909 33 0.068 0.805 0.057 2.592 0.386 2.888 0.914 0.682 0.896 34 0.022 0.840 0.021 2.480 0.318 2.279 0.009 0.161 0.788 35 0.108 0.782 0.003 2.527 0.313 2.286 0.000 0.473 0.726 36 0.091 0.719 0.089 2.604 0.356 2.681 0.000 0.845 0.537 37 0.103 0.813 0.034 2.637 0.359 2.737 0.090 0.893 0.393 39 0.309 0.628 0.101 2.503 0.394 2.847 0.049 – 0.743 40 0.158 0.729 0.021 2.833 0.451 3.692 0.013 – 0.368 41 0.061 0.889 0.012 2.682 0.465 3.603 0.028 0.281 0.767 42 0.088 0.667 0.012 2.450 0.295 2.090 0.041 – 0.564 Notes The Chinese industries and associated codes are classified as follows: processing of foods (13), manufacture of foods (14), beverages (15), textiles (17), apparel (18), leather (19), timber (20), furniture (21), paper (22), printing (23), articles for culture and sports (24), petroleum (25), raw chemicals (26), medicines (27), chemical fibres (28), rubber (29), plastics (30), non‐metallic minerals (31), smelting of ferrous metals (32), smelting of non‐ferrous metals (33), metal (34), general machinery (35), special machinery (36), transport equipment (37), electrical machinery (39), communication equipment (40), measuring instruments (41) and manufacture of artwork (42). I do not report the standard errors for each coefficient in first three columns to save space, which are available upon request. In all estimates, I include a one‐period lag of capital, labour and materials. I also include a pure assembly dummy and its interaction with both current period and a one‐period lag of capital, labour and materials. After obtaining system‐GMM TFP in column (4), I compute the standard deviation of system‐GMM TFP both across firms within an industry and across all firms, divide the industrial average to total average and multiply TFP in column (4) to obtain the weighted TFP in column (5). Numbers are p‐values in Columns (6)‐(8), which report various tests for the system‐GMM TFP estimates. Open in new tab Table C1 Estimates of System‐GMM Firm TFP by Industry . Estimated coefficients . TFP . SD . Weighted TFD . Tests (p‐value) . Labour . Materials . Capital . . . TFP . AR(1) . AR(2) . Hansen . Chinese industry . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . 13 0.094 0.718 0.010 2.575 0.387 2.884 0.000 0.987 0.443 14 0.089 0.828 0.003 2.528 0.380 2.776 0.000 0.396 0.603 15 0.077 0.677 0.152 2.677 0.465 3.599 0.063 0.724 1.00 17 0.065 0.748 0.002 2.523 0.298 2.175 0.007 0.389 0.569 18 0.068 0.724 0.020 2.447 0.326 2.303 0.000 0.317 0.834 19 0.050 0.868 0.029 2.488 0.323 2.320 0.015 0.858 0.676 20 0.015 0.844 0.010 2.851 0.412 3.398 0.011 0.510 0.548 21 0.114 0.795 0.001 2.650 0.309 2.367 0.000 0.051 0.808 22 0.151 0.655 0.011 2.705 0.338 2.644 0.424 0.570 1.00 23 0.178 0.474 0.051 2.618 0.341 2.578 0.036 0.059 0.846 24 0.098 0.609 0.058 2.485 0.281 2.018 0.030 0.411 0.990 25 0.017 0.700 0.173 2.865 0.498 4.127 0.156 0.744 1.00 26 0.142 0.701 0.034 2.669 0.353 2.721 0.000 0.868 0.222 27 0.014 0.748 0.054 2.764 0.350 2.797 0.008 0.988 0.712 28 0.052 0.812 0.088 2.674 0.326 2.520 0.082 0.280 1.00 29 0.165 0.633 0.025 2.593 0.348 2.606 0.015 0.691 0.899 30 0.128 0.865 0.022 2.690 0.335 2.605 0.000 0.303 0.371 31 0.105 0.769 0.019 2.626 0.343 2.600 0.000 0.936 0.034 32 0.001 0.876 0.001 2.864 0.388 3.212 0.060 0.233 0.909 33 0.068 0.805 0.057 2.592 0.386 2.888 0.914 0.682 0.896 34 0.022 0.840 0.021 2.480 0.318 2.279 0.009 0.161 0.788 35 0.108 0.782 0.003 2.527 0.313 2.286 0.000 0.473 0.726 36 0.091 0.719 0.089 2.604 0.356 2.681 0.000 0.845 0.537 37 0.103 0.813 0.034 2.637 0.359 2.737 0.090 0.893 0.393 39 0.309 0.628 0.101 2.503 0.394 2.847 0.049 – 0.743 40 0.158 0.729 0.021 2.833 0.451 3.692 0.013 – 0.368 41 0.061 0.889 0.012 2.682 0.465 3.603 0.028 0.281 0.767 42 0.088 0.667 0.012 2.450 0.295 2.090 0.041 – 0.564 . Estimated coefficients . TFP . SD . Weighted TFD . Tests (p‐value) . Labour . Materials . Capital . . . TFP . AR(1) . AR(2) . Hansen . Chinese industry . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . 13 0.094 0.718 0.010 2.575 0.387 2.884 0.000 0.987 0.443 14 0.089 0.828 0.003 2.528 0.380 2.776 0.000 0.396 0.603 15 0.077 0.677 0.152 2.677 0.465 3.599 0.063 0.724 1.00 17 0.065 0.748 0.002 2.523 0.298 2.175 0.007 0.389 0.569 18 0.068 0.724 0.020 2.447 0.326 2.303 0.000 0.317 0.834 19 0.050 0.868 0.029 2.488 0.323 2.320 0.015 0.858 0.676 20 0.015 0.844 0.010 2.851 0.412 3.398 0.011 0.510 0.548 21 0.114 0.795 0.001 2.650 0.309 2.367 0.000 0.051 0.808 22 0.151 0.655 0.011 2.705 0.338 2.644 0.424 0.570 1.00 23 0.178 0.474 0.051 2.618 0.341 2.578 0.036 0.059 0.846 24 0.098 0.609 0.058 2.485 0.281 2.018 0.030 0.411 0.990 25 0.017 0.700 0.173 2.865 0.498 4.127 0.156 0.744 1.00 26 0.142 0.701 0.034 2.669 0.353 2.721 0.000 0.868 0.222 27 0.014 0.748 0.054 2.764 0.350 2.797 0.008 0.988 0.712 28 0.052 0.812 0.088 2.674 0.326 2.520 0.082 0.280 1.00 29 0.165 0.633 0.025 2.593 0.348 2.606 0.015 0.691 0.899 30 0.128 0.865 0.022 2.690 0.335 2.605 0.000 0.303 0.371 31 0.105 0.769 0.019 2.626 0.343 2.600 0.000 0.936 0.034 32 0.001 0.876 0.001 2.864 0.388 3.212 0.060 0.233 0.909 33 0.068 0.805 0.057 2.592 0.386 2.888 0.914 0.682 0.896 34 0.022 0.840 0.021 2.480 0.318 2.279 0.009 0.161 0.788 35 0.108 0.782 0.003 2.527 0.313 2.286 0.000 0.473 0.726 36 0.091 0.719 0.089 2.604 0.356 2.681 0.000 0.845 0.537 37 0.103 0.813 0.034 2.637 0.359 2.737 0.090 0.893 0.393 39 0.309 0.628 0.101 2.503 0.394 2.847 0.049 – 0.743 40 0.158 0.729 0.021 2.833 0.451 3.692 0.013 – 0.368 41 0.061 0.889 0.012 2.682 0.465 3.603 0.028 0.281 0.767 42 0.088 0.667 0.012 2.450 0.295 2.090 0.041 – 0.564 Notes The Chinese industries and associated codes are classified as follows: processing of foods (13), manufacture of foods (14), beverages (15), textiles (17), apparel (18), leather (19), timber (20), furniture (21), paper (22), printing (23), articles for culture and sports (24), petroleum (25), raw chemicals (26), medicines (27), chemical fibres (28), rubber (29), plastics (30), non‐metallic minerals (31), smelting of ferrous metals (32), smelting of non‐ferrous metals (33), metal (34), general machinery (35), special machinery (36), transport equipment (37), electrical machinery (39), communication equipment (40), measuring instruments (41) and manufacture of artwork (42). I do not report the standard errors for each coefficient in first three columns to save space, which are available upon request. In all estimates, I include a one‐period lag of capital, labour and materials. I also include a pure assembly dummy and its interaction with both current period and a one‐period lag of capital, labour and materials. After obtaining system‐GMM TFP in column (4), I compute the standard deviation of system‐GMM TFP both across firms within an industry and across all firms, divide the industrial average to total average and multiply TFP in column (4) to obtain the weighted TFP in column (5). Numbers are p‐values in Columns (6)‐(8), which report various tests for the system‐GMM TFP estimates. Open in new tab Table C2 Additional One‐step GMM Estimation with Tariffs and Production Functions Regressand: Log of output (lnyit) . (1) . (2) . (3) . (4) . Firm output tariffs −3.272** −3.044** −2.389* −2.726** (−2.15) (−2.11) (−1.85) (−2.03) Firm output tariffs × fitted extent of processing 5.350** 5.012** 3.837 4.408* (2.03) (2.03) (1.63) (1.87) Firm input tariffs −2.700*** −2.707*** −2.121** −2.453** (−2.83) (−2.78) (−2.43) (−2.57) Firm input tariffs × fitted extent of processing 6.408*** 6.035*** 4.212** 4.826** (3.06) (3.02) (2.29) (2.19) Extent of processing −1.062** −1.055** −0.749* −0.933* (−2.03) (−2.11) (−1.65) (−1.96) Log of output at one lag (lnyit−1) 0.376*** 0.357*** 0.414*** 0.358*** (2.90) (2.81) (3.31) (2.80) Log of materials (lnMit) 0.553*** 0.565*** 0.563*** 0.578*** (15.79) (14.60) (15.28) (13.91) Log of materials at one lag (lnMit−1) −0.147 −0.137 −0.161* −0.128 (−1.62) (−1.50) (−1.86) (−1.44) Log of labour (lnLit) 0.145*** 0.145*** 0.130*** 0.129*** (9.19) (8.44) (7.75) (6.75) Log of labour at one lag (lnLit−1) −0.016 −0.014 −0.028 −0.013 (−0.43) (−0.41) (−0.89) (−0.39) Log of capital (lnKit) 0.069*** 0.066*** 0.071*** 0.065*** (5.13) (4.22) (4.95) (3.75) Log of capital at one lag (lnKit−1) −0.003 −0.002 −0.010 −0.007 (−0.36) (−0.26) (−1.06) (−0.70) SOE indicator −0.171*** −0.183*** −0.143*** −0.171*** (−3.12) (−3.15) (−2.88) (−2.95) Foreign ownership indicator 0.113 0.117* 0.082 0.109* (1.62) (1.73) (1.38) (1.73) Year‐specific fixed effects Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Pure domestic firms dropped No Yes No Yes Pure exporting firms dropped No No Yes Yes Observations 15,308 13,675 13,383 11,750 Regressand: Log of output (lnyit) . (1) . (2) . (3) . (4) . Firm output tariffs −3.272** −3.044** −2.389* −2.726** (−2.15) (−2.11) (−1.85) (−2.03) Firm output tariffs × fitted extent of processing 5.350** 5.012** 3.837 4.408* (2.03) (2.03) (1.63) (1.87) Firm input tariffs −2.700*** −2.707*** −2.121** −2.453** (−2.83) (−2.78) (−2.43) (−2.57) Firm input tariffs × fitted extent of processing 6.408*** 6.035*** 4.212** 4.826** (3.06) (3.02) (2.29) (2.19) Extent of processing −1.062** −1.055** −0.749* −0.933* (−2.03) (−2.11) (−1.65) (−1.96) Log of output at one lag (lnyit−1) 0.376*** 0.357*** 0.414*** 0.358*** (2.90) (2.81) (3.31) (2.80) Log of materials (lnMit) 0.553*** 0.565*** 0.563*** 0.578*** (15.79) (14.60) (15.28) (13.91) Log of materials at one lag (lnMit−1) −0.147 −0.137 −0.161* −0.128 (−1.62) (−1.50) (−1.86) (−1.44) Log of labour (lnLit) 0.145*** 0.145*** 0.130*** 0.129*** (9.19) (8.44) (7.75) (6.75) Log of labour at one lag (lnLit−1) −0.016 −0.014 −0.028 −0.013 (−0.43) (−0.41) (−0.89) (−0.39) Log of capital (lnKit) 0.069*** 0.066*** 0.071*** 0.065*** (5.13) (4.22) (4.95) (3.75) Log of capital at one lag (lnKit−1) −0.003 −0.002 −0.010 −0.007 (−0.36) (−0.26) (−1.06) (−0.70) SOE indicator −0.171*** −0.183*** −0.143*** −0.171*** (−3.12) (−3.15) (−2.88) (−2.95) Foreign ownership indicator 0.113 0.117* 0.082 0.109* (1.62) (1.73) (1.38) (1.73) Year‐specific fixed effects Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Pure domestic firms dropped No Yes No Yes Pure exporting firms dropped No No Yes Yes Observations 15,308 13,675 13,383 11,750 Notes This Table reports the one‐step system‐GMM dynamic panel‐data estimation. t‐values in parentheses are obtained using bootstrapped standard errors, corrected for clustering at the firm level. Significant at *10%, **5% and ***1%. Year‐specific fixed effects and industry‐level fixed effects are included. Column (1) includes the whole sample. Column (2) drops pure domestic firms. Column (3) drops pure exporting firms. Column (4) drops both pure domestic firms and pure exporting firms. As in Table 11, firm output (input) tariffs with initial time‐invariant weight and one‐period lag of tariffs are used as instruments for firm output (input) tariffs with initial time‐invariant weight. Similarly, the interactions between fitted extent of processing obtained from the second‐step Heckman estimates in Table 10 and firm output (input) tariffs with initial time‐invariant weight and one‐period lag of tariffs are used as instruments for the interaction between fitted extent of processing and firm output (input) tariffs. Open in new tab Table C2 Additional One‐step GMM Estimation with Tariffs and Production Functions Regressand: Log of output (lnyit) . (1) . (2) . (3) . (4) . Firm output tariffs −3.272** −3.044** −2.389* −2.726** (−2.15) (−2.11) (−1.85) (−2.03) Firm output tariffs × fitted extent of processing 5.350** 5.012** 3.837 4.408* (2.03) (2.03) (1.63) (1.87) Firm input tariffs −2.700*** −2.707*** −2.121** −2.453** (−2.83) (−2.78) (−2.43) (−2.57) Firm input tariffs × fitted extent of processing 6.408*** 6.035*** 4.212** 4.826** (3.06) (3.02) (2.29) (2.19) Extent of processing −1.062** −1.055** −0.749* −0.933* (−2.03) (−2.11) (−1.65) (−1.96) Log of output at one lag (lnyit−1) 0.376*** 0.357*** 0.414*** 0.358*** (2.90) (2.81) (3.31) (2.80) Log of materials (lnMit) 0.553*** 0.565*** 0.563*** 0.578*** (15.79) (14.60) (15.28) (13.91) Log of materials at one lag (lnMit−1) −0.147 −0.137 −0.161* −0.128 (−1.62) (−1.50) (−1.86) (−1.44) Log of labour (lnLit) 0.145*** 0.145*** 0.130*** 0.129*** (9.19) (8.44) (7.75) (6.75) Log of labour at one lag (lnLit−1) −0.016 −0.014 −0.028 −0.013 (−0.43) (−0.41) (−0.89) (−0.39) Log of capital (lnKit) 0.069*** 0.066*** 0.071*** 0.065*** (5.13) (4.22) (4.95) (3.75) Log of capital at one lag (lnKit−1) −0.003 −0.002 −0.010 −0.007 (−0.36) (−0.26) (−1.06) (−0.70) SOE indicator −0.171*** −0.183*** −0.143*** −0.171*** (−3.12) (−3.15) (−2.88) (−2.95) Foreign ownership indicator 0.113 0.117* 0.082 0.109* (1.62) (1.73) (1.38) (1.73) Year‐specific fixed effects Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Pure domestic firms dropped No Yes No Yes Pure exporting firms dropped No No Yes Yes Observations 15,308 13,675 13,383 11,750 Regressand: Log of output (lnyit) . (1) . (2) . (3) . (4) . Firm output tariffs −3.272** −3.044** −2.389* −2.726** (−2.15) (−2.11) (−1.85) (−2.03) Firm output tariffs × fitted extent of processing 5.350** 5.012** 3.837 4.408* (2.03) (2.03) (1.63) (1.87) Firm input tariffs −2.700*** −2.707*** −2.121** −2.453** (−2.83) (−2.78) (−2.43) (−2.57) Firm input tariffs × fitted extent of processing 6.408*** 6.035*** 4.212** 4.826** (3.06) (3.02) (2.29) (2.19) Extent of processing −1.062** −1.055** −0.749* −0.933* (−2.03) (−2.11) (−1.65) (−1.96) Log of output at one lag (lnyit−1) 0.376*** 0.357*** 0.414*** 0.358*** (2.90) (2.81) (3.31) (2.80) Log of materials (lnMit) 0.553*** 0.565*** 0.563*** 0.578*** (15.79) (14.60) (15.28) (13.91) Log of materials at one lag (lnMit−1) −0.147 −0.137 −0.161* −0.128 (−1.62) (−1.50) (−1.86) (−1.44) Log of labour (lnLit) 0.145*** 0.145*** 0.130*** 0.129*** (9.19) (8.44) (7.75) (6.75) Log of labour at one lag (lnLit−1) −0.016 −0.014 −0.028 −0.013 (−0.43) (−0.41) (−0.89) (−0.39) Log of capital (lnKit) 0.069*** 0.066*** 0.071*** 0.065*** (5.13) (4.22) (4.95) (3.75) Log of capital at one lag (lnKit−1) −0.003 −0.002 −0.010 −0.007 (−0.36) (−0.26) (−1.06) (−0.70) SOE indicator −0.171*** −0.183*** −0.143*** −0.171*** (−3.12) (−3.15) (−2.88) (−2.95) Foreign ownership indicator 0.113 0.117* 0.082 0.109* (1.62) (1.73) (1.38) (1.73) Year‐specific fixed effects Yes Yes Yes Yes Industry‐specific fixed effects Yes Yes Yes Yes Pure domestic firms dropped No Yes No Yes Pure exporting firms dropped No No Yes Yes Observations 15,308 13,675 13,383 11,750 Notes This Table reports the one‐step system‐GMM dynamic panel‐data estimation. t‐values in parentheses are obtained using bootstrapped standard errors, corrected for clustering at the firm level. Significant at *10%, **5% and ***1%. Year‐specific fixed effects and industry‐level fixed effects are included. Column (1) includes the whole sample. Column (2) drops pure domestic firms. Column (3) drops pure exporting firms. Column (4) drops both pure domestic firms and pure exporting firms. As in Table 11, firm output (input) tariffs with initial time‐invariant weight and one‐period lag of tariffs are used as instruments for firm output (input) tariffs with initial time‐invariant weight. Similarly, the interactions between fitted extent of processing obtained from the second‐step Heckman estimates in Table 10 and firm output (input) tariffs with initial time‐invariant weight and one‐period lag of tariffs are used as instruments for the interaction between fitted extent of processing and firm output (input) tariffs. Open in new tab Finally, by definition, each delivery of firm k to firm i is also the intermediate input for firm i. That is, Yki = Mik. Or equivalently, d lnYki/dt = dMik/dt. Then I have: ∑i∑kPiYkiPFAFAdlnYkidt=∑k∑iPiMikPFAFAdlnMikdt.(C.10) Table C3 Transitional Probability for State‐owned Enterprises (SOEs) Probability (%) . Next period . Current period . SOEs . Non‐SOEs . Total . SOEs 99.87 0.13 100 Non‐SOEs 13.01 86.99 100 Total 98.21 1.79 100 Probability (%) . Next period . Current period . SOEs . Non‐SOEs . Total . SOEs 99.87 0.13 100 Non‐SOEs 13.01 86.99 100 Total 98.21 1.79 100 Open in new tab Table C3 Transitional Probability for State‐owned Enterprises (SOEs) Probability (%) . Next period . Current period . SOEs . Non‐SOEs . Total . SOEs 99.87 0.13 100 Non‐SOEs 13.01 86.99 100 Total 98.21 1.79 100 Probability (%) . Next period . Current period . SOEs . Non‐SOEs . Total . SOEs 99.87 0.13 100 Non‐SOEs 13.01 86.99 100 Total 98.21 1.79 100 Open in new tab Table C4 Transitional Probability for Foreign Firms Probability (%) . Next period . Current period . Foreign firms . Non‐foreign firms . Total . Foreign firms 98.32 1.62 100 Non‐foreign firms 0.96 99.04 100 Total 38.22 61.78 100 Probability (%) . Next period . Current period . Foreign firms . Non‐foreign firms . Total . Foreign firms 98.32 1.62 100 Non‐foreign firms 0.96 99.04 100 Total 38.22 61.78 100 Open in new tab Table C4 Transitional Probability for Foreign Firms Probability (%) . Next period . Current period . Foreign firms . Non‐foreign firms . Total . Foreign firms 98.32 1.62 100 Non‐foreign firms 0.96 99.04 100 Total 38.22 61.78 100 Probability (%) . Next period . Current period . Foreign firms . Non‐foreign firms . Total . Foreign firms 98.32 1.62 100 Non‐foreign firms 0.96 99.04 100 Total 38.22 61.78 100 Open in new tab Table C5 Transitional Probability for Processing Firms Probability (%) . Next period . Current period . Non‐processing . Processing . Total . Non‐processing firms 85.90 14.10 100 Processing firms 34.14 65.86 100 Total 69.11 30.89 100 Probability (%) . Next period . Current period . Non‐processing . Processing . Total . Non‐processing firms 85.90 14.10 100 Processing firms 34.14 65.86 100 Total 69.11 30.89 100 Open in new tab Table C5 Transitional Probability for Processing Firms Probability (%) . Next period . Current period . Non‐processing . Processing . Total . Non‐processing firms 85.90 14.10 100 Processing firms 34.14 65.86 100 Total 69.11 30.89 100 Probability (%) . Next period . Current period . Non‐processing . Processing . Total . Non‐processing firms 85.90 14.10 100 Processing firms 34.14 65.86 100 Total 69.11 30.89 100 Open in new tab The aggregated productivity measure can be readily obtained by inserting (C.10) into (C.9): dlnπdt=∑iPiYiPFAFAdlnYidt−PiMiPiYidlnMidt−PNiNiPiYidlnNidt−PIMiIMiPiYidlnIMidt.(C.11) All terms in the parentheses of (C.11) are the change in firm productivity, as seen from (C.6). Therefore, I have: dlnπdt=∑iPiYiPFAFAdlnπidt.(C.12) That is, the economy‐wide productivity change can be represented as a weighted sum of firm productivity change in which the weight is calculated by the firm's gross output value divided by the economy‐wide total absorption (i.e. total gross output minus total trade surplus in an open economy like China). As this is initiated by Domar (1961), I hence call (C.12) the Domar‐weight aggregated productivity (Tables C1–C5). Footnotes 1 " Brandt et al. (2012) is an outstanding exception. 2 " The studies focusing on developed countries, among others, include Bernard et al. (2003) for the US and Trefler (2004) for Canada. However, more evidence has been found for developing countries, such as Bustos (2011) for Argentina, Schor (2004) for Brazil, Pavcnik (2002) for Chile, Fernandes (2007) for Colombia, Harrison (1994) for Côte d'Ivoire, Krishna and Mitra (1999) and Topalova and Khandelwal (2011) for India, Amiti and Konings (2007) for Indonesia and Levinsohn (1993) for Turkey. Other research, such as that of Lu et al. (2010), Lu (2011) and Ma et al. (2011), also explores the nexus between export growth and productivity improvement in China. 3 " Such types of processing trade include, among others, foreign aid (code: 12), compensation trade (13), assembly (14), processing with inputs (15), goods on consignment (16), goods on lease (17), border trade (19), contracting projects (20), outward processing (22), barter trade (30), customs warehouse trade (33) and entrepôt trade by bonded area (34). 4 " Processing with assembly is also referred to as ‘processing with supplied materials’, as stated in the official customs reports, or ‘pure assembly’ as adopted in Feenstra and Hanson (2005). Correspondingly, processing with inputs is also referred to as ‘processing with imported materials’ or ‘input and assembly’. 5 " Different from processing importers, non‐processing importers have to pay import tariffs for their imported intermediate inputs, although such imported goods are possibly used as inputs to produce final exportable goods. The key difference is that non‐processing firms cannot show processing contracts/licences to the customs to enjoy the privilege of free duty. 6 " The data are from WTO webpage http://tariffdata.wto.org/ReportersAndProducts.aspx. Note that TRAINS data generally suffer from missing values, particularly regarding the tariffs imposed by other countries for Chinese exports. The product‐destination‐year combinations that have missing tariffs are hence dropped. All data sets and programmes that allow the replication of the results in the article are available online. 7 " Aggregated data on the industrial sector in the annual China's Statistical Yearbook by the NBS are compiled from this data set. 8 " For example, information on some family‐based firms, which usually have no formal accounting system in place, is based on a unit of one RMB, whereas the official requirement is a unit of RMB 1,000. 9 " Note that in the firm‐level production data, a firm's sales to trade intermediaries are accounted for as domestic sales but not exports, following the requirement of the GAAP. 10 " In China, pure trading companies are required to register with a name containing Chinese characters for ‘trading company’ or ‘importing and exporting company’. 11 " In particular, the firm's codes in the product‐level trade data are at the ten‐digit level, whereas those in the firm‐level production data are at the nine‐digit level, with no common elements inside. 12 " The year variable is necessary as an auxiliary identification variable as some firms could change their name in different years and newcomers could possibly take their original name. 13 " For example, the phone numbers in the product‐level trade data include both area phone codes and a hyphen, whereas those in the firm‐level production data do not. 14 " Note that in the merged sample shown in column (7) of Appendix Table A1, exports for some firms reported from the customs trade data set are larger than total sales reported from the NBS production data set. I also drop such firms from the sample in column (8) of Appendix Table A1 to guarantee the quality of my merged data set. 15 " Note that the percentages for ordinary importing firms and processing firms in Table 2 are different from the import volumes for ordinary imports and processing imports shown in Table 1, as a processing importing firm (except pure processing firms) usually also has both processing imports and ordinary imports. 16 " Around 60% of firms are exporters whereas the other 40% are importers. The merged sample also includes entry and exit of firms. The last paragraph of Appendix A provides more detailed descriptions on this. 17 " The customs trade data provide information on unit‐value, which could serve as a proxy for the price for each imported good. However, the prices of imported intermediate inputs could be much different from those of domestic intermediate inputs (Helpern et al., 2010). Using the imported intermediate inputs as a proxy for all intermediate inputs may generate another unnecessary estimation bias. This bias may be exaggerated when the scope of domestic inputs is much different from the scope of foreign inputs. 18 " As in Brandt et al. (2012), the output deflators are constructed using ‘reference price’ information from China's Statistical Yearbooks, whereas input deflators are constructed based on output deflators and China's national input–output table (2002). 19 " By the official definition reported in the China City Statistical Yearbook (2006), SOEs include firms such as domestic SOEs (code: 110), state‐owned joint venture enterprises (141), and state‐owned and collective joint venture enterprises (143) but exclude state‐owned limited corporations (151). Appendix Table C3 presents the transitional probability for all SOEs. 20 " As discussed by Blundell and Bond (1998), even if transient measurement error exists in some of the series (i.e. ωit∼MA(1)), the system‐GMM approach can still provide consistent estimates of the coefficients in (2). 21 " Appendix Table C1 reports the associated specification tests for system‐GMM estimates including AR(1) and AR(2) tests and Hansen over‐identification tests. For most Chinese two‐digit level industries, the system‐GMM estimates have first‐order serial autocorrelation but not second‐order serial autocorrelation. The Hansen over‐identification tests also suggest that the instruments are valid for most industries. 22 " Alternatively, the weighted output tariff index can be written as FOTit=∑k[vi,initial_yeark/(∑kvi,initial_yeark)]τtk and the domestic value of product k for firm i is vi,initial_yeark=(Xii,initial_yearkk/∑kXi,i,initial_yearkk)(Yi−∑kXi,i,initial_yearkk), where Yi is firm i's total sales in its initial year. Therefore, the difference enclosed by the second parentheses measures firm i's total domestic sales. 23 " Besides, when firms sell in both the domestic and export markets, the quality of the products is likely to be different, with better quality products sold to the export markets. As data on unit‐price, a common proxy of product quality, are unavailable for domestic products, here I am not able to distinguish the quality difference between domestic products and exportable products, which is a future research topic once data are available. I thank a referee for correctly pointing this out. 24 " The reason for not using weighted import tariffs, again, is to avoid the endogeneity of tariffs: imports are negatively correlated to tariffs. 25 " China's input–output table is compiled every five years; the most recent updates were in 2007. As my data sample is between 2000 and 2006, I adopt the input–output table from 2002. In particular, I proceed with the following steps to calculate the industry‐specific tariffs. As there are 71 manufacturing sectors reported in China's input–output table (2002) and only 40 manufacturing sectors reported in the CIC, the first step is to find the correspondence between sectors in the input–output table and the CIC. The second step matches the CIC sectors with the International Standard Industrial Classification (ISIC, rev. 3). Note that China's government adjusted its CIC in 2003. I make the same adjustment in the sample. The third step is to link the ISIC and the HS six‐digit classification to find the corresponding tariffs from the WTO. The final step calculates the average industry‐level tariffs, which are aggregated to the CIC sector level. 26 " For example, if firm i in industry f uses 50% lumber with 1% tariffs and 50% steel with 10% tariffs, then the firm‐specific input tariff is 5.5%. However, if industry f uses more domestic lumber, the industrial weight of lumber increases to 70%. Accordingly, the industry‐specific input tariffs are reduced to 0.7 × 1% + 0.3 × 10% = 3.7%, which is significantly lower than its counterpart of firm‐specific input tariffs. 27 " Specifically, foreign‐invested enterprises (FIEs) include the following firms: foreign‐invested joint‐stock corporations (code: 310), foreign‐invested joint venture enterprises (320), fully FIEs (330), foreign‐invested limited corporations (340), Hong Kong/Macao/Taiwan (henceforth, H/M/T) joint‐stock corporations (210), H/M/T joint venture enterprises (220), fully H/M/T‐invested enterprises (230) and H/M/T‐invested limited corporations (240). Appendix Table C4 presents the transitional probability for such foreign firms. 28 " At first glance, these ratios are significantly higher than their counterparts reported in other studies, such as Feenstra et al. (2013a). However, this finding simply reflects the fact that the present article covers only large trading firms. Large, non‐trading firms have been excluded. 29 " Feenstra et al. (2013a) also apply this method to estimate the impact of credit constraints on firm's exports. 30 " Similar to Heckman and Vytlacil (1998), the conditional homoscedasticity of covariance assumption for the term εitμit is needed to ensure that it would not bias the estimates. 31 " As in common, the R2 in all estimates with firm‐specific and year‐specific fixed effects in the artcle is exclusive of both firm‐specific and year‐specific dummies. 32 " Around 40% of the observations are missing investment data. 33 " Appendix Table C5 also reports the transitional probability for processing firms. The switching of processing firms is an interesting topic for future research, although it is beyond the scope of the present article. 34 " Note that even when the firm's age is included, its coefficient in the second‐step Heckman estimate is also statistically insignificant. 35 " Accordingly, the interaction between the firm's input and output one‐period tariff with the time‐invariant weight and the fitted extent of processing trade are adopted as additional instruments in all IV estimates. 36 " Note that adopting firm‐specific fixed effects here would cause a huge loss of observations as most of the firms do not have a continuous panel in the sample. Such a pattern is more pronounced in the 2SLS estimates when using the one‐year lagged tariffs as instruments. I therefore include the disaggregated three‐digit CIC industry‐specific fixed effects and year‐specific fixed effects in all 2SLS estimates. 37 " Note that the Cragg and Donald (1993) F‐statistic is no longer valid because it only works under the i.i.d. assumption. As here I have four (more than three) endogenous variables, STATA does not report the critical values for the Kleibergen–Paap (2006) weak instruments test. In this case, Baum et al. (2007) suggest that one can safely adopt 10 as a critical value as initiated by Staiger and Stock (1997). As all my Kleibergen–Paap (2006) F‐statistics are one‐order much higher than 10, it is safe to reject the null hypothesis of weak instruments in all estimates. 38 " There are in fact four steps to my estimation: the selection (11); the second‐step Heckman equation used to obtain the predicted extent of processing; the first‐step of 2SLS where the predicted extent of processing is a regressor; and the second‐step of 2SLS estimates. Panel bootstrapping by randomly drawing firms is done in the last two steps. 39 " Note that all the main findings are not changed if firm external tariffs are measured using time‐invariant export weights. The reason for choosing a time‐variant export weight is to allow a dynamic response of the firm's exports to a reduction in foreign tariffs. 40 " See columns (5) and (6) of Appendix Table C1. I thank a referee for suggesting this point. 41 " Using the log of firm output as the regressand, both the current period and a one‐period lag realisation of firm inputs – labour, capital and materials – are included as regressors. Simultaneously, firm output and input tariffs based on time‐invariant weights, the extent of processing imports and its interaction with tariffs are included as another set of regressors. To control for possible endogeneity, I adopt a one‐period lag of firm output (input) tariffs with time‐invariant weights as instruments as before. Appendix Table C2 reports the 2SLS fixed‐effects estimates using the one‐step system‐GMM approach. All estimation results are highly consistent with the previous findings: the impact of tariff reductions on productivity improvement shrinks as the firm's processing imports grow. Overall, firm output tariff reduction leads to stronger productivity gains than firm input tariff reductions. 42 " Besides variety, Amiti and Konings (2007) highlight two other possible channels through which cheaper imported inputs can raise productivity: learning and quality effects. 43 " In particular, R&D in 2004 is completely missing. Moreover, around 50% of firms report negative or zero R&D expenses in my sample. 44 " My estimates are also close to other studies such as Amiti and Konings (2007), who find that a 10 percentage point fall in output (input) tariffs leads to a productivity gain of 6.4 (12.7)% using data on Indonesian firms. 45 " It is also interesting to check the productivity gains from tariff reductions for pure processing firms, for which the ratio of processing imports to total imports equals one. As firm input tariffs for pure processing firms reduce to zero, given that processing imports are duty‐free, one cannot directly calculate such productivity gains from column (1) of Table 11. However, as the impact of the input tariff reductions is given by −0.0171+0.0246×E(Pextit|Zit) ⁠, by using a sufficiently high value for the extent of processing (e.g. the 90th percentile of E(Pextit|Zit)=0.69 ⁠) as a proxy of pure processing firms, the impact of input tariff reductions is close to zero, confirming that heavy processing firms rarely gain from input tariff reductions. 46 " For example, if TFP growth for both shoe and rubber firms is 1%, the simple average of such firms' TFP growth will be 1%. However, productivity growth of the integrated rubber and shoe industry will be more than 1%, as the shoe firms' productivity gains cumulate with those of the rubber firms as the latter sells inputs to the former. 47 " To calculate Domar‐weight TFP, the Domar weight is multiplied by four since the gross output of my merged sample only accounts for a quarter of total gross output in the full‐sample data set, as shown in Table 3. 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Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Author notes " I thank Robert Feenstra, Gordon Hanson, Samuel Kortum, Kala Krishna, Zhiyuan Li, Justin Lin, Devashish Mitra, Larry Qiu, Jose Antonio Rodriguez Lopez, Robert Staiger, Joaquim Silvestre, Wei Tian, Heiwai Tang, Yang Yao and Zhihong Yu for their very helpful comments and constructive suggestions. Financial support from China's Natural Science Foundation Grant (No. 71003010) is gratefully acknowledged. I thank the editor, Rachel Griffith, and two anonymous referees for their very insightful suggestions. All errors are mine. © 2014 Royal Economic Society

Wang,, Tianxi

doi: 10.1111/ecoj.12104pmid: N/A

Abstract This study examines the causal effects of bank size on banks' survival, asset quality and leverage. Two forces drive these effects: increasing returns to scale derived from banks' expertise and competition. The first enables bigger banks to survive competition better, have higher asset quality and be more leveraged. It drives banks into a race for expansion. This race toughens competition between banks, which edges out small banks and may worsen all banks' asset quality. Consequently, the banking industry will be dominated by a small number of highly leveraged banks. In this study, financial intermediation arises endogenously and co‐exists with direct finance. Why is the banking industry dominated by a few ‘too big to fail’ institutions? As a step towards answering the question, this study considers the economic implications of bank size, which, unlike bank capitalisation, receives little attention in the academic literature. Specifically, the study examines the causal effects of bank size on banks' survival in a competitive market and on their asset quality. Based on these effects, the study also considers how size affects banks' choice of leverage. This study characterises the banking industry with two assumptions. First, banks have an advantage over the general public (households) in identifying which investment projects of entrepreneurs are profitable. In this study, while households have no way to evaluate projects, banks can attain expertise to evaluate and screen them. Therefore, households invest in a project only if some bank certifies to them that it finds this project profitable. Second, banks can provide this certification not by word of mouth but by investing a sufficient quantity of their own funds in the projects. Thus, informed funds – namely, those provided by banks – earn a higher rate of return than uninformed funds – namely, those of households. These two assumptions imply that the banking industry has increasing returns to scale, which is consistent with the industry's long‐term trend of increasing concentration.1 If a bank has attained a certain level of screening expertise, it can apply this expertise to the deployment of all its funds. Thus, all its funds earn the return of informed funds. The more funds a bank has, the bigger profit it earns from its screening expertise, which induces the bank to attain a higher level of the expertise. Increasing returns to scale may, in general, result from any sort of expertise because the acquisition of expertise is in the nature of fixed costs. For example, learning a widely spoken language delivers bigger benefits than learning a narrowly spoken one; and becoming a top comedian is more profitable where the potential audience is larger.2 To exploit the increasing returns to scale, banks all want to expand. Then, competition becomes fiercer. The larger the quantity of informed funds that banks supply, the lower the return rates these funds earn. Consequently, banks all gain less from screening expertise and thus attain less of it. The two forces, increasing returns to scale and competition, are in conflict. Together they shape the implications of banks' sizes for their survival, asset quality and leverage and drive the banking industry to be dominated by a small number of highly leveraged banks. First, a bank can survive competition only if its size is above a threshold and this threshold is higher when bank finance is more abundant. This result is consistent with the previously mentioned trend of increasing concentration: the pressure to survive drives banks all to expand, which raises the survival threshold, edging smaller banks out. Second, if all banks are enlarged but their shares of the loan market remain fairly stable, then their screening expertise declines and their asset quality falls. This is because, for a bank whose market share is not much increased, the negative effects of competition dominate the positive effects of increasing returns to scale. What banks sell to entrepreneurs, essentially, is certification service rather than financing because households have abundant funds. If a bank cannot sell the service to more entrepreneurs, the scale of its business is not enlarged and the force of increasing returns to scale not unleashed. Third, there is a complementarity between leverage and screening expertise. On the one hand, the higher a bank's level of screening expertise, the greater its leverage. On the other hand, the greater leverage enlarges the scale of its funds more, which, due to increasing returns to scale, induces the bank to attain a higher level of screening expertise. Hence, bigger banks have better screening expertise and are more leveraged. This result is consistent with a trend of growing leverage in the banking industry: with small banks continuously edged out, the remaining banks become bigger and bigger and, thus, the industry‐wide leverage gets higher and higher.3 In this study, financial intermediation arises naturally and co‐exists with direct finance. Screening expertise earns banks the informed funds' rate on the asset side, while they borrow from households at the uninformed funds' rate on the liability side. The difference in rates is the profit margin of financial intermediation. Besides investing in banks, households invest directly in projects that receive bank funding as the bank funding certifies their quality. In this study, the allocation of households' funds between the intermediated and direct finance channels is uniquely determined. The financial intermediation affects the equilibrium outcome by expanding the pool of informed funds.4 The study shows that a universal increase in banks' sizes may cause their asset qualities to decline. It is thus related to the literature that explains why loose lending standards are associated with lending booms; Rajan (1994), Rucks (2004) and Dell'Ariccia and Marquez (2006). A difference is that, in this study, an increase in size drives a fall in quality, whereas in that literature, both are driven by some other factor, such as bank managers' career concerns in Rajan (1994), the distribution of borrowers' quality in Rucks (2004), or the distribution of information in Dell'Ariccia and Marquez (2006). In this study, financial intermediation arises endogenously as banks want to enlarge their scale to exploit the increasing returns to scale. This is related to the literature that endogenises financial intermediation with delegated monitoring and increasing returns to scale regarding the incentive costs of monitoring the bank.5 The source of the increasing returns to scale is different: In this study they are due to a general feature of expertise as mentioned above, while in that literature they are due to cross insurance. Moreover, direct finance and intermediated finance co‐exist in this study, whereas not in that literature. This study finds bigger banks able to grow faster, implying a trend of increasing dominance, which is extensively examined in industrial economics; Flaherty (1980), Gilbert and Newbery (1982), Budd et al. (1993) and, recently, Besanko et al. (2010) and Cabral (2011). Using the terminology of industrial economics, banks here engage in (differentiated product) Bertrand competition with limited capacity à la Kreps and Scheinkman (1983). The rest of the study is organised as follows. Section 1 sets up the basic model in which banks invest only their own funds. This model is analysed in Section 3 and extended in Section 4 to encompass bank leverage. Section 5 lays out empirical implications, whereas Section 5 discusses a modelling choice. Section 6 concludes. All proofs are relegated to the Appendix. 1. Basic Model The economy lasts for two dates (numbered 1 and 2) with no discounting and is populated by many small households, a continuum of [0, 1] of banks, and a continuum of [0, 1] × [0, 1] entrepreneurs. Banks are in perfect competition6 and each serves a continuum of entrepreneurs. All agents are risk neutral and protected by limited liability. Each entrepreneur has a project, while banks and households have funds. Funds are invested at date 1 either in projects or in a risk‐free asset for which the gross rate of return is 1. A project requires an investment of £B and may succeed, yielding £Z, or fail, yielding 0.7 The probability of success is q¯ for high‐type projects and q̲<q¯ for low types. The fraction of high types is n¯ and that of low types is n̲=1−n¯ ⁠. I assume that q¯Z−B>0>(n¯q¯+n̲q̲)Z−B.(1) That is, a high‐type project has positive social value but a randomly selected one does not. Each bank j Ɛ [0, 1] has Kj units of funds, which captures the bank's size. The unit is so defined that out of 1 unit of funds £1 can be invested in each of a continuum of mass 1 of projects. Without loss of generality, I assume that Kj is a continuous, non‐increasing function of j. In the basic model, banks are assumed to invest only their own funds, while bank borrowing will be incorporated in Section 3. Households are small but, overall, their funds are so abundant that not all these funds can be invested in entrepreneurs' projects. The remainder flows to the risk‐free asset. It follows that households are satisfied with a gross rate of return 1. Households know the prior distribution of the types of projects but have no way of evaluating and screening them. However, banks can attain expertise in evaluating and screening projects. By Assumption ((1)), only the high‐type projects yield a surplus, so that the banks' screening services create a social value. 1.1. Banks' Screening Expertise By spending C(p), a bank acquires screening expertise of accuracy p Ɛ [0, 1]. With the expertise of this accuracy, for each project it screens, the bank receives an independent signal s~=g(ood) or b(ad) about the project's type according to Pr(s~=g|q~=q¯)=1,Pr(s~=b|q~=q̲)=p,Pr(s~=g|q~=q̲)=1−p,(2) where q~ is the true probability of success of the project. That is, high‐type projects obtain a good evaluation with certainty, whereas low types receive a bad evaluation with probability p.8 I call projects that obtain a good evaluation good projects and those that receive a bad evaluation bad projects. To focus the analysis on banks, I assume that an entrepreneur does not know the type of his project before a bank evaluates it and that he observes the result of the bank's evaluation. Households never observe this evaluation. Moreover, I assume that the screening accuracy of any bank is publicly observed, so the only information asymmetry between banks and entrepreneurs on the one hand, and households on the other, is over the evaluations of projects. The cost function C(·) is convex over [0, 1] and satisfies C′′ (·) > 0, C(0) = C′ (0) = 0 and C′(1) = ∞ ⁠. Furthermore, C′(p) = o[1/(1 − p)2] around p = 1,9 which ensures that the cost does not grow too fast in p. Let qg(p) denote the posterior probability of success for a good project, qb(p) the posterior probability of success for a bad project and ng(p) the probability of obtaining a good evaluation.10 Then, Vg(p) ≔ qg(p)Z − B is the social value of a good project; S(p) ≔ ng(p)Vg(p) is the ex ante social surplus of a project if it is financed only when it obtains a good evaluation and d(p) ≔ qg(p)/qb(p) measures the difference in quality between good and bad projects. It is straightforward to show that qg′(p)>0,d′(p)>0,d(p)>1forp>0,S′(p)>0andS″(p)≤0.(3) These properties are all that is required for the analysis in this study; the specific modelling of screening accuracy (2) is adopted for its simplicity. Let p̲ be the critical level of accuracy at which the social surplus of a good project just reaches 0, namely, Vg(p̲)=0 ⁠. Then, 0 < p̲ < 1 and a good project has a positive social value if and only if p > p̲ ⁠.11 If the accuracy of evaluation is below p̲ ⁠, then even projects evaluated as being good are not financed and screening service of this accuracy is useless. A bank, if not choosing accuracy above p̲ ⁠, chooses p = 0. If a bank chooses so and thus to be uninformed, then it is identical to a household in making investments. I say that such a bank is edged out of the banking industry. If a bank opts to be informed by choosing some p > p̲ ⁠, I say that this bank stays in business and survives competition. I call funds provided by informed banks informed funds and funds provided by households (and uninformed banks) uninformed funds. As for how banks can credibly communicate their evaluations of the projects to households, I make two assumptions. Assumption 1. Banks' announcements of their evaluations of projects are not credible. This assumption means that to certify good evaluations to households, banks must ‘put their money where their mouth is’. A contract between an entrepreneur and a bank must involve the investment of the bank's funds. Such a contract is characterised by a pair (I, F), where I is the amount invested by the bank in the project and F is the face rate of return, or, simply, the face rate of this investment so that I × F is the amount to be repaid to the bank when the project succeeds; when it fails, by limited liability, no party gets anything. Assumption 2. For a contract (I, F), the amount of investment, I, is observable to households, but the face rate, F, is not. I justify this assumption on two grounds. First, in real life, the amount of investment is usually publicised or reported in the media but the terms of the investment, which F represents, are not. Second, even if the terms are publicised, they are not a reliable guide to the actual rate of return because the entrepreneur could easily sign another contract with the bank (e.g. for consulting services) to arrange a side payment to the bank and it would be too costly for households to check all the contracts between the two. Because of these two assumptions, to certify a project's quality credibly, the bank needs to invest enough of its own funds, as will be shown. This ties banks' certification service, which is what they essentially sell to entrepreneurs, to the investment of their own funds. Then, bank size matters: the more funds a bank has, the more entrepreneurs it sells the certification service to and, thus, the more profits it earns. This is how the study endogenises increasing returns to scale in the banking sector. After an entrepreneur secures £I of funds from an informed bank, he goes to the market for the funds of households. Households observe I, and based on it, infer the quality of the project. Projects that receive no bank funding are of lower than average quality, therefore do not attract household funding; their entrepreneurs withdraw from the market. 1.2. Timing of Events and Information Structure The timing of events at date 1 is as below. Stage 1: For each j Ɛ [0, 1], bank j posts (pj, Rj) – namely, the screening accuracy it attains and the expected rate of return it commits to charge for its funds. Stage 2: Entrepreneurs each go to one informed bank and get their projects evaluated. The evaluation of a project is observed by the entrepreneur and the bank but not by households. Based on the evaluations, the entrepreneurs submit a request for the bank's funds. If the bank has attracted too many entrepreneurs and the total demand for its funds is above its funding capacity, then rationing happens and only a fraction of the entrepreneurs get their requests satisfied. These entrepreneurs then sign a contract (I, F ) with the bank, where I is observed by households. Afterwards, entrepreneurs seek funds from households. If they manage to acquire £B altogether, they start their projects. At date 2, the returns of the projects are realised and distributed to investors according to the contracts signed at date 1. 2. Size, Survival and Asset Quality To find the subgame perfect equilibrium, I analyse the decisions in the following order. First, given (p, R) offered by a bank, the entrepreneurs coming to it decide their demand for the bank's funds, contingent on the evaluations of their projects. Second, given all banks' offers, {pj, Rj}j Ɛ [0,1], each entrepreneur decides to which bank he goes for a certification service. Third, anticipating these decisions by entrepreneurs, banks decide on (p, R). 2.1. Entrepreneurs' Decisions No entrepreneurs would come to an uninformed bank, funding by whom serves no certification purposes. Consider the entrepreneurs who have come to an informed bank of size K that offers (p, R), with p > p̲ and R > 1.12 Then, the demand for the bank's funds by those whose projects receive a good evaluation, henceforth called ‘good entrepreneurs’, is I(p,R)=Vg(p)Rd(p)−1(4) and the demand by bad entrepreneurs, namely, those whose projects receive a bad evaluation, is 0. This result is driven by two considerations. First, any demand of I ≥ I(p, R) of the bank's funds certifies that the entrepreneur is good. Consider a bad entrepreneur who mimics a good one by demanding the same amount of bank funding. The bank, assessing the probability of success as qb, charges him a face rate of R/qb and demands a repayment of I × R/qb when his project succeeds. If this investment I convinces households that the project is good, they are willing to finance the shortfall, B − I, at face rate 1/qg,13 for a face value of (B − I)/qg. If Vm ≔ Z − I × R/qb − (B − I)/qg < 0, the project's revenue in the case of success, Z, is insufficient to cover the liability outlay, I × R/qb + (B − I)/qg. Then, the bank expects not to be fully repaid with I × R/qb, and therefore declines to lend I to the bad entrepreneur. Hence, bank funding certifies only good evaluations if Vm < 0, or equivalently I ≥ I(p,R). Second, because bank funding is more costly than household funding, as R > 1, a good entrepreneur demands only the minimum bank funding necessary to certify the quality of his project – namely, I(p,R) – and finances the shortfall with the cheaper household funding. Consider now the aggregation of the individual demands for the bank's funds. Given an entrepreneur receives a good evaluation with probability ng(p) and the evaluations come independently, by the law of large number, the total demand by one unit of entrepreneurs is ng(p)I(p,R). The bank can thus serve β(p,R)=Kng(p)I(p,R),(5) units of entrepreneurs. If M > β(p,R) units of entrepreneurs come to the bank, rationing happens and each entrepreneur is served with probability l = β(p,R)/M. Otherwise, all the entrepreneurs are served by the bank. Hence, the probability of an entrepreneur being served is l=min1,β(p,R)M.(6) Conditional on being served, an entrepreneur obtains the difference in the social surplus of his project, S(p), minus the profit surrendered to the bank. This profit is ng(p)I(p,R)(R − 1) because the expected demand by him for the bank's funds is ng(p)I(p,R) and each pound of them earns the bank a net profit of R − 1. With I(p,R) given by (4) and S(p) = ng(p)Vg(p), the expected pay‐off of a served entrepreneur is Π(p,R)=S(p)[d(p)−1]Rd(p)R−1.(7) Now, consider entrepreneurs' decisions on which banks to borrow from, given all banks' deals, {pj, Rj}j Ɛ [0,1]. All the banks that attract entrepreneurs to come offer an incoming entrepreneur the same expected pay‐off, Π^ ⁠, because no entrepreneurs would go to a bank that offers less if they can get Π^ from some other banks. Therefore, if a bank offering (p,R) attracts M > 0 units of entrepreneurs, then lΠ(p,R)=Π^,(8) where l is given by (6). The allocation of entrepreneurs to banks, {Mj}jƐ[0,1], and Π^ are determined by (8) and the following market‐clearing condition: ∫j∈[0,1]Mj=1.(9) 2.2. Banks' Decisions: Increasing Returns to Scale and Competition Having examined entrepreneurs' decisions, I move on to find the best response of a bank given the choices of (p, R) by all the other banks. Given there is a continuum of banks, each bank has only a negligible effect on Π^, so each bank takes Π^ as given when choosing (p, R). Given Π^, a bank can choose to be uninformed and thereby get zero economic profit. If it chooses to be informed, it has to ensure that its offer, (p, R), can attract entrepreneurs, i.e. that Π(p,R)≥Π^ by (8). The bank has no incentives to make Π(p,R)>Π^ and induce excess demand.14 Therefore, at the optimum, Π(p,R)=Π^, which, together with (7), implies that after attaining accuracy p, the bank charges the following interest rate: R(p;Π^)=Π^d(p)Π^−[d(p)−1]S(p).(10) Neither does the bank induce underdemand.15 It follows that if a bank of size K chooses to be informed at accuracy p, all its funds are lent out at return rate R(p;Π^) and its value is K[R(p;Π^)−1]−C(p).(11) The bank's decision problem is to find a p Ɛ [0, 1] to maximise this value, taking Π^ as given. Let ρ(K,Π^) be the optimal p and Θ(K,Π^) the optimal value of this decision problem. Then, the bank chooses to be informed only if Θ(K,Π^)≥0 and, if it does so, it picks p=ρ(K,Π^)(12) R=R[ρ(K,Π^),Π^],(13) where R(p,Π^) is given by (10). As the bank induces neither excess demand nor under demand, it receives the following number (units) of entrepreneurs and serves them all: β(K,Π^):=β(ρ(K,Π^),R[ρ(K,Π^),Π^])(14) where β(p,R) is given by (5). The Lemma below establishes a lower bound for the equilibrium pay‐off of entrepreneurs. Lemma 1. If Π^≤[d(1)−1/d(1)]S(1), then Θ(K,Π^)=∞ for any K > 0. The lower bound for Π^ exists because of competition between banks. If Π^ were below the bound, then a particular bank would undercut all the other banks and obtain a big profit by providing a screening service of such accuracy p that the pay‐off to entrepreneurs, Π(p,R), is above Π^, namely, what they can obtain elsewhere, for even R → ∞ ⁠. By this Lemma, any function of Π^, such as Θ(K,Π^), is meaningfully defined only for Π^>[d(1)−1]S(1)/d(1) ⁠, which, therefore, is a pre‐condition for any Proposition below where Π^ is taken as given. Now I can explain the two forces that shape the economic implications of bank size: increasing returns to scale and competition. Each force presents itself at both profit and marginal profit levels. These two forces, at these two levels, are formalised as follows. Proposition 1. For (K,Π^) at which ∞>Θ(K,Π^)>0 (namely, the bank chooses to be informed), (i) ∂Θ/∂K>0,∂2Θ/∂K2>0and∂ρ/∂K>0; (ii) ∂Θ/∂Π^<0and∂ρ/∂Π^<0. Result (i) is concerned with increasing returns to scale at the two levels. First, at the level of bank profit, I have ∂Θ/∂K>0, that is, bigger banks get higher value from being informed. A bigger capacity benefits informed banks by raising both the extensive margin and profit (or intensive) margin. A bank becomes informed only if doing so enables it to charge R > 1 for its funds; thus, the more funds a bank deploys, the higher profit it earns on becoming informed. Also, the bigger the bank, the higher the profit margin, as by (10), R increases with the screening accuracy,16 which, as ∂ρ/ ∂K > 0, increases with size. Second, increasing returns to scale at the level of marginal profit drive ∂ρ/ ∂K > 0, that is, bigger banks choose higher screening accuracy. The choice of screening accuracy, ρ(K,Π^), which maximises (11), satisfies the following first‐order condition for p: KΠ^[d(p)−1]S′(p)+d′(p)[S(p)−Π^]{d(p)Π^−[d(p)−1]S(p)}2=C′(p).(15) The marginal profit from higher accuracy, which appears on the left‐hand side of the equation, is proportional to the size, K. Intuitively, screening expertise of higher accuracy enables the bank to charge a higher interest rate, which augments the bank's profit further if the bank is larger. From ∂ρ/ ∂K > 0 it follows that the larger the bank, the bigger the marginal value of size (i.e. ∂ Θ/ ∂K) because by the envelope theorem ∂Θ/∂K=R[ρ(K,Π^),Π^]−1 and thus ∂2Θ/∂K2=Rp′×∂ρ/∂K>0. Result (ii) is concerned with two effects of competition, where competition is represented by Π^ ⁠, the pay‐off a bank has to give entrepreneurs for attracting them to come. First, at the level of profit, I have ∂Θ/∂Π^<0; that is fiercer competition lowers banks' profit. This is because it forces banks to surrender more pay‐off to entrepreneurs and, thus, to obtain less from the certification service. Second, I have ∂ρ/∂Π^<0, that is, fiercer competition drives banks to lower screening accuracy. It does so by squeezing the marginal profit to a bank from an increment in screening accuracy. The increment both widens the bank's profit margin and enlarges its business scale. But the gain from both channels is abated by a higher Π^. First, screening of higher accuracy increases the social value of a project, S(p), and thereby widens the profit margin from each project screened, S(p)−Π^. The total marginal profit so generated is proportional to the number of projects screened, which decreases with Π^ ⁠: the higher the pay‐off to the entrepreneurs (Π^), the lower the rate charged (R); and the more the bank's funds demanded by each good entrepreneur (⁠ IR′<0 by (4)) and the fewer the entrepreneurs served. Second, higher screening accuracy enables the bank to serve more entrepreneurs because entrepreneurs, when more accurately evaluated as being good, need less of the bank's funds for certification (⁠ Ip′<0 by (4)). The gain to the bank from this increase in business scale is proportional to the profit margin, S(p)−Π^, which decreases with Π^. The increasing returns to scale imply that only big banks survive competition, as stated in the following Proposition. Proposition 2. Let K̲(Π^) be the largest root of Θ(K,Π^)=0 ⁠. (i) K̲(Π^) exists and is strictly positive, and Θ(K,Π^)>0 if and only if K > K. (ii) K̲′(Π^)>0. Result (i) says that a bank survives competition (i.e. Θ ≥ 0) only if its size is above a threshold (i.e. K ≥ K), whereas smaller banks are edged out. Result (ii) says that when competition becomes fiercer (i.e. a larger Π^ ⁠, the threshold of survival rises. This is because the more pay‐off needed to give entrepreneurs, the less a bank gets from serving one entrepreneur on becoming informed; to cover the cost of attaining screening expertise, therefore, the more entrepreneurs it needs to serve, which requires a larger funding capacity. The two results together suggest that the banking industry is subject to a trend of increasing concentration.17 The pressure to survive and the motive to exploit the increasing returns to scale drive banks into a race for expansion, which raises Π^ (as I show), so that all banks face even fiercer competition. Moreover, if the extent of expansion is proportional to the marginal value of size, namely, to ∂ Θ/ ∂K, then result ∂2 Θ/ ∂K2 > 0 (Proposition 1(i)) suggests that smaller banks are able to expand less, thus losing out in the race. Altogether, smaller banks are continuously edged out, leaving the banking industry more and more concentrated. Having examined the decisions of entrepreneurs and banks, I now proceed to define the equilibrium formally and examine its existence and properties. 2.3. Equilibrium Banks choose to be informed if the value from doing so, Θ, is non‐negative, which, by Proposition 2, is the case if and only if K≥K̲(Π^). As I assume Kj to be non‐increasing in j, it follows that there is a threshold t Ɛ [0, 1], such that bank j chooses to be informed if and only if j ≤ t. There are two cases for the value of t. One is t = 1, namely, all banks become informed; in this case K1≥K̲(Π^). The other is t < 1, namely, smaller banks are edged out; in this case, Kt=K̲(Π^), because Kj is assumed continuous in j. Note that in the latter case, there may be a,b Ɛ (0, 1) such that a < t < b and Ka=Kb=K̲ ⁠; that is, the marginal banks, which are of size K, play a mixed strategy, some of them being informed, the rest uninformed. As banks' decisions on whether to be informed are summarised by variable t and each of them takes Π^ as given, an equilibrium is thus represented by a pair of (t,Π^) ⁠, defined as follows. Definition 1. A pair of (t,Π^) is an equilibrium if (i) Given Π^ ⁠, Kt=K̲(Π^)ift<1orK1≥K̲(Π^)ift=1;(16) (ii) the market clears: ∫0tβ(Kj,Π^)dj=1.18(17) 18 Condition (17) is derived from (9) and the fact that no banks induce excess demand or underdemand. Once (t,Π^) is pinned down, the equilibrium decisions of banks and entrepreneurs follow straightforwardly. Bank j chooses to be informed if and only if j ≤ t; informed banks choose (p, R) as given by (12) and (13); β(K,Π^) (given by (14)) units of entrepreneurs go to an informed bank of size K and demand I(p, R) (given by (4)) of its funding when their projects receive a good evaluation and demand nothing otherwise. The equilibrium market share of bank j is thus: β^j=β(Kj,Π^)ifj≤t0ifj>t. Proposition 3. A unique equilibrium exists and has the following properties. (i) If a positive measure of informed banks increase their sizes, then the pay‐off of entrepreneurs (Π^) increases and all the other informed banks lower their screening accuracies and interest rates. (ii) If all banks increase size without changing their market shares, {β^j}j∈[0,1], then they all lower screening accuracy and interest rate. Result (i) says that expansion by some banks toughens competition (i.e. Π^ higher) and weakens their competitors, who consequently lower the quality of their screening expertise and the price of their funds. The expanded banks face the same negative effects of competition but are blessed by the increasing returns to scale, which enable them to choose greater screening accuracy (i.e. ∂ρ/ ∂K > 0) and thereby charge a higher price. For an enlarged bank, these positive effects dominate the negative effects if it expands far more than all the other expanding banks, in which case its expansion not only weakens the competitors but also strengthens itself. Therefore, not only do banks want to expand but they also want to expand far more than all the others. Result (ii) gives one condition under which this race for expansion weakens all the banks: it adds no market share to any bank. For an intuition, note that expansion always toughens competition, which tends to decrease the profit margin of providing certification service for all banks. If this decrease in profit margin is not compensated by an increase in business scale – namely, if banks cannot sell certification service to more entrepreneurs – banks gain less from screening expertise and therefore attain less of it. With all banks' screening expertise weakened, the default risks of their assets rise.19 Note that banks still invest only in projects evaluated as good but, with the evaluations becoming less accurate, the composition of these projects gets worse: a smaller fraction of them are high types, a bigger one low types. 2.4. Welfare Properties of the Equilibrium The equilibrium is ex post efficient, because all the good projects are financed and none of the bad ones is. As for ex ante efficiency, I define the first best allocation as the choice of the social planner if she can allocate entrepreneurs to banks and pick a level of accuracy for each bank and the second best allocation as the planner's choice if she has to respect the equilibrium market shares of banks but picks screening accuracy for each informed bank. The first best allocation is more efficient than the second best one, which, I show below, is more efficient than the equilibrium allocation. Suppose the planner chooses accuracy p for bank j. Then the bank's service generates a social value of S(p) from each of the .5.5β^j units of entrepreneurs whom it serves in equilibrium. Overall, thus, the bank generates a social value of .8.8β^jS(p), while the social cost of attaining the accuracy is C(p). The social planner's problem for bank j is therefore: max0≤p≤1β^jS(p)−C(p). The second best choice of quality, denoted by pj*, satisfies the following first‐order condition: β^jS′(p*)=C′(p*).(18) Proposition 4. For any informed bank j, p^j>pj*. That is, compared to the second best allocation, banks overspend on screening expertise. For an intuition, refer back to the discussion of why ∂ρ/∂Π^<0 (see Proposition 1(ii)). There I show that for a bank, higher screening quality generates two benefits: one from more entrepreneurs to be served and the other from a widened profit margin S(p)−Π^. As [S(p)−Π^]′=S′(p) ⁠, the latter benefit accrues equally to the social planner. However, the former benefit does not accrue to the social planner because the planner takes as given the number of entrepreneurs allocated to each bank. It is the motive of attracting more entrepreneurs with higher screening quality that drives banks to overspend on screening expertise. Essential to the Proposition is the model's feature that banks' funds serve mainly certification purposes and the investments of the projects are mainly financed by households. Should the household sector be absent, an entrepreneur's demand for the bank's funds would be fixed at B, the investment need. The bank would serve K/B units of entrepreneurs, independent of its screening quality. Improved screening quality would not bring more entrepreneurs to the bank, therefore, the overspending result would not arise. For an implication of this result, consider where the resources are spent to attain or improve screening expertise. A fraction of the resources might be spent on IT infrastructure. But for the banking sector, probably a bigger fraction is spent in attracting human capital with high salaries and bonuses. For example, over 2005–10, the average ratio of compensation and benefits to overall non‐interest expenses were 65% for Goldman Sachs and 64% for Morgan Stanley,20 whereas, in contrast, this ratio for US manufacturing sector is 11%.21 Moreover, from an economics point of view, these payments of compensation and benefits may work more as a fixed cost than as a marginal cost because it seems that very often they are outlaid independently of the banks' performance rather than anchored to it.22 If I accept that a big fraction of C(p) is thus spent, then the Proposition suggests that the banking industry indeed hands out excessive payments and that a policy to cap them could improve ex ante efficiency. The next Section extends the model to encompass bank leverage, whereby I show that there is a complementarity between leverage and screening quality. 3. Bank Leverage In the analysis thus far, banks do not borrow from households; they invest only their own funds. In this Section, I assume that before banks choose screening accuracy, they can borrow funds from households, while using their own funds as the equity.23 The investing households, as debt holders, are repaid prior to the banks, the equity holders. Banks' advantage over households in screening expertise drives banks to borrow. It enables them to earn the rate of informed funds on the asset side, while they repay the rate of uninformed funds to households on the liability side, the gap between these two rates producing the profit margin of borrowing. Individual banks take this profit margin as given and, so long as it is positive, they want to borrow as much as possible. To limit their borrowing, I introduce risk‐shifting problems in the manner of Jensen and Meckling (1976). This requires the risks of the projects to be correlated,24 while the analysis so far is independent of such correlation. To simplify the exposition, I assume that the risks of projects are perfectly correlated. Specifically, foreseen at date 1, the economy is in one of three possible states, {ϕ,1,2}, occurring with probability 1−q¯ ⁠, q¯−q̲ ⁠, and q̲ respectively. In state ϕ, no projects succeed; in state 1, only high‐type projects succeed and low types fail; and in state 2, both types succeed. So high‐type projects succeed in both states 1 and 2 with probability q¯ and low types succeed in state 2 only, with probability q̲ ⁠. 3.1. The Risk‐shifting Problem and Leverage Ratio Suppose a bank borrows D units of funds from households at face rate f, so that its liability is Df. The risk‐shifting problem of the bank is that if D is too large, the bank may want to invest in bad projects at a lower expected return rate but a higher face rate than it obtains by investing in good projects. Let F be the face rate of investing in good projects, that is, F = R/qg, and F′ be that of investing in bad projects, which succeed with probability qb. The value of F′ lies between F and F × qg/qb. On the one hand, the bank rejects any face rate below F. On the other hand, bad entrepreneurs cannot afford a face rate above F × qg/qb, namely an expected rate of return above R, by the argument leading to (4). Then, for some α Ɛ (0, 1), F′=(1−α)qgqb+αF.(19) Note that qbF′ < qgF but F′ > F, that is, weighted against the risk, the investment in bad projects is worse than that in good projects but, contingent on success, the former delivers a higher return than the latter does. This hallmarks a typical circumstance liable to risk‐shifting problems. To prevent them, the leverage ratio, L := D/K, should be capped, as the following Lemma shows. Lemma 2. If a bank with K units of equity funds borrows D at face rate f, attains accuracy p and charges return rate R, then it does not invest in bad projects if and only if the leverage ratio L satisfies: L≤α1−1d(p)R(q¯−q̲)f−1−1d(p)R.(20) If the inequality holds, the bank repays its debt with probability q¯ and f=1/q¯.(21) The bank repays its debt with probability q¯ because it does so whenever high‐type projects succeed, which is in turn because the debt claims are senior to the equity claim held by the bank. The debt holders earn an expected rate of return of q¯f and are satisfied with a return rate of 1. Hence, (23) holds. 3.2. Complementarity Between Leverage and Screening Accuracy In this subsection I find the equilibrium leverage of banks and show a complementarity between leverage and screening accuracy. Banks, taking the profit margin of borrowing as given, want to borrow as much as they can fend off the risk‐shifting problems. They are thus leveraged to the upper bound given by (22). With f given by (23) and R as a function of p given by (10), bank j's leverage ratio, Lj, is Lj(pj)=α1−1d(pj)R(pj,Π^)[(q¯−q̲)/q¯]−1−1d(pj)R(pj,Π^).(22) This equation describes how the leverage ratio of a bank depends on its screening accuracy.25 Furthermore, leverage feeds back to screening accuracy. If bank j is leveraged at ratio Lj, then its screening accuracy is given by pj(Lj)=ρ[Kj(1+Lj),Π^].(23) That is, in determining the choice of screening accuracy, debt‐financed funds, D = KL, play an equal role as the equity funds, K. This is because borrowed funds and equity funds contribute in equal terms to the marginal value of higher accuracy, for which an intuition is as follows. By the preceding Lemma, the probability of debt being repaid, fixed at q¯, is independent of the accuracy choice, p. Hence, so is the marginal cost of debt (D), as is the marginal cost of equity (K). Moreover, debt and equity contribute in equal terms to a bank's revenue at given p (i.e. (K + D)R(p)) and thus to its marginal revenue (i.e. (K+D)Rp′. Therefore, D and K contribute in equal terms to both the marginal benefit and the marginal cost of higher p. Equations (24) and (25) together yield the following. Proposition 5. There is a complementarity between screening accuracy and leverage: for any bank j, Lj′(pj)>0 and pj′(Lj)>0. That is, on the one hand, a bank with more‐accurate screening expertise is leveraged at a higher ratio; on the other hand, higher leverage leads to greater accuracy. Intuitively, that Lj′(pj)>0 because the greater the screening accuracy p, the starker the difference in quality between good and bad projects (i.e. d) and the higher the rate charged (because Rp′>0. Both lead to bigger value destruction by risk shifting (as R − qbF′ = α(1 − 1/d)R by (21)), thus inducing smaller incentives to do that, which allows for higher leverage. That pj′(Lj)>0 because higher leverage augments the bank's size further, which, due to the increasing returns to scale at the level of marginal profit, induces the bank to attain greater accuracy. The next Section presents empirical implications of the study 4. Empirical Implications 4.1. Bigger Banks Are Leveraged at Higher Ratios The bigger the bank, the higher the screening accuracy due to increasing returns to scale and, by the complementarity, the higher the leverage.26 I show in Proposition 1 ∂2 Θ/ ∂K2 > 0, which suggests that bigger banks are able to enlarge their equity (i.e. K) further if the extent of equity enlargement increases with the marginal value of equity, ∂ Θ/ ∂K. Together with implication A, therefore, the study suggests that bigger banks can expand more, both by enlarging equity further and by being leveraged higher. This suggests a trend of increasing dominance in the banking industry. Implication A is consistent with many empirical findings. Liang and Rhoades (1991), with a sample of 4,751 US banking firms over 1979–86, report in their Table II that the total asset negatively and significantly affects the equity/asset ratio (E/A). Akhavein et al. (1997), analysing big US banks over 1980–90, find that after merger and acquisitions (M&A), consolidated banks widen the negative difference in E/A from their peer banks by six basis points. Demsetz and Strahan (1997), examining large US bank holding companies (BHC) over 1980–93, document in their Table 3 a strong, significant and negative correlation between size and E/A. This strong negative correlation is also found, more recently, by Lepetit et al. (2008) for a set of European banks over 1996–2002 and by Haq and Heaney (2012) for 117 European financial institutions over 1996–2010. While this empirical literature often attributes the higher leverage of bigger banks to the benefits of diversification, this study finds that it may come, orthogonal to diversification, from the complementarity between leverage and screening expertise. The two arguments diverge in the implication of the size of a bank for the quality of its individual loans. No link is implied by the diversification argument, whereas this study implies the following. 4.2. Obtaining Funding from Bigger Banks Certifies that the Borrowing Firms Are of Higher Quality Bigger banks have more accurate screening expertise. Thus, the projects that they evaluate as being good and then invest in are more likely to be among the high types, therefore of higher quality.27 For this result, direct empirical support is provided by Ross (2010). He documents that loans from three dominant banks ( JP Morgan Chase, Bank of America and Citigroup, accounting for more than 55% of the US commercial loan market) over 2000–3 induced their borrowers' stock prices to jump higher, were issued at lower interest rates and were ‘less likely to be protected by a borrowing base’, altogether suggesting that these banks ‘provide a higher level of certification’ (Ross, 1990, p. 2731). Also, Hao (2003) documents, using a sample of US banks over 1988–99, an inverse link between bank size and loan yield spread, which, the author suggests, may be explained by bigger banks picking borrowers of higher credit quality. Indirectly, Billett et al. (1995) report, using a sample of corporate loans over 1980–9, that greater abnormal returns of the borrowers’ shares are associated with loans from banks with higher credit ratings,28 which Poon et al. (2009) show is strongly and positively correlated with bank size.29 4.3. The Banking Industry Displays a Trend of Growing Leverage This is because small banks keep being edged out and only big banks remain, which, by implication A, are more leveraged. This trend is empirically well documented; see, among others, Berger et al. (1995, Figure 1 for the US over 1840–1990, Saunders and Wilson (1999, Figures 4–6) for the UK, Canada and US over 1893–1991, Hortlund (2005, Figure 2 for Sweden over 1870–2001 and Miles et al. (2013, Figure 1 for the UK over 1880–2010. 4.4. An Industry‐wide Rise in Leverage Tends to Increase Concentration in the Banking Industry This is because it enlarges the capacity of all banks and thereby intensifies competition between them, consequently edging out more small banks. Berger et al. (1999) document that an important factor contributing to the substantial consolidation in US banking industry over 1988–97 (when the number of banks fell by 30%) was the improvement in financial conditions, such as low interest rates, which made it more profitable for banks to increase leverage. Moreover, they find it puzzling that ‘M&A activity in banking appears to respond more to low interest rates … than does M&A activity in non‐financial industries, despite the fact that stock deals are more common than cash acquisitions in banking’ (Berger et al., 1999, p. 149). This study gives an account for this phenomenon by showing that a larger capacity delivers greater benefits in the financial sector than it does in a non‐financial sector. In the former, I show in the discussion of Proposition 1(i) that a larger capacity benefits a bank in both extensive margin and profit margin, whereas in the latter, it usually delivers no benefit in the profit margin. Lastly, Berger et al. (1995) find that deregulation of deposit ceiling rates contributed to the consolidation of the US banking industry over 1979–94. That is consistent with the study's findings because the deregulation allowed banks to absorb more deposits, which increased leverage. An industry‐wide rise in leverage could be induced by a regulatory loophole; for example, the off‐balance investment vehicles to the Basel II accords.30 The ‘shadow banking system’ thus created contributed substantially to the massive increase in bank leverage over the decade leading up to the 2008 crisis. 5. Oligopolistic Banks In this study, banks are modelled in perfect competition in the sense that each bank takes entrepreneurs’ equilibrium pay‐off, Π^ ⁠, as given. This approach offers a simple way to disentangle the two forces that this study identifies as important in shaping the economic implications of bank size, namely, increasing returns to scale and competition. In many real‐life economies, however, the banking industry is an oligopoly. This Section briefly discusses how the main results of this study apply under such circumstances. To simplify the exposition, consider a two‐bank case, where banks 1 and 2 have funds K1 and K2, respectively, and compete for one unit of entrepreneurs; all the other aspects are as they were modelled in Section 1. Following the analysis of subsection 2.1, after banks have chosen (pj, Rj)j =1,2, the allocation of entrepreneurs to banks, (M1,M2), and the pay‐off of entrepreneurs, Π^ ⁠, are determined by the following three equations: l1Π(p1,R1)=l2Π(p2,R2)=Π^,(24) M1+M2=1,(25) where lj=max1,KjMjng(pj)I(pj,Rj).(26) Let the solution for Mj be Mj(pj, Rj;p−j,R−j) for j = 1,2. Now consider the decisions of the two banks. As in the case of perfect competition, it is not optimal for a bank to induce overdemand. However, unlike the previous case, a bank can now be too large, in the sense that it is optimal for it to induce underdemand. The study assumes this case away because it hardly captures a real‐life circumstance. Then, given the other bank's choice (p−j,R−j), bank j chooses (pj, Rj) such that Mj(pj,Rj;p−j,R−j)=K1ng(pj)I(pj,Rj),(27) and its best response is to be found by solving the following problem: maxpj,RjKj(Rj−1)−C(pj),s.t.(27).(28) The best responses pin down the subgame perfect equilibrium. In this setting of oligopolistic banks, as in that of perfect competition, there exist the forces of increasing returns to scale and competition, as formally presented in Proposition 1. Because lj = 1 for j = 1,2, it follows from (26) that Rj=R(pj,Π^), with R(p,Π^) given by (10). Thus the objective in problem (28) is the same as that in the case of perfect competition, given by (11). Therefore, if Π^ could be taken as a given to a bank, then Proposition 1 would hold here also. However, here Π^ cannot be taken as given when the size of one bank is changed because now any bank has a non‐negligible effect on Π^ ⁠. Therefore, in this setting of oligopolistic banks, the effects of increasing returns to scale and those of competition cannot be disentangled. However, some results parallel to Proposition 3 can be derived, as follows. Proposition 6. Assume that the two banks are of such sizes that they both choose to be informed and not to induce underdemand. The subgame perfect equilibrium uniquely exists and has the following properties. (i) If a bank increases its size, then the pay‐off of entrepreneurs (Π^) increases and the other bank lowers its screening accuracy and interest rate. (ii) If both banks increase size without changing their market shares, then they both lower screening accuracy and interest rate. The Proposition can be proved in the same way in which Proposition 3 is proved. 6. Conclusion In a framework where banks can attain expertise to screen projects, whereas the general public (namely, households) cannot, this study examines the causal effects of bank size for banks’ survival, asset quality and leverage. The study finds the following. First, banks’ screening expertise generates increasing returns to scale, which help bigger banks survive competition better and drive banks into a race for expansion. This race intensifies competition between banks and thereby dampens their incentives to attain or improve screen expertise. Moreover, in this race, small banks tend to lose out, subjecting the banking industry to a trend of increasing concentration. Second, if all banks expand without much changing their market shares, then as a consequence of fiercer competition, their screening expertise all decline and asset quality all fall, in spite of the increasing returns to scale. Third, there is a complementarity between a bank's leverage and the level of its screening expertise. Fourth, there is a sense in which banks overspend on screening expertise. If, as seems plausible, a big fraction of this spending is used to attract human capital with high bonuses or salaries, then this overspending result suggests that the banking industry indeed hands out excessive payments. Appendix A. Proofs For Lemma 1: Proof There are two cases. First, I show that if Π^<{[d(1)−1]/d(1)}S(1) ⁠, a bank gets an infinitely large profit, namely, Θ(K,Π^)=∞. Note first that {[d(·)−1]/d(·)}S(·) is an increasing function because both d′(·) > 0 and S′(·) > 0. If Π^<{[d(1)−1]/d(1)}S(1), there exists some p′∈[p̲,1] such that Π^={[d(p′)−1]/d(p′)}S(p′) ⁠. Then, a bank can both charge R = ∞ ⁠, thus reaping Θ = ∞ ⁠, and attract all the entrepreneurs by giving them more than Π^ ⁠, as follows. Entrepreneurs’ pay‐off from a deal (p,R) is Π(p,R) = {[d(p)−1]R/[d(p)R−1]}S(p). It increases with p, decreases with R, and {[d(p)−1]/d(p)}S(p) = limR→∞Π(p,R). If the bank offers p = p′ + ε < 1 and R = ∞ ⁠, the entrepreneurs coming to the bank get more than Π^ ⁠: Π(∞,p′+ϵ)=d(p)−1d(p)S(p)|p=p′+ϵ>d(p′)−1d(p′)S(p′)=Π^, where the inequality applies the fact that {[d(·)−1]/d(·)}S(·) is increasing as noted above. Second, I show that Θ(K,Π^)=∞ if Π^={[d(1)−1]/d(1)}S(1). Let f(Π^,K;p):=K[R(p;Π^)−1]−C(p). Then, Θ(K,Π^)=maxp∈[0,1]f(Π^,K;p) and fp′=KΠ^d(p)−1S′(p)+d′(p)S(p)−Π^d(p)Π^−[d(p)−1]S(p)2−C′(p). Of these two terms, at p ≈ 1, the first one is in the order of 1/(1−p)2 if Π^={[d(1)−1]/d(1)}S(1), whereas the second one, C′(p), by assumption, is in the order of o[1/(1−p)2], dominated by the first term. It follows that fp′>A[1/(1−p)]2 if p > p0 for some p0 Ɛ (0, 1) and A > 0. Then, limp→1f(Π^,K;p)=limp→1[f(Π^,K;p0)+∫p0pfp′dp]>f(Π^,K;p0)+Alimp→1∫p0p1/(1−s)2ds=∞. That is, at Π^={[d(1)−1]/d(1)}S(1), the value Θ(K,Π^)=∞ and the optimal choice ρ(K,Π^)=1. For Proposition 1: Proof (i) If Θ(K,Π^)>0, then K(R−1)>C[ρ(K,Π^)]>0, therefore R − 1 > 0. By (11) and the envelope theorem, then, ∂Θ(K,Π^)/∂K=R[ρ(K,Π^),Π^]−1>0, and ∂2Θ(K,Π^)/∂K2=Rp′∂ρ/∂K ⁠. I saw Rp′>0 if Θ > 0 at footnote 13. Therefore, it suffices to show ∂ρ/ ∂K > 0, for which denote the left‐hand side (LHS) term of (15) by Y(p,Π^) ⁠. Then, by the implicit function theorem, ∂ρ(K,Π^)/∂K=∂Y/∂K/{−[(∂Y/∂p)−C″]} and ∂ρ(K,Π^)/∂Π^=∂Y/∂Π^/{−[(∂Y/∂p)−C″]}. The second‐order condition of the maximisation problem (11) implies that (⁠ ∂Y/ ∂p) − C′′ < 0 at p=ρ(K,Π^):=p~ ⁠. Therefore, to prove ∂ρ/ ∂K > 0, it suffices to show ∂Y/ ∂K > 0, which holds true as Y is positively proportional to K. (ii) By the envelope theorem ∂Θ(K,Π^)∂Π^=K−[d(p)−1]S(p){d(p)Π^−[d(p)−1]S(p)}2|p=p~<0. To show ∂ρ/∂Π^<0, by the argument above, it suffices to prove that ∂Y/∂Π^<0 ⁠. For this purpose, note that YK=[d(p~)−1]S′(p~)Π^d(p~)Π^−[d(p~)−1]S(p~)2+d′(p~)(S−Π^)Π^dΠ^−(d−1)S2; both terms are to be shown decreasing with Π^. For the first, [d(p~)−1]S′(p~)>0 and Π^/{d(p~)Π^−[d(p~)−1]S(p~)}2 decreases with Π^ for Π^>{[d(1)−1]/d(1)}S(1) (by Lemma 1) and thus bigger than {[d(p~)−1]/d(p~)}S(p~). For the second, d′(p~)>0 and {(S−Π^)Π^/[dΠ^−(d−1)S]2}Π^′<0⇔(S−2Π^)[dΠ^−(d−1)S]<2d(S−Π^)Π^. Note that Θ > 0 only if S>Π^ ⁠, namely if the surplus, S−Π^, that the bank gets from each project screened is positive; and also note that dΠ^−(d−1)S>0 by Lemma 1 Therefore, the last inequality of the chain above holds true if S−2Π^<0. If S−2Π^>0, the left‐hand side of that inequality is smaller than (S−2Π^)dΠ^<d(S−Π^)Π^<2d(S−Π^)Π^, the right‐hand side. For Proposition 2 Proof (i) From the proof of Proposition 1, I know that ∂Θ(K,Π^)/∂K=R−1>0 if Θ(K,Π^)>0 ⁠. Thus, Θ(K,Π^) is an increasing function of K over the range where Θ(K,Π^)>0. And Θ(K,Π^)→∞ if K→ ∞ ⁠. Therefore, K̲(Π^)=inf{K|Θ(K,Π^)>0} is well defined. By this definition and the increasing of Θ with K, K is the largest root of Θ(K,Π^)=0 and Θ(K,Π^)>0 if and only if K > K.  I then proceed to show K > 0. For this purpose, note that Θ(K,Π^)=0 and ρ(K,Π^)=0 at K = 0. Also for Π^>{[d(1)−1]/d(1)}S(1), Θ(K,Π^) and ρ(K,Π^) are continuous with K. By the continuity there exists some ɛ > 0 such that p~:=ρ(K,Π^)<p̲ if K < ɛ. Then, for these K, S(p~)<0. It follows that R(p~,Π^)=Π^d(p)Π^−[d(p)−1]S(p)|p=p~<1. Therefore, for K < ɛ, ∂Θ(K,Π^)/∂K=R−1<0. As Θ(0,Π^)=0, I have Θ(K,Π^)<0 if 0 < K < ɛ; that is, a bank of size K makes loss if it chooses to invest in screening expertise and compete for entrepreneurs. Therefore, K̲(Π^)>ε>0. First, ∂Θ(K,Π^)/∂K|K=K̲=R−1>0 ⁠: for banks with K = K, they get 0 value if becoming informed; thus (R−1)K̲=C[ρ(K̲,Π^)]>0. Second, by Proposition 1(ii), ∂Θ(K,Π^)/∂Π^<0 ⁠. Then, by the implicit function theorem, K̲′(Π^)=−∂Θ(K,Π^)/∂Π^∂Θ(K,Π^)/∂K|K=K̲>0. For Proposition 3: Proof To prove the existence and uniqueness of equilibrium, it suffices to show that (16) and (17) have a unique solution of (t,Π^). For this purpose, a key role is played by the following Claim, which is proved after the Proposition. Claim A. The mass of entrepreneurs served by a bank of size K, β(K,Π^), given by (14), satisfies: ∂β(K,Π^)/∂Π^<0, ∂β(K,Π^)/∂K>0, and β(K,Π^)→∞ if Π^→{[d(1)−1]/d(1)}S(1). To simplify notation, let βj(Π^):=β(Kj,Π^). With the Claim, g(Π^):=∫01βj(Π^) decreases with Π^ and goes to ∞ if Π^→{[d(1)−1]/d(1)}S(1). Obviously limΠ^→∞g(Π^)=0, namely, if entrepreneurs demand a too large pay‐off no banks are willing to serve them. Therefore, g(Π^)=1 has a unique solution, denoted by Π^a. Two cases may arise. Case 1. K1≥K̲(Π^a). Then, the unique solution to (16) and (17) is t = 1 and Π^=Π^a. Case 2. K1<,K̲(Π^a). That is, not all banks become informed, namely, t < 1. Thus (16) is reduced to Kt=K̲(Π^). To this equation and (17) I show there is a unique solution. By Proposition 2(ii), K̲(Π^) is a strictly increasing function. Therefore, the inverse function of K̲(Π^) exists, denoted by Ψ(K); thus, Ψ(K) is the level of entrepreneurs’ pay‐off at which banks of size K are indifferent between being informed and staying out. From (16) Π^=Ψ(Kt). Substitute it into (17), which then becomes: f(t):=∫0tβj[Ψ(Kt)]=1. I now prove f(t) = 1 has a unique solution t < 1 by noting or showing the following four points. First, f(·) is continuous. Second, f(0) = 0. Third, f′(t) > 0 because f′(t)=βt[Ψ(Kt)]+∫0tβj′(Π^)Ψ′(Kt)dKt/dt>0 ⁠, as βj′(Π^)<0 by Claim A, dKt/dt ≤ 0 by assumption, and Ψ′(Kt) > 0 as Ψ(K) is an inverse function of K̲(Π^) and K̲′(Π^)>0 by Proposition 2(ii). Fourth, f(1) > 1. In this case, K1<K̲(Π^a), which is equivalent to Ψ(K1)<Π^a ⁠. Then f(1)=∫01βj[Ψ(K1)]βj′(Π^)<0>∫01βj(Π^a)definition ofΠ^a=1. (i) That is, if for any m Ɛ Ω, Km is increased, where Ω is a positively measured subset of [0,t], then Π^ goes up and for any j ∉ Ω and j ≤ t, Rj goes down. To prove the former result, I apply reductio ad absurdum. If, on the contrary, Π^ non‐increases, then, K non‐decreases by Proposition 2, therefore the number of informed banks is not reduced, namely, the threshold t is not decreased. For each of the informed banks, say bank j, by Claim A, its market share, βj(Π^), non‐decreases. But the market share of all banks in set Ω strictly increases because their capacities are enlarged. Moreover, set Ω has a positive measure. It follows that ∫0tβj(Π^) is strictly increased, thus above 1, which contradicts the market clearing condition (17). To prove that for any j ∉ Ω, pj and Rj both fall, first note that as pj=ρ(Kj,Π^) and ∂ρ/∂Π^<0, with Π^ rising and Kj fixed, pj falls. Second, Rj as a function of pj and Π^, given by (10), increases with pj and decreases with Π^ ⁠. Then, with pj going down and Π^ up (as shown above), Rj decreases. (ii) To prove the result, it suffices to show that if β^j is given, then dp^j/dKj<0. Note that p^j satisfies the first‐order condition, (15), with K replaced by Kj, namely, KjΠ^[d(p)−1]S′(p)+d′(p)[S(p)−Π^]d(p)Π^−[d(p)−1]S(p)2=C′(p).(A.1) To get Π^ as a function of K and β, note that for any informed bank Π(p,R)=Π^ ⁠, which, with Π(p,R) given by (7), becomes: S(p)[d(p)−1]Rd(p)R−1=Π^.(A.2) Solving R from (5) as a function of β, and substituting it into (A.2), I find Π^ related to Kj and β^j through Π^=d(p)−1d(p)S(p)+Kjβ^j|p=p^j.(A.3) Substitute it for Π^ into (A.1) and then find p^j solely determined by Kj through β^jd(p)β^jS(p)Kj+1S′(p)+d′(p)d(p)S(p)d(p)−1−Kjβ^j=C′(p).(A.4) To prove that given β^j ⁠, dp^j/dKj<0 ⁠, denote the LHS of the above equation by X(p,Kj). Then, by the implicit function theorem, dp^j/dKj=(∂X/∂Kj)/{−[(∂X)/∂p−C″]}. Obviously ∂X/ ∂Kj < 0, because S(p) > 0 and d′(·) > 0. Therefore, it suffices to prove (⁠ ∂X/ ∂p)−C′′ < 0. Let g(p,K):={[d(p)−1]/d(p)}[S(p)+(K/β^j.)] Then, Π^=g(p^j,Kj) by (A.3). Thus, X(p,K) = Y[p,g(p,K)], where Y(p,Π^) was used to denote the LHS of (15) (namely, (A.1) here) in the proof of Proposition 1. Therefore, ∂X∂p=∂Y∂p+∂Y∂Π^∂g∂p<∂Y∂p. because (∂Y/∂Π^)<0, as was shown in the proof of Proposition 1(ii) and (⁠ ∂g/ ∂p) > 0. It follows that ∂X/ ∂p − C′′ < (⁠ ∂Y/ ∂p) − C′′ < 0, which was shown in the proof of Proposition 1(i).  For each informed bank j, as p^j decreases and also Π^ increases (by result (i)), the interest rate it charges, R(p^j,Π^), decreases, as was noted above. For Claim A, which is used for proving Proposition 3: Proof By (5), β(K,Π^)=K/E(K,Π^), where E(K,Π^):=ng(p)I(p,R), with p and R as functions of (K,Π^) given respectively by (12) and (13). To prove that ∂β(K,Π^)/∂Π^<0 and ∂β(K,Π^)/∂K>0, it suffices to prove that EΠ^′>0 and EK′<0 ⁠. Substituting (10) for R in I(p,R) which is given by (4), I find E=[d/(d−1)]Π^−S:=f(p,Π^). Note that d and S are both functions of p only, while p=ρ(K,Π^) by (12). Therefore, EΠ^′=(∂f/∂p)(∂ρ/∂Π^)+∂f/∂Π^ and EK′=(∂f/∂p)(∂ρ/∂K) ⁠. Straightforwardly ∂f/∂Π^>0. By Proposition 1(ii), ∂ρ/∂Π^<0. Then, EΠ^′>0 follows ∂f/∂p<0⇔dd−1p′Π^−S′(p)<0|S′>0,Π^>0⇐dd−1p′<0⇔−d′(p)(d−1)2<0 which holds true because d′ > 0 by (3). Moreover, as ∂f/ ∂p < 0, and ∂ρ/ ∂K > 0 by Proposition 1(i), EK′=(∂f/∂p)(∂ρ/∂K)<0. For limΠ^→{[d(1)−1]/d(1)}S(1)β(K,Π^)=∞ ⁠, it suffices to prove that I(p,R)=Vg(p)Rd(p)−1|p,Rgivenby(12),(13)→0. if Π^→{[d(1)−1]/d(1)}S(1), which follows from limΠ^→{[d(1)−1]/d(1)}S(1)R[ρ(K,Π^),Π^]=∞. For this, note first that by the proof of Lemma 1, if Π^→{[d(1)−1]/d(1)}S(1), then p=ρ(K,Π^)→1. The LHS of (A.2) thus goes to S(1)[(d(1)−1]R/[d(1)R − 1], while the right‐hand side, namely Π^, goes to [d(1)−1/d(1)]S(1), which implies R → ∞ ⁠. For Proposition 4: Proof To compare pj* with the equilibrium quality, p^j, rewrite (A.4) that characterises p^j ⁠. By (10), R^j=Π^d(p^j)Π^−[d(p^j)−1]S(p^j). Substitute for Π^ from (A.3), then, R^j=1d(p^j)β^jS(p^j)Kj+1.(A.5) It follows that S(p^j)=(d^jR^j−1)Kj/β^j. Substitute it for S(p) in (A.4), and p^j is then characterised by: R^jβ^jS′(p)+d′(p)Kjd(p)−1(R^j−1)=C′(p).(A.6) Now I come to compare pj* with p^j. Let η:=[d′(p^j)Kj]/[d(p^j)−1]>0 be a constant and p(x) be the function implicitly defined by U(x,p):=x[β^jS′(p)+η(x−1)]−C′(p)=0 for x ≥ 1. Then, p(1)=pj* and p(R^j)=p^j. As informed banks charge R^j>1, the Proposition is equivalent to p′(x) > 0. By the implicit function theorem, p′(x)=−Ux′/Up′. Obviously, Ux′>0 ⁠. Moreover, Up′=xβ^jS″−C″≤−C″<0 ⁠, as S′ ≤ 0 (by (3)). Thus, p′(x) > 0. For Lemma (2): Proof If out of K + D units of funds under its deployment, the bank invests M in bad projects and K + D − M in good ones, what the bank gets in each state is as follows. In state ϕ, no projects succeed and the bank gets nothing. In state 1, high‐type projects succeed but low types fail. By the LLN, out of all the good projects, the fraction of high types is Pr(q~=q¯|s~=g):=hg ⁠, whereas the fraction of the bad projects is Pr(q~=q¯|s~=b):=hb. Hence, in state 1, fraction hg of the investment in good projects and hb of that in bad ones succeed. Success delivers a return rate F in the former investment and F′ in the latter investment. Therefore, the revenue of the bank in state 1 is (K + D − M)hgF + MhbF′ := Q(M ) and its liability duty is Df. The bank might default in this state. Its profit is then max[Q(M)−Df,0] := Θ1(M). In state 2, all the bank's projects succeed. The bank does not default in the state, or it gets 0 profit, certainly not the case. Hence, the bank's profit is (K + D − M)F + MF′−Df := Θ2(M). Altogether, the expected profit of the bank is Θ(M)=(q¯−q̲)Θ1(M)+q̲Θ2(M) ⁠. I show this function has two local maximisers, M = 0 and M = K + D. If M is small enough such that Θ1(M) ≥ 0, then Θ(M)=(K+D−M)F[(q¯−q̲)hg+q̲]+MF′[(q¯−q̲)hb+q̲]−Dq¯f. Note that (q¯−q̲)hg+q̲=q¯Pr(q~=q¯|s~=g)+q̲(q~=q̲|s~=g)=qg and, similarly, (q¯−q̲)hb +q̲=qb. Therefore, Θ(M)=(K+D)qgF−M(qgF−qbF′)−Dq¯f. It decreases with M because qgF > qbF′ by (21). Thus the maximum occurs at M = 0. If M is so big that Θ1(M) = 0, then Θ(M)=q̲[(K+D)F+M(F′−F)−Df]. It increases with M because F′ > F by (21). Thus, the maximum occurs for this case at M = K + D. To prevent the bank from investing in the evaluated bad projects, it commands Θ(0) ≥ Θ(K + D), which, by substituting (21) for F′ and noting F = R/qg, gives rise to (22). Θ(0) > Θ(K + D) if and only if Θ1(M) > 0. That is, if the bank is prevented from risk shifting, it repays the debt in both states 1 and 2, thus with probability q¯. For Proposition 5: Proof By (24), L increases with d and R, both in turn increasing with p. Therefore, L′(p) > 0. And by (25), pj′(Lj)=(∂ρ/∂K)Kj>0, because ∂ρ/ ∂K > 0 by Proposition 1(ii). Footnotes 1 " This trend is well documented in the empirical literature; see, among others, Berger et al. (1995) (for US over 1979–94), Saunders and Wilson (1999) (Figures 2 and 3 for Canada and UK over 1893–1991), Berger et al. (1999) (for US over 1988–97) and Jones and Critchfield (2005) (for US over 1984–2003). 2 " Williamson (1986b), Carter and Manaster (1990), and Hauswald and Marquez (2006) consider similar cases of increasing returns to application scale but not in relation to bank size. 3 " See Section 5 for a detailed discussion of empirical implications and the relevant empirical studies. 4 " In studies by Besanko and Kanatas (1993) and Holmstrom and Tirole (1997), entrepreneurs (firms) receive funds from both households and banks, as in this study. However, Besanko and Kanatas (1993) do not consider financial intermediation (i.e. banks drawing funds from households). While Holmstrom and Tirole include financial intermediation, they find that it does not affect the equilibrium outcome and that the allocation of funds between it and direct finance is indeterminate. 5 " See Diamond (1984), Gale and Hellwig (1985), Williamson (1986a), Krasa and Villamil (1992), Winton (1995) and Cantillo (2004), among others, and see Gorton and Winton (2003) for a survey. 6 " The case of oligopolistic banks is briefly discussed in Section 6. 7 " If projects yield a return of Z′ Ɛ (0, B) in the event of failure, then entrepreneurs can borrow up to Z′ in a risk‐free manner, which will not change the study's results. 8 " Note that C(p) is the cost of obtaining screening expertise of accuracy p, not that of evaluating a single project to this accuracy. That is, once a bank has paid C(p), it can evaluate all projects at accuracy p. Allowing the accuracy of evaluating one project to depend on the resources spent for this particular evaluation would not qualitatively change the study's results. 9 " That is, C′(p)(1−p)2 → 0 if p → 1. 10 " With (2), qg(p)=[nq¯+n̲(1−p)q̲]/[n¯+n̲(1−p)] ⁠, qb(p)=q̲ and ng(p)=n¯+n̲(1−p). 11 " This follows from the fact that Vg′(p)=qg′Z>0 ⁠, Vg(0)=(nq¯+n̲q̲)Z−B<0 and Vg(1)=q¯Z−B>0 by the assumption in (1). 12 " As noted earlier, informed banks choose p > p̲ ⁠. Also, no informed bank posts R ≤ 1, a return rate no higher than that of uninformed funds. 13 " So the expected rate of return is qg × 1/qg = 1, which satisfies households. 14 " Otherwise, it would increase the interest rate, R, which decreases Π(p,R), and lend all its funds out at this increased rate. 15 " If the bank has only part of its funds earn return rate R(p;Π^), it would lower the rate a little to R − ɛ, which induces over demand, as Π(p,R−ε)>Π^, so all its funds earn return rate R − ɛ. 16 " Note that −{d(p)Π^−[d(p)−1]S(p)}p′=(d−1)S′+d′(S−Π^), which is positive because d > 1, d′ > 0 and S′ > 0 by (3), and S−Π^, the surplus the bank gets from each project evaluated, is positive if the bank chooses to be informed. 17 " For supportive empirical evidence, see the empirical studies cited in footnote 1. 18 " As is shown in the proof of Proposition 3, ∂β/ ∂K > 0, that is, β(Kj,Π^) is non‐increasing with j. Thus, as a function of j, β(Kj,Π^) is integrable. 19 " Mathematically, the default probability of the projects bank j invests is 1 − qg(pj), which increases when pj decreases, because qg′(·)>0 ⁠. 20 " Calculated from the annual reports of the two banks published on their websites. 21 " See ‘The misery of manufacturing’, The Economist, pages 75–76, 27/09–03/10, 2003. 22 " For example, for the year 2008, Citigroup and Merrill Lynch both paid their employees each a bonus of one million or more dollars, although the banks suffered a huge loss; see ‘Million‐dollar bonus breakdown to reignite US bank controversy’ (Financial Times, 31 July 2009). Banks often resort to the need to retain key human capital, rather than that to provide incentives, when coming to justify high bonuses. 23 " This study assumes that banks do not issue outside equities to households, possibly due to some friction of costly state verification in the manner of Townsend (1979), Diamond (1984) and Gale and Hellwig (1985). 24 " Otherwise, as each bank finances a continuum of projects, the risks on its asset side will be completely diversified away and no risk‐shifting problems will arise. 25 " Or, rather on the rational expectation of the accuracy, as the leverage choice is decided before the accuracy choice. 26 " Mathematically, if Ai := Ki + Di > Aj := Kj + Dj, then by (25) and ∂ρ/ ∂K > 0, pi > pj, hence Li > Lj as L′(p) > 0. 27 " Mathematically, if Ai := Ki + Di > Aj := Kj + Dj, then by (25) and ∂ρ/ ∂K > 0, pi > pj. 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I thank, for their helpful comments, Roy Bailey, Sanjay Banerji, Hans Gersbach, Xuewen Liu, John Moore, Jean Rochet, Zhen Song, Tuomas Takalo, Huainan Zhao and seminar participants at Zhejiang University, Fudan University, Essex University, Bank of Finland, ETH Zurich and Nottingham Business School. © 2013 Royal Economic Society

Freeman, Mark, C.;Groom,, Ben

doi: 10.1111/ecoj.12129pmid: N/A

Abstract The aggregated term structure of social discount rates that results from Weitzman's (2001) survey of expert opinion is shown to be highly sensitive to the nature of the responses. If variation reflects irreducible differences in ethical judgments, the term structure can decline rapidly. If variation occurred because respondents were forecasting future rates under uncertainty, the term structure is much flatter because additional experts provide new information. The former approach triples the social cost of carbon when compared to the latter. The distinction between heterogeneity and uncertainty illustrates the need for a nuanced treatment of survey data in intergenerational policy making. The issue of social discounting has long been a major source of disagreement amongst economists and philosophers, with some perspectives being described as not simply myopic but ‘ethically indefensible’, ‘rapacious’ and ‘defective’. Such strong sentiments arise from the fact that the estimated present values of very long‐term projects are generally highly sensitive to the choice of discount rates that are deployed. For example, the present value of £1 in 100 years is 50 times higher when discounted at 1% than at 5%, and this ratio increases exponentially with the time horizon. As a consequence, the policy prescriptions on intergenerational projects are often determined by the rate at which the costs and benefits are discounted. Indeed, some have argued that the immediate and dramatic action on climate change recommended by the Stern Review on the Economics of Climate Change (Stern, 2007) resulted solely from the inappropriately low social discount rate (SDR) that was used, a position which evoked accusations of ‘perhaps stoking the dying embers of the British Empire’ (Nordhaus, 2007, p. 691). In recent years, however, something resembling a consensus has emerged in the field of social discounting. A recent Policy Forum article in Science summarises the case for using lower rates for discounting long‐term costs and benefits than their short‐term equivalent counterparts (Arrow et al., 2013). Reviews of the relevant academic literature that supports this approach have also recently been provided by Gollier (2012) and Arrow et al. (2012). The UK, French, Norwegian and Danish governments all now recommend schedules of declining discount rates (DDRs) as the time horizon increases.1 Similar policy recommendations are currently being considered by the authorities in the US. In this article we return to a study that has been highly influential in shaping the policy landscape for DDRs. Weitzman (2001) sought the opinion of a large number of economists on the appropriate discount rate that should be used for calculating the present value of future global warming damages. The responses to the survey were widely dispersed, with the sample frequency distribution closely resembling the probability density function of a gamma distribution. When these responses were aggregated according to Professor Weitzman's preferred method, known as ‘gamma discounting’, the resulting SDR schedule declined sharply. We focus here on the interpretation of the expert responses and the method by which they were combined to calculate the SDR within gamma discounting. Specifically, the question that Professor Weitzman asked contained a significant ambiguity. As a consequence, experts might have interpreted the survey in one of two distinct ways. Our central point is that the appropriate method of aggregation depends critically on which of these interpretations each expert had in mind when responding. Under the first interpretation, experts might have revealed their individual ethical views concerning intergenerational justice. Differences in such subjective opinions are essentially irreducible. In this case, sampling additional experts only serves to characterise better the extent of disagreement and does nothing to diminish the variation in responses. This may reasonably lead the social planner to construct the same schedule of SDRs as reported by Weitzman (2001). Under the second interpretation, the variation in survey responses might instead have reflected forecasting errors about some objective ‘true’ value. These responses then reveal the nature of our uncertainty about the future rather than the extent of heterogeneity in ethical positions. In this case, increasing the sample of experts provides additional information to the social planner, improving the quality of the aggregated forecast. We show that this generates a term structure of SDRs which declines slowly and, in some cases, is essentially flat. This distinction between uncertainty and heterogeneity has important implications for economic valuations made across a wide range of key policy areas. We demonstrate this through four examples; the social cost of carbon (SCC), the costs of teenage obesity, nuclear decommissioning costs and the economic benefits of the HS2 rail link. 1. The Survey Question The basis of the gamma discounting framework of Weitzman (2001) was an emailed survey to PhD‐level economists that generated n = 2,160 responses, ri for i ∈ [1, n], to the question ‘Taking all relevant considerations into account, what real interest rate do you think should be used to discount over time the (expected) benefits and (expected) costs of projects being proposed to mitigate the possible effects of global climate change?’ The individual responses were widely dispersed with a range from −3% to +27%. The problem that then faces the social planner is how to aggregate this range of values into a single SDR, R(H), to apply when discounting a certainty‐equivalent cash flow that will arrive at time H. Without fully explaining his rationale, Weitzman (2001) proposed taking the simple average of individual discount factors: exp[−HR(H)]=1n∑i=1nexp(−Hri).(1) Our analysis centres on (1). For now we note that R(H) when defined this way has the following properties: limH→0R(H)=r¯i ⁠, where r¯i is the mean of the individual responses, lim H→∞R(H) = min{ri} and dR(H)/dH < 0. The decline in the term structure occurs since, ‘from today's perspective, the only relevant limiting scenario is the one with the lowest interest rate – all of the other states at that far distant time, by comparison, are relatively much less important now because their present value has been reduced by the power of compound discounting at a higher rate’ (Weitzman, 1998, p. 205). The reason for the DDR also has a simple mathematical explanation. Exponential functions are convex, so by Jensen's inequality n−1∑i=1nexp(−Hri)>exp(−Hr¯i) ⁠. The greater H, the more curved the exponential function, hence the appropriate SDR declines with the maturity of the project. Furthermore, the more dispersed the responses, the greater the magnitude of the Jensen's inequality. These points can be simply demonstrated by example. Suppose that there were only two responses; r1 = 3% and r2 = 5%. Then, R(H) = [4.0%, 3.9%, 3.6%, 3.2%] for H = [1, 30, 100, 400]. By contrast, if we preserve the mean but increase the spread of the responses by setting r1 = 1% and r2 = 7%, R(H) = [4.0%, 2.8%, 1.7%, 1.2%] for the same values of H. Rather than applying (1) directly, Weitzman (2001) took the following elegant analytical approximation. He noticed that the sample frequency of responses, ϕ(ri), closely resembled the probability density function of a gamma distribution with shape parameter α and rate parameter β. Using this continuous distribution to describe the individual responses, from (1): R(H)=−1Hln∫0∞e−HrβαΓ(α)rα−1e−βrdr=−αHlnββ+H,(2) thus, giving a simple closed‐form solution for the H‐period discount rate. To determine the term structure of the SDR that arises from gamma discounting, we estimate α, β using a maximum likelihood method based on the strictly positive responses that Weitzman received, giving α^=2.54 and β^=63.08 ⁠. This results in a sharply declining term structure of SDRs; R(H) = 4.00%, 3.29%, 2.41% and 1.27% for H = [1, 30, 100, 400] years respectively. We now consider how experts might have interpreted the survey question that Weitzman posed. The most prominent distinction here, as highlighted by Arrow et al. (1996), is between those who view long‐term discounting as a fundamentally ethical issue (Stern, 2008) and others who prefer to calibrate the SDR to reflect ‘market and policy factors as they currently exist’ (Nordhaus, 2007, p. 692). Respectively, these are commonly referred to as normative and positive (or prescriptive and descriptive) positions on social discounting. This dichotomy was central to the controversial aftermath of the Stern Review as the low ethically derived rates applied by Stern prescribe a more urgent response to climate change than many market‐based discount rate schedules (e.g. Nordhaus, 2007; Stern, 2008). Our contention in this study is that the wording of the survey leaves sufficient scope to be interpreted within either of these paradigms. 2 The use of the term ‘interest rate’ points towards a positivist framework. Alternatively, it seems equally likely that the reference to climate change may have evoked ethical considerations, leading to normative responses. We do not take a stance on which was the appropriate reading of the survey question. Instead, we demonstrate that the term structure of SDRs that emerges from this survey depends crucially on whether the exercise elicited ethical responses reflecting heterogeneity, or market‐based responses reflecting uncertainty. 2. Ethical Responses Assume first that each expert took an ethical position concerning intergenerational justice when determining his or her response. Combining these heterogeneous preferences is essentially a social choice problem for which there is no uncontentious solution and not all approaches lead to a declining SDR. For instance, Heal (2012) reminds us that the median value will be the outcome of a number of plausible social choice rules, including majority voting. That said, a number of different theoretical frameworks exist that might provide justification for the use of (1) when determining the SDR in a normative context. We briefly note one example here and engage in a fuller discussion of this point in our online Appendix A. Following Gollier and Zeckhauser (2005) and Jouini et al. (2010) imagine a pure exchange economy with different agents who all have logarithmic utility. These individuals differ in three respects: their beliefs about future consumption growth, their initial endowment levels, wi, and their rates of pure time preference (utility discount rate), δi > 0. Acting atomistically, each agent would choose a different path of consumption, so disagreement arises over how to share the exogenous consumption stream. Within the framework proposed by Jouini et al. (2010) the agents are experts who disagree on consumption growth and δi. The social planner resolves this disagreement by invoking an ‘as‐if’ market between the experts. The resulting intertemporal Arrow–Debreu equilibrium, and hence socially efficient, discount rate is given by: R(H)=−1Hln∑i=1nwiδi∑j=1nwjδjexp−Hri.(3) Equation (1) follows from here provided that wiδi is constant across experts. The efficient term structure is declining as intertemporal trade between experts leads those with a high‐utility discount rate to prefer paths with more consumption early on, leaving experts with low δi as the chief determinants of the long‐run discount rate. This framework neatly captures the idea that variation in ethical judgments reflects fundamental and irreducible disagreements. This follows by virtue of R(H) being independent of the number of experts surveyed, n, above the minimum threshold required for the sample to be representative of the population. Therefore, with ethical responses, gamma discounting can be theoretically defended, although its usage remains contentious. 3. The Positivist Interpretation Suppose instead that the survey responses resulted from within a purely positivist framework, where experts were requested to ‘look carefully at the returns on alternative investments – at the real real interest rate – as the benchmarks for climatic investments’ (Nordhaus, 2007, p. 692). In this setting, the standard model for determining the term structure of SDRs, which has been highly influential in determining international governmental policy in this area, is the expected net present value (ENPV) condition: R(H)=−1HlnEexp−Hr¯H,(4) where r¯H=H−1∑t=0H−1rft and rft is the yield on a Treasury bond at time t. r¯H can be interpreted as the average future risk‐free rate over the horizon of interest. Empirical schedules of the SDRs using (4) have been provided by Newell and Pizer (2003), Groom et al. (2007) and Freeman et al. (2013) amongst others.3 Within this setting, the most natural interpretation of each individual's survey response is that it reflects his or her own personal estimate of the future realised value of r¯H ⁠; ri=Ei(r¯H) ⁠. This is fundamentally different from the information content of a survey eliciting ethical opinions; ri is now a forecast of r¯H ⁠. Variations in response arise from asymmetric information or differences in professional judgment over how best this forecasting process might be undertaken. These different choices do not reflect fundamentally different ethical stances and are not irreducible in this sense. At time H−1 the true value of r¯H will be revealed and, with the benefit of hindsight, we will all agree on which respondents gave the most accurate forecast. The social planner must now decide how to combine these different forecasts. The simplest method is to take a statistical approach. Let ei=ri−r¯H denote the forecast error of expert i. First, assume that all experts are unbiased, E(ei) = 0, that the forecast error variance of each is identical, Var(ei) = σ2, and that experts are independent. As Weitzman provides us with such a large sample, irrespective of the sample frequency distribution, ϕ(ri), under weak regularity conditions, the central limit theorem tells us that the probability density function of r¯H ⁠, f(r¯H)=N(r¯i,σ2/n) ⁠. The H‐period discount rate from (4) is now: R(H)=r¯i−0.5σ2Hn.(5) In contrast to (1), the appropriate measure of uncertainty is the standard error, not the standard deviation, of the sample distribution. More experts provide additional information to the social planner. This reduces her uncertainty over the ‘true’ value of r¯H ⁠, which in turn lessens the Jensen's inequality effects that drive declining schedules of SDRs. Of course, expert opinions are not independent; see, for example, Clemen and Winkler (1985) and Graham (1996). To account for this correlation, we generalise the statistical approach by turning to the substantial literature on combining probability distributions (Genest and Zidek, 1986). To capture the fact that each additional expert now brings less fresh information than the previous one, we follow Clemen and Winkler (1985) by mapping the total number surveyed, n, onto N, the effective number of independent experts. We discuss in detail in our online Appendix C how this relatively technical exercise can be undertaken and how this then influences the social planner's probability density function of r¯H , f(r¯H), for use in (4). Unfortunately, Weitzman's survey does not ask experts how they arrived at their responses, and therefore it is not possible to estimate the correlation between different expert forecasts empirically. We therefore take a range of possible values; N ∈ {1,000, 100, 50, 25, 12}.4 For the final case, this means that the information content of Weitzman's survey of 2,000+ economists is the same as could be found in a sample of 12 truly independent experts. The derived term structures of SDRs are presented in Figure 1. Fig. 1. Open in new tabDownload slide The H‐Period Social Discount Rate, R(H) Fig. 1. Open in new tabDownload slide The H‐Period Social Discount Rate, R(H) These results differ markedly from those reported by Weitzman (2001). The 400‐year (30‐year) discount rate is only 32% (82%) of the short‐term rate under gamma discounting but this increases to 72% (97%) when placing a positivist interpretation on the survey with N = 12. Increasing N to 1,000 leads to a term structure that is essentially flat. This reflects the fact that, with more information, forecast errors are reducible in a way that ethical opinions are not. 4. Applications To demonstrate the importance of this point for decision‐making across a range of key policy areas, we consider four examples of long‐term cash flows. First, we use the profile of damages from Newell and Pizer (2003) associated with each marginal ton of carbon emitted to estimate the SCC. Next, we take the schedule of estimated costs from decommissioning 19 now non‐operational nuclear power stations in the UK as given in the Nuclear Decommissioning Authority (NDA) Report and Accounts 2012/13. Third, we use estimates from Wang et al. (2003) of the incremental costs that arise for a woman between the ages of 41–65 conditional on her being obese at the age of 14. Finally, we use official estimates of the benefits that are expected to arise between 2026 and 2085 from Phase 1 (London to Birmingham) of the HS2 rail link. A detailed description of each of these schedules of cash flows is given in our online Appendix D and their profile is illustrated in Figure 2. We estimate present values for each of these examples using all of the term structures of the SDR presented in Figure 1. We also use the schedule recommended by the Green Book (UK Treasury, 2003) as well as a non‐declining 4% rate (which is very close to r¯i) ⁠. Results are reported in Table 1. The policy decisions taken by the social planner will often be significantly influenced by the schedule of discount rates selected. The greatest sensitivities are for the longest horizon cash flows, as reflected by the SCC. Here gamma discounting gives a present value (PV) that is three times as great as the N = 12 case. For both HS2 and teenage obesity, the gamma discounting PV is more than half as much again as the PV when N = 12. The PV of decommissioning costs for the previous generation of nuclear power stations is most robust to different possible choices. However, if these cash flows are delayed by 50 years, to broadly capture the costs of decommissioning the next generation of power stations, the present value estimated from gamma discounting is 175% greater than the PV calculated from the N = 12 case. By contrast, in no case does the N = 12 PV differ by more than 10% from that calculated using a non‐DDR of 4% for all horizons. For larger N, the results become even closer to the non‐DDR values. Fig. 2. Open in new tabDownload slide The Proportion of the Total Cash Flow that is Estimated to Arise in Each Year Fig. 2. Open in new tabDownload slide The Proportion of the Total Cash Flow that is Estimated to Arise in Each Year Table 1 The Present Value of Intergenerational Projects . SCC ($/tC) . Current NDA (£ bn) . Delayed NDA (£ bn) . Teenage obesity ($) . HS2 benefits (£ bn) . N = 1,000 5.692 39.06 5.27 14,753 13.91 N = 100 5.738 39.08 5.32 14,780 13.95 N = 50 5.706 38.93 5.28 14,660 13.84 N = 25 5.543 38.42 5.10 14,238 13.46 N = 12 5.346 37.56 4.88 13,560 12.86 Gamma 15.928 44.97 13.41 20,767 21.01 Green Book 10.154 43.84 9.57 19,181 18.68 Flat 4% 5.713 39.11 5.29 14,796 13.95 . SCC ($/tC) . Current NDA (£ bn) . Delayed NDA (£ bn) . Teenage obesity ($) . HS2 benefits (£ bn) . N = 1,000 5.692 39.06 5.27 14,753 13.91 N = 100 5.738 39.08 5.32 14,780 13.95 N = 50 5.706 38.93 5.28 14,660 13.84 N = 25 5.543 38.42 5.10 14,238 13.46 N = 12 5.346 37.56 4.88 13,560 12.86 Gamma 15.928 44.97 13.41 20,767 21.01 Green Book 10.154 43.84 9.57 19,181 18.68 Flat 4% 5.713 39.11 5.29 14,796 13.95 Notes This Table presents net present values for five different cost schedules. ‘SCC’ is the social cost of carbon in year 2000 US dollars per ton of carbon. ‘current NDA’ costs are taken from the Nuclear Decommissioning Authority annual report and accounts 2012/3. The ‘delayed NDA’ costs are the same as those for current decommissioning, but delayed by 50 years to reflect the process of running down a new generation of nuclear power plants. Teenage obesity costs monetise the estimated impacts that are realised between ages 41 and 65 from being obese at the age of 14. These values are based on the calibration of Wang et al. (2003). The HS2 benefits are taken from the estimates of net transport benefits provided at the hs2.org.uk website. N denotes the equivalent number of independent observers, ‘Gamma’ refers to the gamma discounting schedule provided by Weitzman (2001). ‘Green Book’ applies the UK Treasury's current recommended schedule of discount rates. ‘Flat 4%’ applies a non‐declining discount rate of 4% at all horizons. Open in new tab Table 1 The Present Value of Intergenerational Projects . SCC ($/tC) . Current NDA (£ bn) . Delayed NDA (£ bn) . Teenage obesity ($) . HS2 benefits (£ bn) . N = 1,000 5.692 39.06 5.27 14,753 13.91 N = 100 5.738 39.08 5.32 14,780 13.95 N = 50 5.706 38.93 5.28 14,660 13.84 N = 25 5.543 38.42 5.10 14,238 13.46 N = 12 5.346 37.56 4.88 13,560 12.86 Gamma 15.928 44.97 13.41 20,767 21.01 Green Book 10.154 43.84 9.57 19,181 18.68 Flat 4% 5.713 39.11 5.29 14,796 13.95 . SCC ($/tC) . Current NDA (£ bn) . Delayed NDA (£ bn) . Teenage obesity ($) . HS2 benefits (£ bn) . N = 1,000 5.692 39.06 5.27 14,753 13.91 N = 100 5.738 39.08 5.32 14,780 13.95 N = 50 5.706 38.93 5.28 14,660 13.84 N = 25 5.543 38.42 5.10 14,238 13.46 N = 12 5.346 37.56 4.88 13,560 12.86 Gamma 15.928 44.97 13.41 20,767 21.01 Green Book 10.154 43.84 9.57 19,181 18.68 Flat 4% 5.713 39.11 5.29 14,796 13.95 Notes This Table presents net present values for five different cost schedules. ‘SCC’ is the social cost of carbon in year 2000 US dollars per ton of carbon. ‘current NDA’ costs are taken from the Nuclear Decommissioning Authority annual report and accounts 2012/3. The ‘delayed NDA’ costs are the same as those for current decommissioning, but delayed by 50 years to reflect the process of running down a new generation of nuclear power plants. Teenage obesity costs monetise the estimated impacts that are realised between ages 41 and 65 from being obese at the age of 14. These values are based on the calibration of Wang et al. (2003). The HS2 benefits are taken from the estimates of net transport benefits provided at the hs2.org.uk website. N denotes the equivalent number of independent observers, ‘Gamma’ refers to the gamma discounting schedule provided by Weitzman (2001). ‘Green Book’ applies the UK Treasury's current recommended schedule of discount rates. ‘Flat 4%’ applies a non‐declining discount rate of 4% at all horizons. Open in new tab 5. Conclusion We have shown that the term structure of discount rates that results from Weitzman's (2001) survey is highly dependent on whether the responses reflect forecasts of future risk‐free interest rates or the ethics of intergenerational equity. In the former case, very long‐term present value calculations barely differ from those calculated using a flat‐term structure. In the latter case, the term structure can decline rapidly. We have demonstrated that this has important implications across a wide range of policy areas for those making decisions with intergenerational consequences. In our online Appendix E we make a further point. Even in a purely normative world, standard approaches potentially exaggerate the decline. This is because such responses frequently contain ex post verifiable elements, such as the growth rate of per capita consumption. A mixed normative–positivist approach is therefore recommended in this case, which again flattens the term structure. Gamma discounting has been highly influential in shaping the international policy landscape on DDRs. This study shows that closer scrutiny on both the motives behind individual responses and the empirics of aggregation is required before any further policy changes can be justified on the basis of such surveys of expert opinion. Footnotes 1 " Within the UK, the Treasury‐recommended DDR schedule forms the basis for the economic evaluation of the High Speed 2 (HS2) rail link by government. It is also used for capital budgeting purposes by the Nuclear Decommissioning Authority. 2 " There are, of course, a range of other possible interpretations between these two extremes. The supplementary wording of the survey included such terms as ‘gut feeling’, ‘back‐of‐the‐envelope guesstimate’ and ‘off the top of your head’, making it clear that a variety of rationales could have underpinned any given expert's response. In addition, the sample may mix some purely normative responses with others that were purely positive. We dichotomise the debate for reasons of simplicity and clarity and note that this distinction is well understood in the literature. 3 " The theoretical case for using ENPVs within environmental economics was presented by Weitzman (1998) and subsequently discussed in detail by Gollier (2009), Freeman (2010), Gollier and Weitzman (2010) and Traeger (2013), amongst others. In our online Appendix B we note that (4) also has a long tradition in financial economics. In particular, the Local Expectations Hypothesis of Cox et al. (1981) is equivalent to the ENPV condition, but the underlying assumptions concerning the stochastic nature of r¯H and the resolution of uncertainty are much less stylised than in the original thought experiment of Weitzman (1998). 4 " We concentrate on exponentially correlated forecasts, which have been used in the context of gamma distributions by Kotz and Adams (1964). Forecasts are ranked in ascending order and the correlation between the ith and jth expert is assumed to be ρi−j for constant ρ. The cases N ∈ {1,000, 100, 50, 25, 12} correspond to ρ ∈ {35.71%, 90.95%, 95.37%, 97.66%, 98.87%}. See online Appendix C for further details. References Arrow , K. , Cline , W., Maler , K.G., Munasinghe , M., Squitieri , R. and Stiglitz , J. ( 1996 ). ‘Intertemporal equity, discounting, and economic efficiency’, in ( H.L.J.P. Bruce and E. Haites, eds.), Climate Change 1995: Economic and Social Dimensions of Climate Change, Contribution of Working Group III to the Second Assessment Report of the Intergovernmental Panel on Climate Change , pp. 125 – 44 , Cambridge : Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Arrow , K. , Cropper , M., Gollier , C., Groom , B., Heal , G., Newell , R., Nordhaus , W., Pindyck , R., Pizer , W., Portney , P., Sterner , T., Tol , R. and Weitzman , M. ( 2012 ). ‘ How should benefits and costs be discounted in an intergenerational context? The views of an expert panel ’, Discussion Paper, RFF DP 12‐53, Resources for the Future. Arrow , K. , Cropper , M., Gollier , C., Groom , B., Heal , G., Newell , R., Nordhaus , W., Pindyck , R., Pizer , W., Portney , P., Sterner , T., Tol , R. and Weitzman , M. ( 2013 ). ‘ Determining benefits and costs for future generations ’, Science , vol. 341 ( 6144 ), pp. 349 – 50 . 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( 2013 ). ‘ Declining discount rates and the Fisher Effect: Inflated past, discounted future? ’, Grantham Research Institute on Climate Change and the Environment Working Paper 109, London School of Economics. Genest , C . and Zidek , J.V. ( 1986 ). ‘ Combining probability distributions: a critique and an annotated biliography ’, Statistical Science , vol. 1 ( 1 ), pp. 114 – 35 . Google Scholar Crossref Search ADS WorldCat Gollier , C . ( 2009 ). ‘ Should we discount the far‐distant future at its lowest possible rate? ’, Economics ‐ The Open‐Access, Open‐Assessment E‐Journal , vol. 3 ( 2009–25 ), pp. 1 – 14 . Google Scholar Crossref Search ADS WorldCat Gollier , C . ( 2012 ). Pricing the Planet's Future: The Economics of Discounting in an Uncertain World , Princeton, NJ : Princeton University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Gollier , C . and Weitzman , M.L. ( 2010 ). ‘ How should the distant future be discounted when discount rates are uncertain? ’, Economics Letters , vol. 107 ( 3 ), pp. 350 – 3 . Google Scholar Crossref Search ADS WorldCat Gollier , C . and Zeckhauser , R. ( 2005 ). ‘ Aggregation of heterogeneous time preferences ’, Journal of Political Economy , vol. 113 ( 4 ), pp. 878 – 96 . Google Scholar Crossref Search ADS WorldCat Graham , J.R . ( 1996 ). ‘ Is a group of economists better than one? Than none? ’, Journal of Business , vol. 69 ( 2 ), pp. 193 – 232 . Google Scholar Crossref Search ADS WorldCat Groom , B. , Koundouri , P., Panopoulou , E. and Pantelidis , T. ( 2007 ). ‘ Discounting the distant future: how much does model selection affect the certainty equivalent rate? ’, Journal of Applied Econometrics , vol. 22 ( 3 ), pp. 641 – 56 . Google Scholar Crossref Search ADS WorldCat Heal , G . ( 2012 ). ‘ Discounting – uncertainty, disagreement or both? ’, mimeo, Columbia Business School . Jouini , E. , Marin , J.M. and Napp , C. ( 2010 ). ‘ Discounting and divergence of opinion ’, Journal of Economic Theory , vol. 145 ( 2 ), pp. 830 – 59 . Google Scholar Crossref Search ADS WorldCat Kotz , S . and Adams , J.W. ( 1964 ). ‘ Distribution of sum of identically distributed exponentially correlated gamma‐variables ’, Annals of Mathematical Statistics , vol. 35 ( 1 ), pp. 277 – 83 . Google Scholar Crossref Search ADS WorldCat Newell , R.G . and Pizer , W.A. ( 2003 ). ‘ Discounting the distant future: How much do uncertain rates increase valuations? ’, Journal of Environmental Economics and Management , vol. 46 ( 1 ), pp. 52 – 71 . Google Scholar Crossref Search ADS WorldCat Nordhaus , W.D . ( 2007 ). ‘ A review of the “Stern Review on the Economics of Climate Change ”’, Journal of Economic Literature , vol. 45 ( 3 ), pp. 686 – 702 . Google Scholar Crossref Search ADS WorldCat Stern , N . ( 2007 ). The Economics of Climate Change: The Stern Review , Cambridge and New York : Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Stern , N . ( 2008 ). ‘ The economics of climate change ’, American Economic Review: Papers & Proceedings , vol. 98 ( 2 ), pp. 1 – 37 . Google Scholar Crossref Search ADS WorldCat Traeger , C.P. ( 2013 ). ‘Discounting under uncertainty: disentangling the Weitzman and the Gollier effect ’, Journal of Environmental Economics and Management , vol. 66 ( 3 ), pp. 573 – 82 . Google Scholar Crossref Search ADS WorldCat UK Treasury . ( 2003 ). The Green Book: Appraisal and Evaluation in Central Government , London : UK Treasury . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Wang , L.Y. , Yang , Q., Lowry , R. and Wechsler , H. ( 2003 ). ‘ Economic analysis of a school‐based obesity prevention program ’, Obesity Research , vol. 11 ( 11 ), pp. 1313 – 24 . 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Spenkuch, Jörg, L.

doi: 10.1111/ecoj.12131pmid: N/A

Abstract Whether individuals vote strategically is one of the most important questions at the intersection of economics and political science. Exploiting a flaw in the German electoral system by which a party may gain seats by receiving fewer votes, this article documents patterns of preference misrepresentation in a large, real‐world election. During the 2005 elections to the Bundestag, the sudden death of a right‐wing candidate necessitated a by‐election in one electoral district. Knowing the results in all other districts and given the paradoxical incentives in place, a substantial fraction of the electorate voted for a party other than their most preferred one, or abstained. Elections are a cornerstone of any modern democracy. Social choice theory, however, has shown almost all voting systems to be susceptible to strategic manipulation (Arrow etal., 1951; Gibbard, 1973; Satterthwaite, 1975). That is, in theory, voters have a systematic incentive to misrepresent their true preferences in order to affect the result of an election. Although tactical voting is individually rational, it may be socially undesirable, as it precludes the proper aggregation of preferences, and can thereby lead to inferior electoral outcomes. Hence, with a significant number of voters behaving strategically, lack of strategy‐proofness in social choice mechanisms may have important real‐world ramifications.1 Yet, whether individuals actually cast strategic ballots to avoid ‘wasting their vote’ (Duverger, 1954) remains one of the most important open questions at the intersection of economics and political science. Although strategic voting is well understood in theory (Myerson and Weber, 1993; Cox, 1994; Carroll, 2011), the unobservability of preferences has made it very difficult to document tactical behaviour empirically – despite substantial anecdotal evidence.2 In fact, in an important paper, Degan and Merlo (2009) prove that one can always find preference profiles to rationalise any cross section of votes. Some even argue that if there are any (psychic) costs associated with performing the ‘calculus of voting’ (Riker and Ordeshook, 1968), then individuals cannot possibly be expected to behave strategically (Downs, 1957; Green and Shapiro, 1994). After all, in large elections the probability of casting the pivotal vote becomes vanishingly small. Of course, how voters actually behave is ultimately an empirical question. Given the difficulties in identifying strategic voting, it may not be surprising that the existing literature draws mixed conclusions. Although Coate et al. (2008) reject the pivotal‐voter model, Cox (1997) presents a swath of evidence broadly consistent with strategic behaviour in different electoral systems. More recently, Kawai and Watanabe (2013) estimate a structural model of preferences and voting decisions, from which they infer that between 1.4% and 4.2% of voters actually misrepresent their preferences. At the same time, Kawai and Watanabe (2013) argue that at least 63.4% of voters would do so if their preferred candidate was not in contention for victory. Relying on survey responses about preferences, expectations and votes, a large number of earlier studies argue that the prevalence of instrumentally rational voting is actually very low. Estimates typically range from 3% to 17% (Abramson et al., 1992; Niemi et al., 1992; Blais et al., 2001), but may be subject to severe survey biases (Alvarez and Nagler, 2000).3 Instead of taking survey responses at face value or imposing strong assumptions about voters' underlying preferences, this article contributes to the existing literature by providing evidence from a natural experiment. The empirical strategy relies on a geographically localised, temporary reversal of incentives due to a flaw in the German electoral system by which a party may actually gain seats if it receives fewer votes. Using multiple years of election data and comparing the behaviour of affected voters with that of unaffected ones (exploiting both geographic variation and variation over time), identification in this article is not subject to the problems that plague most of the existing literature.4 Eleven days before the 2005 federal elections, the candidate (Direktkandidat) of the far right‐wing National Democratic Party (NPD) in Electoral District 160 (Dresden I) passed away unexpectedly. Although she was never expected to win, electoral law required a by‐election. But in contrast to prior cases, her death occurred too close to election day for the by‐election to be held on the originally scheduled date. Consequently, almost 220,000 eligible voters in Saxony's District 160 were given the opportunity to cast their ballots two weeks after everybody else, and after the Federal Returning Officer (Bundeswahllleiter) had announced the preliminary results of the election in all other districts. Based on these results, the Christian Democratic Union (CDU) and its Bavarian sister party, the Christian Social Union (CSU), won 225 seats in the Bundestag, whereas the rival Social Democratic Party (SPD) received only 222. Under ordinary circumstances, one would hardly expect less than 0.5% of the electorate to influence the outcome of an election, especially not when the allocation of seats is approximately proportional to the number of votes. Yet, in 2005, a few hundred votes could make the difference between a two, three or even four‐seat lead for the CDU/CSU. Although the German electoral system aims for proportional representation of all parties clearing a 5% threshold, determining a party's exact number of seats is substantially more complicated. The crucial point is that individuals cast two different votes – a list vote and a candidate vote – and that a party may lose a seat in the legislature by obtaining ‘too many’ list votes in states in which it won the plurality of the candidate vote in sufficiently many districts (see Section 1 for a detailed explanation). In 2005, exactly this situation occurred. Figure 1 illustrates the paradoxical incentives in place during the by‐election. If the CDU were to win more than approximately 41,000 list votes in Saxony's District 160 – it had received 49,638 in the 2002 election – then it would lose a seat in parliament. On the other hand, it might even increase its seat total by one if it garnered few list votes, but the plurality of the candidate vote.5 Fig. 1. Open in new tabDownload slide Seats Accruing to the CDU in the 16th Bundestag as a Function of List Votes in District 160 Notes. Figure shows the total number of seats accruing the CDU as a function of its number of list votes in District 160, and whether it wins the outstanding direct mandate. Although not essential for the location of the discontinuty or the general shape of the seat–vote curve, the Figure is drawn under the assumption that all other parties receive a number of list votes similar to that in the 2002 election. As the CSU won 46 seats, the total number of seats of the CDU/CSU faction could range from 224 to 226. Fig. 1. Open in new tabDownload slide Seats Accruing to the CDU in the 16th Bundestag as a Function of List Votes in District 160 Notes. Figure shows the total number of seats accruing the CDU as a function of its number of list votes in District 160, and whether it wins the outstanding direct mandate. Although not essential for the location of the discontinuty or the general shape of the seat–vote curve, the Figure is drawn under the assumption that all other parties receive a number of list votes similar to that in the 2002 election. As the CSU won 46 seats, the total number of seats of the CDU/CSU faction could range from 224 to 226. Critically, as the by‐election was conducted almost two weeks after the preliminary results had been announced, the electorate in District 160 was potentially aware of the peculiar circumstances before going to the polls.6 In fact, due to the closeness of the national race, the by‐election received considerable media coverage with pundits lamenting the perverse incentives. Consequently, if voters reacted to the change in incentives, one would expect the following:7 (i) The CDU should garner fewer list votes in the by‐election than under ordinary circumstances; but (ii) it should receive a larger share of the candidate vote. In the end, the CDU did win the plurality of the candidate vote, while receiving ‘only’ 38,208 list votes. It was, therefore, able to extend its narrow lead over the SPD. A priori, however, it is not clear whether this would have happened even in the absence of the by‐election, or whether voters misrepresented their preferences to affect the outcome of the election. The empirical evidence presented in this article points clearly towards misrepresentation of preferences. That is, some individuals did not vote for the party or candidate that was ideologically closest to them. For instance, the upper panel in Table 1 shows that between 2002 and 2005, the CDU's share of list votes declined by 6.1 percentage points in District 160, whereas it decreased only 3.4 percentage points in Saxony's other districts, or 3.0 percentage points in East Germany as a whole. Yet, between 2005 and 2009, the CDU gained 9.9 percentage points in District 160, relative to 5.3 and 3.4 percentage points in the comparison groups. By contrast, the lower panel in Table 1 indicates that going from the 2002 to the subsequent 2005 elections, the CDU's share of the candidate vote actually increased in District 160, while it decreased in the remainder of Saxony.8 The opposite is true for the difference between 2005 and 2009. Thus, consistent with voters behaving tactically, the raw data suggest that the CDU won an unusually low fraction of list votes but received a higher than normal share of the by‐election's candidate vote. Table 1 List Votes Accruing to the CDU in the 2002, 2005 and 2009 Elections to the Bundestag . Share of list vote . Δ Share of list vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (a) List vote Electoral district 160 30.5 24.4 34.3 −6.1 9.9 Other districts in Saxony 33.8 30.4 35.7 −3.4 5.3 Other states in East Germany 27.0 24.0 27.4 −3.0 3.4 . Share of list vote . Δ Share of list vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (a) List vote Electoral district 160 30.5 24.4 34.3 −6.1 9.9 Other districts in Saxony 33.8 30.4 35.7 −3.4 5.3 Other states in East Germany 27.0 24.0 27.4 −3.0 3.4 . Share of candidate vote . Δ Share of candidate vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (b) Candidate vote Electoral district 160 33.8 37.0 36.6 3.2 −0.4 Other districts in Saxony 37.3 35.4 39.7 −1.8 4.3 Other states in East Germany 28.6 27.2 29.6 −1.5 2.4 . Share of candidate vote . Δ Share of candidate vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (b) Candidate vote Electoral district 160 33.8 37.0 36.6 3.2 −0.4 Other districts in Saxony 37.3 35.4 39.7 −1.8 4.3 Other states in East Germany 28.6 27.2 29.6 −1.5 2.4 Source. Author's calculations based on Bundeswahlleiter (2002a; 2005a; 2009a). Open in new tab Table 1 List Votes Accruing to the CDU in the 2002, 2005 and 2009 Elections to the Bundestag . Share of list vote . Δ Share of list vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (a) List vote Electoral district 160 30.5 24.4 34.3 −6.1 9.9 Other districts in Saxony 33.8 30.4 35.7 −3.4 5.3 Other states in East Germany 27.0 24.0 27.4 −3.0 3.4 . Share of list vote . Δ Share of list vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (a) List vote Electoral district 160 30.5 24.4 34.3 −6.1 9.9 Other districts in Saxony 33.8 30.4 35.7 −3.4 5.3 Other states in East Germany 27.0 24.0 27.4 −3.0 3.4 . Share of candidate vote . Δ Share of candidate vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (b) Candidate vote Electoral district 160 33.8 37.0 36.6 3.2 −0.4 Other districts in Saxony 37.3 35.4 39.7 −1.8 4.3 Other states in East Germany 28.6 27.2 29.6 −1.5 2.4 . Share of candidate vote . Δ Share of candidate vote . . 2002 (%) . 2005 (%) . 2009 (%) . 2005−2002 (%) . 2009−2005 (%) . (b) Candidate vote Electoral district 160 33.8 37.0 36.6 3.2 −0.4 Other districts in Saxony 37.3 35.4 39.7 −1.8 4.3 Other states in East Germany 28.6 27.2 29.6 −1.5 2.4 Source. Author's calculations based on Bundeswahlleiter (2002a; 2005a; 2009a). Open in new tab Using hitherto unavailable official election data on the sub‐district level, the main result of this aticle establishes that the disparities documented in Table 1 are quite robust (even to the inclusion of municipality specific trends) and unlikely to be due to chance. Moreover, exploiting the different incentive structures associated with candidate and list votes, the evidence presented below indicates that supporters of the rival SPD voted for the CDU (in an attempt to hurt it), whereas adherents of the latter either abstained or substituted towards the libertarian Free Democratic Party (FDP) – the CDU's traditional coalition partner. Although the by‐election was a highly unusual event, the point estimates indicate that at least 8.8% of the electorate in District 160 misrepresented their preferences when faced with the new incentive structure. Moreover, the findings in this article confirm the long‐standing concern that voters’ misrepresentation of preferences might distort electoral outcomes. It is important to note at the outset that the main estimates in this article refer to voters misrepresenting their preferences, and that this behaviour can, but need not, be strategically motivated. For instance, some voters may be naïve or ‘behavioural’ and simply follow the official party line. These voters might still misrepresent their true preferences if party leaders made such a request, but one would not necessarily want to call them ‘strategic’. These agents do, however, vote ‘non‐ideologically’ in the sense of Degan and Merlo (2009). That is, they do not vote for the candidate or party that is ideologically closest to them. For the issue of strategy‐proofness in social choice, it is inconsequential why voters misrepresent their preferences. After all, the possibility that misaligned votes may affect electoral outcomes remains regardless of voters’ motivation. It is, therefore, important to empirically document patterns of preference misrepresentation. Ancillary results suggest that individuals did not merely follow parties’ official recommendations as to how to cast their votes, but at least some agents acted ‘strategically’ in the narrower sense of the word. That is, the estimates presented below speak not only to the question of whether voters do, in fact, misrepresent their true preferences but also as to why they might do so. The remainder of the article proceeds as follows. Section 1 provides background information on the electoral system used in Germany's elections to the Bundestag, as well as further details on the natural experiment. Section 1 describes the data and Section 3 contains the empirical results. The last Section concludes. An online Data Appendix with the precise definitions and sources of all variables is also provided. 1. Germany's Electoral System and the Case of District 160 Elections of representatives to the Federal Diet of Germany (Bundestag) are held according to a mixed member system with approximately proportional representation. Except for minor modifications, the same system has been in place since 1957; see Bawn, (1993) for an account of its genesis. This Section explains the specifics of the German voting system in which a party may gain seats by receiving fewer votes.9 Furthermore, it provides additional information on the by‐election in District 160 and the perverse incentives in place. Readers satisfied with the description in the introduction may skip the section without much loss in continuity. 1.1. Perverse Incentives in Germany's Federal Elections As mentioned before, each voter in Germany casts two votes. The first, or candidate vote (Erststimme), is used to elect a constituency representative in each of 299 single‐member districts according to plurality rule.10 The winners of these races hold direct mandates, and are automatically awarded a seat in the legislature. More importantly, the second, or list vote (Zweitstimme), is cast for a party list, and the total number of party members in the Bundestag is roughly proportional to a party's share of the national list vote among parties clearing a 5% threshold.11 Approximately, proportional representation is achieved by deducting direct mandates from the number of mandates to which parties are entitled based on the list vote – their list mandates.12 Deviations from strict proportionality are due to three factors: the 5% threshold, rounding and overhang mandates (Überhangmandate). The latter are a peculiarity of the German system that arises when in some state a party wins more direct mandates than seats under proportional representation. In such cases, the total number of seats in the Bundestag is raised and said party gets to keep all direct mandates without losing seats in other states.13 More formally, let dp,s denote the number of direct mandates accruing to party p in state s. vp,s is the number of list votes that p received in s, with the equivalent number on the national level given by v~p ⁠, that is, v~p=∑svp,s ⁠. With this notation in hand, party p's seat total is calculated in three steps: Step 1: Proportional Allocation of List Mandates to Parties. In the absence of overhang mandates, there are 598 seats in the Bundestag. These are allocated by proportionality rule to the set of parties clearing the 5% threshold or winning at least three direct mandates. That is, the number of list mandates of party p equals l~p≅598×v~p/∑p′∈Pv~p′ifp∈P0ifp∉P, where P={p|v~p/∑p′v~p′≥0.05∨∑sdp,s≥3} and ≅ represents equality after rounding according to the method of Hare–Niemeyer.14 Also known as the ‘largest remainder method’, Hare–Niemeyer first assigns each party a number of seats equal to the integer part of 598×v~p/∑p′∈Pv~p′ ⁠. The parties with the largest remainders are then allocated one additional list mandate until all available seats have been distributed. This ensures that ∑pl~p=598 ⁠. Step 2: Proportional Allocation of Mandates to State Lists. By law, parties are required to decide on different lists in each state. Hence, l~p needs to be broken down to the state level, lp,s, with l~p=∑slp,s ⁠. Again, this is done in approximately proportional fashion. More precisely, for all s and all p, lp,s≅l~p×vp,s/v~pifp∈P0ifp∉P,(1) where ≅ is defined as above. Step 3: Determination of the Actual Number of Seats. The actual number of seats that party p receives in state s is given by np,s=maxdp,s,lp,s∀p.(2) If dp,s < lp,s then, in addition to the district winners, the first lp,s − dp,s candidates on p's list in s are elected to the Bundestag as well. Otherwise, only holders of direct mandates receive a seat. Note that unless a party can secure overhang mandates, that is, unless dp,s > lp,s for some s, its seat total, n~p=∑snp,s ⁠, equals the number of seats it would be assigned under proportional representation, l~p ⁠. Empirically, overhang mandates are not uncommon – 15 occurred after the 2005 elections and 24 after those in 2009.15 Importantly for the purposes of this article, there also exist scenarios in which a party loses a seat by gaining list votes. To see this, consider a small increase in vp,s, not large enough to affect l~p ⁠. Although the total number of p's list mandates does not change, even a small gain in vp,s may be enough for l~p×vp,s/v~p in (1) to be rounded upwards instead of downwards. But for l~p=∑slp,s to continue to hold, an increase in lp,s must result in a corresponding reduction in lp,s′ for some s′ ≠ s. If dp,s > lp,s and dp,s′<lp,s′ ⁠, then according to (2) that small increase in votes would actually lower party p's seat total. In other words, a small gain in the number of list votes may lead to the reassignment of a list mandate from a state in which a party won few direct mandates to one in which it secured more direct mandates than seats under proportional representation. But as the actual number of seats it receives in any state equals the maximum of direct and list mandates (Step 3), reallocating a list mandate from one state to another may lead to a lower seat total. In general, the occurrence of such a situation is very difficult to predict in advance – primarily because the inherent uncertainty in vp,s makes rounding in (1) almost impossible to anticipate. This was not the case, however, leading up the 2005 elections in District 160. 1.2. The Case of District 160 On 5 September, 2005, Kerstin Lorenz, running as the NPD's direct candidate in Electoral District 160 (Dresden I), suffered a stroke during a campaign event. As a result of the stroke she passed away on 7 September – 11 days before the elections to the Bundestag. Although Lorenz had virtually no chance of winning, German electoral law requires a by‐election whenever a direct candidate dies prior to election day (§ 43 BWG and § 82 BWO). In similar instances, by‐elections were usually held on election day itself. But in the case of District 160, Lorenz's death occurred too close to the originally scheduled date for the NPD to be given sufficient time to nominate another candidate and for new ballots to be printed. Thus, the electorate in District 160 was asked to go to the polls on 2 October, while elections in all other districts took place on 18 September.16 Yet, by 2 October, the Federal Returning Officer (Bundeswahlleiter) had already announced the preliminary official results of the election (Bundeswahlleiter, 2005c). According to these results, the CDU was entitled to 179 seats in the Bundestag, and its Bavarian sister party, the CSU, would be represented by 46 delegates. The rival SPD received only 222 mandates. Hence, prior to the by‐election in District 160, the CDU/CSU had a narrow three‐seat lead over the SPD, making it the largest faction.17 Table 2 displays the preliminary results (excluding District 160), as well as the calculations used to determine the CDU's seat total. Based on these results and past outcomes in District 160, no party could hope to win an additional list mandate on the national level (Step 1). Focusing on the second step, however, it is straightforward to verify that the CDU found itself in the perverse situation described above. If say, 42,000 voters cast their list votes for the CDU, then it would receive 11 instead of 10 list mandates in Saxony, and 46 instead of 47 in the state of North Rhine‐Westphalia. But given the number of direct mandates it had already won in these states, receiving ‘too many’ list votes in the by‐election would actually cost the CDU a seat in the Bundestag (Step 3). Table 2 Determining the Number of Seats for the CDU in the 16th Bundestag . Preliminary result (excluding district 160) . Final result . Party . Number of list votes . Equivalent number of seats . Rounded number of seats . Number of list votes . Equivalent number of seats . Rounded number of seats . Step 1: proportional allocation of list mandates to parties SPD 16,148,240 213.307 213 16,194,665 213.170 213 CDU 13,096,556 172.996 173 13,136,740 172.919 173 CSU 3,494,564 46.161 46 3,494,309 45.996 46 Green party 3,826,194 50.541 51 3,838,326 50.524 51 FDP 4,619,519 61.021 61 4,648,144 61.184 61 The Left 4,086,134 53.975 54 4,118,194 54.208 54 Total 45,271,207 598.000 598 45,430,378 598.000 598 . Preliminary result (excluding district 160) . Final result . Party . Number of list votes . Equivalent number of seats . Rounded number of seats . Number of list votes . Equivalent number of seats . Rounded number of seats . Step 1: proportional allocation of list mandates to parties SPD 16,148,240 213.307 213 16,194,665 213.170 213 CDU 13,096,556 172.996 173 13,136,740 172.919 173 CSU 3,494,564 46.161 46 3,494,309 45.996 46 Green party 3,826,194 50.541 51 3,838,326 50.524 51 FDP 4,619,519 61.021 61 4,648,144 61.184 61 The Left 4,086,134 53.975 54 4,118,194 54.208 54 Total 45,271,207 598.000 598 45,430,378 598.000 598 . Preliminary result (excluding district 160) . Final result . State . Number of list votes . Equivalent number of seats . Seats under proportionality . Number of list votes . Equivalent number of seats . Seats under proportionality . Step 2: proportional allocation of mandates to state lists (CDU) Schleswig‐Holstein 623,922 8.242 8 624,510 8.224 8 Hamburg 272,798 3.604 4 272,418 3.588 4 Lower Saxony 1,599,867 21.134 21 1,599,947 21.070 21 Bremen 82,411 1.089 1 82,389 1.085 1 North Rhine‐Westphalia 3,524,374 46.555 47 3,524,351 46.413 46 Hesse 1,130,099 14.928 15 1,131,496 14.901 15 Rhineland‐Palatinate 877,213 11.588 12 877,632 11.558 12 Baden‐Württemberg 2,282,729 30.154 30 2,283,085 30.066 30 Saarland 191,065 2.524 2 191,067 2.516 3 Berlin 408,809 5.400 5 408,715 5.382 5 Mecklenburg‐Vorpommern 293,278 3.874 4 293,316 3.863 4 Brandenburg 322,394 4.259 4 322,400 4.246 4 Saxony‐Anhalt 357,638 4.724 5 357,663 4.710 5 Thuringia 372,593 4.922 5 372,435 4.905 5 Saxony 757,366 10.004 10 795,316 10.474 10 Total 13,096,556 173.000 173 13,136,740 173.00 173 . Preliminary result (excluding district 160) . Final result . State . Number of list votes . Equivalent number of seats . Seats under proportionality . Number of list votes . Equivalent number of seats . Seats under proportionality . Step 2: proportional allocation of mandates to state lists (CDU) Schleswig‐Holstein 623,922 8.242 8 624,510 8.224 8 Hamburg 272,798 3.604 4 272,418 3.588 4 Lower Saxony 1,599,867 21.134 21 1,599,947 21.070 21 Bremen 82,411 1.089 1 82,389 1.085 1 North Rhine‐Westphalia 3,524,374 46.555 47 3,524,351 46.413 46 Hesse 1,130,099 14.928 15 1,131,496 14.901 15 Rhineland‐Palatinate 877,213 11.588 12 877,632 11.558 12 Baden‐Württemberg 2,282,729 30.154 30 2,283,085 30.066 30 Saarland 191,065 2.524 2 191,067 2.516 3 Berlin 408,809 5.400 5 408,715 5.382 5 Mecklenburg‐Vorpommern 293,278 3.874 4 293,316 3.863 4 Brandenburg 322,394 4.259 4 322,400 4.246 4 Saxony‐Anhalt 357,638 4.724 5 357,663 4.710 5 Thuringia 372,593 4.922 5 372,435 4.905 5 Saxony 757,366 10.004 10 795,316 10.474 10 Total 13,096,556 173.000 173 13,136,740 173.00 173 . Preliminary result (excluding district 160) . Final result . State . Number of direct mandates . Seats under proportionality . Actual number of seats . Number of direct mandates . Seats under proportionality . Actual number of seats . Step 3: determination of the actual number of seats (CDU) Schleswig‐Holstein 6 8 8 6 8 8 Hamburg 0 4 4 0 4 4 Lower Saxony 4 21 21 4 21 21 Bremen 0 1 1 0 1 1 North Rhine‐Westphalia 24 47 47 24 46 46 Hesse 8 15 15 8 15 15 Rhineland‐Palatinate 10 12 12 10 12 12 Baden‐Württemberg 33 30 33 33 30 33 Saarland 0 2 2 0 3 3 Berlin 1 5 5 1 5 5 Mecklenburg‐Vorpommern 3 4 4 3 4 4 Brandenburg 0 4 4 0 4 4 Saxony‐Anhalt 0 5 5 0 5 5 Thuringia 3 5 5 3 5 5 Saxony 13 10 13 14 10 14 Total 105 173 179 106 173 180 . Preliminary result (excluding district 160) . Final result . State . Number of direct mandates . Seats under proportionality . Actual number of seats . Number of direct mandates . Seats under proportionality . Actual number of seats . Step 3: determination of the actual number of seats (CDU) Schleswig‐Holstein 6 8 8 6 8 8 Hamburg 0 4 4 0 4 4 Lower Saxony 4 21 21 4 21 21 Bremen 0 1 1 0 1 1 North Rhine‐Westphalia 24 47 47 24 46 46 Hesse 8 15 15 8 15 15 Rhineland‐Palatinate 10 12 12 10 12 12 Baden‐Württemberg 33 30 33 33 30 33 Saarland 0 2 2 0 3 3 Berlin 1 5 5 1 5 5 Mecklenburg‐Vorpommern 3 4 4 3 4 4 Brandenburg 0 4 4 0 4 4 Saxony‐Anhalt 0 5 5 0 5 5 Thuringia 3 5 5 3 5 5 Saxony 13 10 13 14 10 14 Total 105 173 179 106 173 180 Notes The Table shows the calculations in determining the CDU's number of seats in the 16th Bundestag. The columns on the left are based on the preliminary results of the 2005 election, that is, before the constituency of District 160 voted, whereas the columns on the right use the final vote counts in all districts. As explained in the main text, Step 1 calculates the number of parties’ list mandates on the national level. Step 2 assigns these to party lists in individual states, and Step 3 determines the actual number of seats by taking the maximum of list and direct mandates. Rounding in Step 1 and 2 is done according to the method of Hare–Niemeyer. Source. Based on Bundeswahlleiter (2005a,2005c). Open in new tab Table 2 Determining the Number of Seats for the CDU in the 16th Bundestag . Preliminary result (excluding district 160) . Final result . Party . Number of list votes . Equivalent number of seats . Rounded number of seats . Number of list votes . Equivalent number of seats . Rounded number of seats . Step 1: proportional allocation of list mandates to parties SPD 16,148,240 213.307 213 16,194,665 213.170 213 CDU 13,096,556 172.996 173 13,136,740 172.919 173 CSU 3,494,564 46.161 46 3,494,309 45.996 46 Green party 3,826,194 50.541 51 3,838,326 50.524 51 FDP 4,619,519 61.021 61 4,648,144 61.184 61 The Left 4,086,134 53.975 54 4,118,194 54.208 54 Total 45,271,207 598.000 598 45,430,378 598.000 598 . Preliminary result (excluding district 160) . Final result . Party . Number of list votes . Equivalent number of seats . Rounded number of seats . Number of list votes . Equivalent number of seats . Rounded number of seats . Step 1: proportional allocation of list mandates to parties SPD 16,148,240 213.307 213 16,194,665 213.170 213 CDU 13,096,556 172.996 173 13,136,740 172.919 173 CSU 3,494,564 46.161 46 3,494,309 45.996 46 Green party 3,826,194 50.541 51 3,838,326 50.524 51 FDP 4,619,519 61.021 61 4,648,144 61.184 61 The Left 4,086,134 53.975 54 4,118,194 54.208 54 Total 45,271,207 598.000 598 45,430,378 598.000 598 . Preliminary result (excluding district 160) . Final result . State . Number of list votes . Equivalent number of seats . Seats under proportionality . Number of list votes . Equivalent number of seats . Seats under proportionality . Step 2: proportional allocation of mandates to state lists (CDU) Schleswig‐Holstein 623,922 8.242 8 624,510 8.224 8 Hamburg 272,798 3.604 4 272,418 3.588 4 Lower Saxony 1,599,867 21.134 21 1,599,947 21.070 21 Bremen 82,411 1.089 1 82,389 1.085 1 North Rhine‐Westphalia 3,524,374 46.555 47 3,524,351 46.413 46 Hesse 1,130,099 14.928 15 1,131,496 14.901 15 Rhineland‐Palatinate 877,213 11.588 12 877,632 11.558 12 Baden‐Württemberg 2,282,729 30.154 30 2,283,085 30.066 30 Saarland 191,065 2.524 2 191,067 2.516 3 Berlin 408,809 5.400 5 408,715 5.382 5 Mecklenburg‐Vorpommern 293,278 3.874 4 293,316 3.863 4 Brandenburg 322,394 4.259 4 322,400 4.246 4 Saxony‐Anhalt 357,638 4.724 5 357,663 4.710 5 Thuringia 372,593 4.922 5 372,435 4.905 5 Saxony 757,366 10.004 10 795,316 10.474 10 Total 13,096,556 173.000 173 13,136,740 173.00 173 . Preliminary result (excluding district 160) . Final result . State . Number of list votes . Equivalent number of seats . Seats under proportionality . Number of list votes . Equivalent number of seats . Seats under proportionality . Step 2: proportional allocation of mandates to state lists (CDU) Schleswig‐Holstein 623,922 8.242 8 624,510 8.224 8 Hamburg 272,798 3.604 4 272,418 3.588 4 Lower Saxony 1,599,867 21.134 21 1,599,947 21.070 21 Bremen 82,411 1.089 1 82,389 1.085 1 North Rhine‐Westphalia 3,524,374 46.555 47 3,524,351 46.413 46 Hesse 1,130,099 14.928 15 1,131,496 14.901 15 Rhineland‐Palatinate 877,213 11.588 12 877,632 11.558 12 Baden‐Württemberg 2,282,729 30.154 30 2,283,085 30.066 30 Saarland 191,065 2.524 2 191,067 2.516 3 Berlin 408,809 5.400 5 408,715 5.382 5 Mecklenburg‐Vorpommern 293,278 3.874 4 293,316 3.863 4 Brandenburg 322,394 4.259 4 322,400 4.246 4 Saxony‐Anhalt 357,638 4.724 5 357,663 4.710 5 Thuringia 372,593 4.922 5 372,435 4.905 5 Saxony 757,366 10.004 10 795,316 10.474 10 Total 13,096,556 173.000 173 13,136,740 173.00 173 . Preliminary result (excluding district 160) . Final result . State . Number of direct mandates . Seats under proportionality . Actual number of seats . Number of direct mandates . Seats under proportionality . Actual number of seats . Step 3: determination of the actual number of seats (CDU) Schleswig‐Holstein 6 8 8 6 8 8 Hamburg 0 4 4 0 4 4 Lower Saxony 4 21 21 4 21 21 Bremen 0 1 1 0 1 1 North Rhine‐Westphalia 24 47 47 24 46 46 Hesse 8 15 15 8 15 15 Rhineland‐Palatinate 10 12 12 10 12 12 Baden‐Württemberg 33 30 33 33 30 33 Saarland 0 2 2 0 3 3 Berlin 1 5 5 1 5 5 Mecklenburg‐Vorpommern 3 4 4 3 4 4 Brandenburg 0 4 4 0 4 4 Saxony‐Anhalt 0 5 5 0 5 5 Thuringia 3 5 5 3 5 5 Saxony 13 10 13 14 10 14 Total 105 173 179 106 173 180 . Preliminary result (excluding district 160) . Final result . State . Number of direct mandates . Seats under proportionality . Actual number of seats . Number of direct mandates . Seats under proportionality . Actual number of seats . Step 3: determination of the actual number of seats (CDU) Schleswig‐Holstein 6 8 8 6 8 8 Hamburg 0 4 4 0 4 4 Lower Saxony 4 21 21 4 21 21 Bremen 0 1 1 0 1 1 North Rhine‐Westphalia 24 47 47 24 46 46 Hesse 8 15 15 8 15 15 Rhineland‐Palatinate 10 12 12 10 12 12 Baden‐Württemberg 33 30 33 33 30 33 Saarland 0 2 2 0 3 3 Berlin 1 5 5 1 5 5 Mecklenburg‐Vorpommern 3 4 4 3 4 4 Brandenburg 0 4 4 0 4 4 Saxony‐Anhalt 0 5 5 0 5 5 Thuringia 3 5 5 3 5 5 Saxony 13 10 13 14 10 14 Total 105 173 179 106 173 180 Notes The Table shows the calculations in determining the CDU's number of seats in the 16th Bundestag. The columns on the left are based on the preliminary results of the 2005 election, that is, before the constituency of District 160 voted, whereas the columns on the right use the final vote counts in all districts. As explained in the main text, Step 1 calculates the number of parties’ list mandates on the national level. Step 2 assigns these to party lists in individual states, and Step 3 determines the actual number of seats by taking the maximum of list and direct mandates. Rounding in Step 1 and 2 is done according to the method of Hare–Niemeyer. Source. Based on Bundeswahlleiter (2005a,2005c). Open in new tab However, if it received fewer than 41,000 list votes, then the number of list mandates in the state of Saxony would remain at 10 (and potentially increase from two to three in the Saarland offset by a corresponding reduction in North Rhine‐Westphalia). Thus, by winning the direct mandate in District 160 while not receiving ‘too many’ list votes, the CDU/CSU faction could even gain a seat (Figure 1). Due to the closeness of the national race, the by‐election received substantial attention from the media. Although coverage focused mostly on the competition for the outstanding direct mandate, a number of pundits also commented on the perverse incentives associated with the list vote. Moreover, some parties adapted their campaign strategies.18 The CDU posted banners promoting its direct candidate, whereas the SPD contender used an interview with the local newspaper to ask supporters of the Green Party, whose nominee had virtually no chance of being elected, for their support (Sächsische Zeitung, 2005a). Only the libertarian FDP, however, came out with an explicit recommendation as to how their supporters should behave. Despite the fact that the FDP could not gain an additional seat in the Bundestag, it printed more than 1,000 new posters prompting voters to cast their first vote for the CDU candidate and their second vote for the FDP (see Figure 1).19 All other parties, including the CDU and the SPD, both of which stood to gain the most from voters’ strategic behaviour, were very hesitant to blatantly ask their supporters to cast tactical list votes – though, if prompted, party volunteers would explain the peculiar incentives. When asked about why his party did not endorse strategic voting, the CDU candidate, Andreas Lämmel, responded that he did not approve of such ‘games’ (Sächsische Zeitung, 2005c). And the SPD candidate, Marlies Volkmer, answered that the SPD ‘wants to be strongest party and doesn’t engage in any tactical considerations’ (Sächsische Zeitung, 2005c). Presumably sincere voters might penalise the party for attempting to ‘manipulate’ the by‐election. For similar reasons, the Green Party completely abstained from making any recommendations. Its representative only said Green Party supporters were smart enough to know what the best decision would be (Frankfurter Allgemeine Zeitung, 2005). The Left explicitly asked its followers to vote for them with both votes, just as in any other year. Fig. 2. Open in new tabDownload slide Campaign Poster Used by the Free Democratic Party Notes. The text translates to ‘Candidate Vote CDU’, ‘List Vote FDP’, and ‘Typical Dresden;‐) Black & Yellow. Good for Germany!’ Fig. 2. Open in new tabDownload slide Campaign Poster Used by the Free Democratic Party Notes. The text translates to ‘Candidate Vote CDU’, ‘List Vote FDP’, and ‘Typical Dresden;‐) Black & Yellow. Good for Germany!’ The columns on the right of Table 2 show the final result of the election. In the end, the CDU received ‘only’ 38,208 list votes, and won the plurality of the candidate vote.20 Therefore, it ended up gaining one seat. Given the intense media coverage, it seems reasonable to assume that a non‐trivial fraction of voters were aware of the reversed incentives. But as a single vote has almost no chance of being decisive, it is not clear whether individual voters would indeed react by misrepresenting their preferences, or whether the CDU would have won the additional seat even under ordinary circumstances. 2. Data Sources and Summary Statistics To answer this question, the present article relies on data from several sources. The primary data set consists of the official results of the 2002, 2005 and 2009 elections to the Bundestag by polling precinct (Wahlbezirk). In Germany, polling precincts are the smallest administrative units at which votes are counted. Each precinct is fully contained in an electoral district, and is associated with one polling station, where a returning officer is to ensure the lawfulness of the election. As a rule, no precinct should consist of more than 2,500 eligible voters. The data include information on the number of list and candidate votes for each party, the number of eligible voters, as well as the number of invalid votes. They have been obtained from the Federal Statistical Office and were, until recently, not publicly available. Although the total number of districts remained unchanged between 2002 and 2009, migration led to the redrawing of electoral districts in a small number of instances. In particular, the state of Saxony contained 17 districts in 2002 and 2005, but only 16 in 2009 (see Figure 1 for a map of electoral districts in the state of Saxony as of 2005). To ensure comparability over time, this article constructs a consistent mapping between polling precincts and electoral districts. The mapping relies on municipality identifiers contained in the raw data, as well as the appendices to the Bundeswahlgesetz, which in every election year list all municipalities in any given district. Thus, knowing in which municipality a precinct is located and given the information on the district to which this municipality belonged in 2005, it is relatively straightforward to map precincts into geographically constant districts. All results build on this mapping.21 Fig. 3. Open in new tabDownload slide Electoral Districts in the State of Saxony as of the 2005 Elections to the Bundestag Source. Based on Bundeswahlleiter (2005b) and Landesvermessungsamt Sachsen (2006). Fig. 3. Open in new tabDownload slide Electoral Districts in the State of Saxony as of the 2005 Elections to the Bundestag Source. Based on Bundeswahlleiter (2005b) and Landesvermessungsamt Sachsen (2006). Information on demographic as well as socio‐economic characteristics of districts is provided by Bundeswahlleiter (2002b; 2005d; 2009b). These publications rely on official numbers from the Federal Statistical Office, aggregated to the level of the electoral district. Unfortunately, there exists no comparable information on polling precincts. Differentiating between District 160 and Saxony's other districts, Table 3 presents summary statistics for all variables used throughout the analysis. Not surprisingly, given that District 160 consists of the southern parts of the city of Dresden, there exist important differences. Not only is District 160 substantially more urban, but in contrast to other districts its population is actually growing. Moreover, residents of District 160 are less likely to work in manufacturing, and experience lower rates of unemployment – although unemployment is still a major problem. In terms of election results, the CDU receives on average somewhat lower vote shares in District 160, whereas the SPD and the Green Party fare slightly worse in the remainder of Saxony. Although differences in political preferences appear to be less stark than those in socio‐economic characteristics, District 160 is clearly not perfectly representative. It is, therefore, important to account for district‐specific idiosyncrasies in determining the electorate's reaction to the paradoxical incentives it faced during the by‐election. 3. Evidence of Preference Misrepresentation 3.1. Econometric Approach The empirical strategy in this article builds on the classical difference‐in‐differences (DD) approach, with District 160 being ‘treated’ and all other districts in the state of Saxony serving as the ‘control group’.22 The most important difference is that there are three instead of the usual two periods. As voters in District 160 faced disparate incentives only during the 2005 elections, it is useful to think of 2005 as the ‘treatment’ period, whereas 2002 and 2009 should be regarded as pre and post‐treatment periods, respectively. Table 3 Summary Statistics of District and Precinct‐level Variables for the 2002, 2005 and 2009 Elections to the Bundestag . Full sample . Electoral district 160 . Other districts in Saxony . . Mean . SD . Mean . SD . Mean . SD . District‐level variables Population (in 1,000) 260.7 22.5 272.1 7.3 260.0 23.0 Population density (residents per km) 656.6 879.8 3,330 70.5 476.5 549.4 Population growth (per 1,000 residents) −6.642 7.139 4.967 7.836 −7.424 6.461 Number of cars (per 1,000 residents) 589.4 76.7 477.5 31.5 596.9 72.9 Percent of labour force in manufacturing 33.63 7.51 22.52 1.56 34.34 7.16 Percent of labour force in service industry 63.66 8.61 76.82 1.78 62.81 8.17 Unemployment rate (in %) 17.31 3.21 14.20 2.05 17.53 3.18 Precinct‐level variables Share of candidate vote (in %) CDU 37.25 8.17 35.75 5.02 37.35 8.33 SPD 24.66 9.19 26.79 7.68 24.52 9.27 FDP 7.76 3.50 6.57 2.64 7.84 3.53 The Left 21.28 5.93 21.35 5.23 21.27 5.97 Green party 4.28 3.43 6.07 3.74 4.16 3.38 Others 4.78 2.98 3.47 1.55 4.86 3.03 Share of list vote (in %) CDU 32.92 7.41 29.71 6.09 33.14 7.44 SPD 24.57 9.06 25.62 7.82 24.50 9.13 FDP 10.10 3.71 12.23 5.40 9.96 3.52 The Left 21.02 6.05 19.80 5.18 21.10 6.10 Green party 5.30 4.11 8.60 4.46 5.08 3.99 Others 6.09 2.78 4.05 1.67 6.22 2.78 Number of eligible voters 1,112 391 1,170 238 1,109 398 Turnout (in %) 72.29 7.58 73.12 7.48 72.24 7.59 Absentee precinct 0.144 0.351 0.233 0.423 0.138 0.345 Number of districts 17 1 16 Number of observations 13,107 774 12,363 . Full sample . Electoral district 160 . Other districts in Saxony . . Mean . SD . Mean . SD . Mean . SD . District‐level variables Population (in 1,000) 260.7 22.5 272.1 7.3 260.0 23.0 Population density (residents per km) 656.6 879.8 3,330 70.5 476.5 549.4 Population growth (per 1,000 residents) −6.642 7.139 4.967 7.836 −7.424 6.461 Number of cars (per 1,000 residents) 589.4 76.7 477.5 31.5 596.9 72.9 Percent of labour force in manufacturing 33.63 7.51 22.52 1.56 34.34 7.16 Percent of labour force in service industry 63.66 8.61 76.82 1.78 62.81 8.17 Unemployment rate (in %) 17.31 3.21 14.20 2.05 17.53 3.18 Precinct‐level variables Share of candidate vote (in %) CDU 37.25 8.17 35.75 5.02 37.35 8.33 SPD 24.66 9.19 26.79 7.68 24.52 9.27 FDP 7.76 3.50 6.57 2.64 7.84 3.53 The Left 21.28 5.93 21.35 5.23 21.27 5.97 Green party 4.28 3.43 6.07 3.74 4.16 3.38 Others 4.78 2.98 3.47 1.55 4.86 3.03 Share of list vote (in %) CDU 32.92 7.41 29.71 6.09 33.14 7.44 SPD 24.57 9.06 25.62 7.82 24.50 9.13 FDP 10.10 3.71 12.23 5.40 9.96 3.52 The Left 21.02 6.05 19.80 5.18 21.10 6.10 Green party 5.30 4.11 8.60 4.46 5.08 3.99 Others 6.09 2.78 4.05 1.67 6.22 2.78 Number of eligible voters 1,112 391 1,170 238 1,109 398 Turnout (in %) 72.29 7.58 73.12 7.48 72.24 7.59 Absentee precinct 0.144 0.351 0.233 0.423 0.138 0.345 Number of districts 17 1 16 Number of observations 13,107 774 12,363 Notes Entries are weighted means and standard deviations of district and precinct‐level variables, pooled across years. Only observations with nonmissing information are used in the calculations. See the Data Appendix for the precise definition and source of each variable. Open in new tab Table 3 Summary Statistics of District and Precinct‐level Variables for the 2002, 2005 and 2009 Elections to the Bundestag . Full sample . Electoral district 160 . Other districts in Saxony . . Mean . SD . Mean . SD . Mean . SD . District‐level variables Population (in 1,000) 260.7 22.5 272.1 7.3 260.0 23.0 Population density (residents per km) 656.6 879.8 3,330 70.5 476.5 549.4 Population growth (per 1,000 residents) −6.642 7.139 4.967 7.836 −7.424 6.461 Number of cars (per 1,000 residents) 589.4 76.7 477.5 31.5 596.9 72.9 Percent of labour force in manufacturing 33.63 7.51 22.52 1.56 34.34 7.16 Percent of labour force in service industry 63.66 8.61 76.82 1.78 62.81 8.17 Unemployment rate (in %) 17.31 3.21 14.20 2.05 17.53 3.18 Precinct‐level variables Share of candidate vote (in %) CDU 37.25 8.17 35.75 5.02 37.35 8.33 SPD 24.66 9.19 26.79 7.68 24.52 9.27 FDP 7.76 3.50 6.57 2.64 7.84 3.53 The Left 21.28 5.93 21.35 5.23 21.27 5.97 Green party 4.28 3.43 6.07 3.74 4.16 3.38 Others 4.78 2.98 3.47 1.55 4.86 3.03 Share of list vote (in %) CDU 32.92 7.41 29.71 6.09 33.14 7.44 SPD 24.57 9.06 25.62 7.82 24.50 9.13 FDP 10.10 3.71 12.23 5.40 9.96 3.52 The Left 21.02 6.05 19.80 5.18 21.10 6.10 Green party 5.30 4.11 8.60 4.46 5.08 3.99 Others 6.09 2.78 4.05 1.67 6.22 2.78 Number of eligible voters 1,112 391 1,170 238 1,109 398 Turnout (in %) 72.29 7.58 73.12 7.48 72.24 7.59 Absentee precinct 0.144 0.351 0.233 0.423 0.138 0.345 Number of districts 17 1 16 Number of observations 13,107 774 12,363 . Full sample . Electoral district 160 . Other districts in Saxony . . Mean . SD . Mean . SD . Mean . SD . District‐level variables Population (in 1,000) 260.7 22.5 272.1 7.3 260.0 23.0 Population density (residents per km) 656.6 879.8 3,330 70.5 476.5 549.4 Population growth (per 1,000 residents) −6.642 7.139 4.967 7.836 −7.424 6.461 Number of cars (per 1,000 residents) 589.4 76.7 477.5 31.5 596.9 72.9 Percent of labour force in manufacturing 33.63 7.51 22.52 1.56 34.34 7.16 Percent of labour force in service industry 63.66 8.61 76.82 1.78 62.81 8.17 Unemployment rate (in %) 17.31 3.21 14.20 2.05 17.53 3.18 Precinct‐level variables Share of candidate vote (in %) CDU 37.25 8.17 35.75 5.02 37.35 8.33 SPD 24.66 9.19 26.79 7.68 24.52 9.27 FDP 7.76 3.50 6.57 2.64 7.84 3.53 The Left 21.28 5.93 21.35 5.23 21.27 5.97 Green party 4.28 3.43 6.07 3.74 4.16 3.38 Others 4.78 2.98 3.47 1.55 4.86 3.03 Share of list vote (in %) CDU 32.92 7.41 29.71 6.09 33.14 7.44 SPD 24.57 9.06 25.62 7.82 24.50 9.13 FDP 10.10 3.71 12.23 5.40 9.96 3.52 The Left 21.02 6.05 19.80 5.18 21.10 6.10 Green party 5.30 4.11 8.60 4.46 5.08 3.99 Others 6.09 2.78 4.05 1.67 6.22 2.78 Number of eligible voters 1,112 391 1,170 238 1,109 398 Turnout (in %) 72.29 7.58 73.12 7.48 72.24 7.59 Absentee precinct 0.144 0.351 0.233 0.423 0.138 0.345 Number of districts 17 1 16 Number of observations 13,107 774 12,363 Notes Entries are weighted means and standard deviations of district and precinct‐level variables, pooled across years. Only observations with nonmissing information are used in the calculations. See the Data Appendix for the precise definition and source of each variable. Open in new tab In the standard DD approach, the key identifying assumption is fairly restrictive: in the absence of treatment, that is, without a by‐election, District 160 would have followed the same path as districts in the comparison group. Although this assumption is not directly testable, one might be willing to judge its reasonability by comparing outcomes in the pre and post‐treatment periods. To this end, consider the upper panel in Figure 1. The graph on the left plots the CDU's share of the list vote in 2002 against that in 2009, and the graph on the right does so for its share of the candidate vote. Each dot corresponds to an electoral district in Saxony. Evidently the CDU's 2009 vote share in District 160 is in both cases very close to what one would predict based on that in 2002. That is, in non‐treatment years, District 160 appears to conform to the same general pattern as other districts in the same state. Fig. 4. Open in new tabDownload slide Share of List and Candidate Votes Accruing to the CDU in the State of Saxony Notes. Graphs on the left plot the CDU's share of the list vote in one election year against its share in a subsequent election. Graphs on the right use the CDU's share of the candidate vote instead. Each dot represents an electoral district in the state of Saxony. Fig. 4. Open in new tabDownload slide Share of List and Candidate Votes Accruing to the CDU in the State of Saxony Notes. Graphs on the left plot the CDU's share of the list vote in one election year against its share in a subsequent election. Graphs on the right use the CDU's share of the candidate vote instead. Each dot represents an electoral district in the state of Saxony. By contrast, the middle and bottom panels in Figure 1 plot the CDU's vote shares in 2005 against those in 2002 and 2009, respectively. In each of these graphs, District 160 is a clear outlier. Although the CDU received substantially fewer list votes in 2005 than predicted based on the outcome of the previous election, the opposite is true for 2009. Conversely, it received a larger than predicted share of the candidate vote in the year of the by‐election but a smaller one thereafter. The pattern of deviations conforms, therefore, exactly to what one would expect if a substantial fraction of the electorate in District 160 did, indeed, cast misaligned votes. Although the patterns in Figure 1 are certainly suggestive, quantifying the extent of misaligned voting more precisely, accounting for district‐specific idiosyncrasies and testing additional predictions require some ‘econometric machinery’. Therefore, consider the following specification: vp,i,t=τp,t+μp,d+δ1(d=160)×1(t=2005)+εp,i,t,(3) where vp,i,t denotes party p's vote share in polling precinct i during election year t. τp,t and μp.d mark comprehensive sets of time and district fixed effects, with d indexing electoral districts. 1(d = 160) is an indicator variable equal to one if precinct i is part of District 160, and zero otherwise. Similarly, 1(t = 2005) equals one for the 2005 elections. The parameter of interest is δ. It indicates what effect the incentives described above had on the voting behaviour of the constituency in District 160. 3.2. Empirical Evidence Table 4 presents the main empirical results. The numbers therein correspond to DD estimates of δ, obtained from estimating (3) by weighted least squares, with weights corresponding to the number of voters. Results in the top panel are based on district‐level data, whereas the bottom one uses all available information by relying on individual polling precincts as the unit of observation. To allow for arbitrary forms of autocorrelation in the residuals as well as for correlation across precincts, standard errors are clustered by electoral district. Moving from the left to the right within each group of regressions, the set of included fixed effects steadily grows. As one would expect, point estimates based on district and precinct level data align quite closely, but the latter are estimated more precisely. Quantitatively, δ is estimated to be fairly large. For instance, in the absence of the by‐election, the CDU is predicted to have garnered about 4.9% less of the candidate vote but an additional 3.6% of the list vote.23 Under most reasonable assumptions on turnout, the latter would have resulted in substantially more than 41,000 list votes, and, therefore, cost the CDU a seat in parliament.24 Of course, it is far from certain whether the key identifying assumption of the DD approach is, in fact, satisfied. For instance, compared to Saxony's other electoral districts, CDU vote shares in District 160 might have been on a different trajectory, even in the absence of a by‐election. If districts do, indeed, follow differential trends, then the estimates in Table 4 might be biased.25 Table 4 Evidence of Preference Misrepresentation . CDU share of list vote . CDU share of candidate vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) District‐level regressions District 160 × 2005 −8.700 −5.995 −3.560 −0.261 1.537 4.918 (0.975) (0.855) (0.417) (1.140) (1.088) (0.430) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.071 0.270 0.960 0.000 0.106 0.943 Number of observations 51 51 51 51 51 51 . CDU share of list vote . CDU share of candidate vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) District‐level regressions District 160 × 2005 −8.700 −5.995 −3.560 −0.261 1.537 4.918 (0.975) (0.855) (0.417) (1.140) (1.088) (0.430) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.071 0.270 0.960 0.000 0.106 0.943 Number of observations 51 51 51 51 51 51 . CDU share of list vote . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Precinct‐level regressions District 160 × 2005 −8.715 −6.007 −3.568 −0.266 1.535 4.921 (0.966) (0.829) (0.329) (1.129) (1.055) (0.340) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.028 0.107 0.381 0.000 0.039 0.345 Number of observations 13,107 13,107 13,107 13,107 13,107 13,107 . CDU share of list vote . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Precinct‐level regressions District 160 × 2005 −8.715 −6.007 −3.568 −0.266 1.535 4.921 (0.966) (0.829) (0.329) (1.129) (1.055) (0.340) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.028 0.107 0.381 0.000 0.039 0.345 Number of observations 13,107 13,107 13,107 13,107 13,107 13,107 Notes Entries are coefficients and standard errors on δ, obtained by estimating (3) using weighted least squares with weights corresponding to the number of voters. The respective dependent variables are listed at the top of each column. The upper panel uses only district‐level data, whereas the lower one relies on individual polling precincts as the level of observation. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab Table 4 Evidence of Preference Misrepresentation . CDU share of list vote . CDU share of candidate vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) District‐level regressions District 160 × 2005 −8.700 −5.995 −3.560 −0.261 1.537 4.918 (0.975) (0.855) (0.417) (1.140) (1.088) (0.430) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.071 0.270 0.960 0.000 0.106 0.943 Number of observations 51 51 51 51 51 51 . CDU share of list vote . CDU share of candidate vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) District‐level regressions District 160 × 2005 −8.700 −5.995 −3.560 −0.261 1.537 4.918 (0.975) (0.855) (0.417) (1.140) (1.088) (0.430) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.071 0.270 0.960 0.000 0.106 0.943 Number of observations 51 51 51 51 51 51 . CDU share of list vote . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Precinct‐level regressions District 160 × 2005 −8.715 −6.007 −3.568 −0.266 1.535 4.921 (0.966) (0.829) (0.329) (1.129) (1.055) (0.340) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.028 0.107 0.381 0.000 0.039 0.345 Number of observations 13,107 13,107 13,107 13,107 13,107 13,107 . CDU share of list vote . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Precinct‐level regressions District 160 × 2005 −8.715 −6.007 −3.568 −0.266 1.535 4.921 (0.966) (0.829) (0.329) (1.129) (1.055) (0.340) Year fixed effects No Yes Yes No Yes Yes District fixed effects No No Yes No No Yes R2 0.028 0.107 0.381 0.000 0.039 0.345 Number of observations 13,107 13,107 13,107 13,107 13,107 13,107 Notes Entries are coefficients and standard errors on δ, obtained by estimating (3) using weighted least squares with weights corresponding to the number of voters. The respective dependent variables are listed at the top of each column. The upper panel uses only district‐level data, whereas the lower one relies on individual polling precincts as the level of observation. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab Fortunately, the fact that the current set‐up contains three instead of the usual two periods allows for econometric models that control quite flexibly for any possible (unobserved) trends. More specifically, the following specification explicitly assigns each village, or municipality m its own linear trend γp,m: vp,i,t=τp,t+μp,m+δ1(d=160)×1(t=2005)+γp,mt+εp,i,t.(4) Here, the key identifying assumption is substantially weaker than in the standard DD approach. For estimates of δ to be unbiased it needs to be the case that the one‐time reversal in incentives did not coincide with larger deviations from trend in affected municipalities than in unaffected ones. Results obtained from estimating (4) by weighted least squares are displayed in the upper two panels of Table 5. Within each group the leftmost point estimate corresponds to the usual DD estimate, that is, the baseline result from the previous table. The second estimate accounts for district‐specific trends, whereas the rightmost allows for each municipality in the data to follow its own trajectory. The results in columns (1)–(3) and (7)–(9) are based on the main precinct‐level data set covering the period from 2002 to 2009. Additional results based on municipality‐level data from 1994 to 2009 appear in columns (4)–(6) as well as (10)–(12). The latter data offer a longer pre‐treatment period (making it easier to pick up trends), but they only exist at a higher level of aggregation. Table 5 Accounting for Trends . CDU share of list vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) List vote District 160 × 2005 −3.568 −3.473 −3.459 −4.623 −4.364 −4.066 (0.329) (0.348) (0.288) (0.494) (0.298) (0.373) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.381 0.389 0.616 0.737 0.758 0.948 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . CDU share of list vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) List vote District 160 × 2005 −3.568 −3.473 −3.459 −4.623 −4.364 −4.066 (0.329) (0.348) (0.288) (0.494) (0.298) (0.373) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.381 0.389 0.616 0.737 0.758 0.948 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Candidate vote District 160 × 2005 4.921 4.925 4.479 3.619 4.169 4.079 (0.340) (0.363) (0.430) (0.432) (0.306) (0.311) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–2009 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.345 0.360 0.621 0.672 0.689 0.924 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Candidate vote District 160 × 2005 4.921 4.925 4.479 3.619 4.169 4.079 (0.340) (0.363) (0.430) (0.432) (0.306) (0.311) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–2009 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.345 0.360 0.621 0.672 0.689 0.924 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . Difference between CDU share of candidate vote and list vote . . (13) . (14) . (15) . (16) . (17) . (18) . (c) Difference between CDU share of candidate vote and list vote District 160 × 2005 8.488 8.398 7.938 8.241 8.533 8.145 (0.265) (0.270) (0.373) (0.280) (0.254) (0.398) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.275 0.311 0.550 0.341 0.366 0.645 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . Difference between CDU share of candidate vote and list vote . . (13) . (14) . (15) . (16) . (17) . (18) . (c) Difference between CDU share of candidate vote and list vote District 160 × 2005 8.488 8.398 7.938 8.241 8.533 8.145 (0.265) (0.270) (0.373) (0.280) (0.254) (0.398) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.275 0.311 0.550 0.341 0.366 0.645 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 Notes Entries are coefficients and standard errors on δ, obtained by estimating (3), (4), and (5) using weighted least squares with weights corresponding to the number of voters. The respective dependent variables are listed at the top of each column. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab Table 5 Accounting for Trends . CDU share of list vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) List vote District 160 × 2005 −3.568 −3.473 −3.459 −4.623 −4.364 −4.066 (0.329) (0.348) (0.288) (0.494) (0.298) (0.373) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.381 0.389 0.616 0.737 0.758 0.948 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . CDU share of list vote . . (1) . (2) . (3) . (4) . (5) . (6) . (a) List vote District 160 × 2005 −3.568 −3.473 −3.459 −4.623 −4.364 −4.066 (0.329) (0.348) (0.288) (0.494) (0.298) (0.373) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.381 0.389 0.616 0.737 0.758 0.948 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Candidate vote District 160 × 2005 4.921 4.925 4.479 3.619 4.169 4.079 (0.340) (0.363) (0.430) (0.432) (0.306) (0.311) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–2009 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.345 0.360 0.621 0.672 0.689 0.924 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . CDU share of candidate vote . . (7) . (8) . (9) . (10) . (11) . (12) . (b) Candidate vote District 160 × 2005 4.921 4.925 4.479 3.619 4.169 4.079 (0.340) (0.363) (0.430) (0.432) (0.306) (0.311) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–2009 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.345 0.360 0.621 0.672 0.689 0.924 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . Difference between CDU share of candidate vote and list vote . . (13) . (14) . (15) . (16) . (17) . (18) . (c) Difference between CDU share of candidate vote and list vote District 160 × 2005 8.488 8.398 7.938 8.241 8.533 8.145 (0.265) (0.270) (0.373) (0.280) (0.254) (0.398) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.275 0.311 0.550 0.341 0.366 0.645 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 . Difference between CDU share of candidate vote and list vote . . (13) . (14) . (15) . (16) . (17) . (18) . (c) Difference between CDU share of candidate vote and list vote District 160 × 2005 8.488 8.398 7.938 8.241 8.533 8.145 (0.265) (0.270) (0.373) (0.280) (0.254) (0.398) Unit of observation Precinct Precinct Precinct Municipality Municipality Municipality Time period 2002–9 2002–9 2002–9 1994–2009 1994–2009 1994–2009 Year fixed effects Yes Yes Yes Yes Yes Yes District fixed effects Yes Yes No Yes Yes No District‐specific trends No Yes No No Yes No Municipality fixed effects No No Yes No No Yes Municipality‐specific trends No No Yes No No Yes R2 0.275 0.311 0.550 0.341 0.366 0.645 Number of observations 13,107 13,107 13,107 3,404 3,404 3,404 Notes Entries are coefficients and standard errors on δ, obtained by estimating (3), (4), and (5) using weighted least squares with weights corresponding to the number of voters. The respective dependent variables are listed at the top of each column. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab Reassuringly, independent of the level of aggregation and the length of the time period under consideration, estimates of δ that explicitly control for trends are very similar to their DD counterparts in Table 4. This suggests that differential trends cannot explain the patterns in the data.26 Another way to account for the pitfalls of the DD approach in isolating the effect of the by‐election and its unusual incentive structure is to pursue a triple‐differencing strategy. That is, to use the difference in the CDU's share of the candidate and list vote as dependent variable, and to estimate vp,i,tC−vp,i,tL=τp,t+μp,m+δ1(d=160)×1(t=2005)+γp,mt+εp,i,t,(5) where vp,i,tC refers to the CDU's share of the candidate vote in precinct i during election year t, and vp,i,tL denotes its share of the list vote. Note well that even if there was an unobserved deviation from trend specific to District 160 and the 2005 election, as long as it did not have a differential impact on voters’ perceptions of the CDU's direct candidate and the CDU as a whole, the estimates reported in the lower panel of Table 5 would still be consistent. It may therefore be comforting to know that the triple‐difference estimate of δ continues to be large – about 8 percentage points. Moreover, the precision of the estimates makes it extremely unlikely that the results in Table 5 are due to chance. Instead, it appears that at least some voters in District 160 misrepresented their preferences. 3.3. Strategic Voting? Table 6 exploits the precinct‐level nature of the data, and presents additional evidence in favour of this assertion. The evidence therein also indicates that supporters of parties other than the CDU misrepresented their preferences, and that at least some voters did so strategically. First, however, columns (1) and (2) explore the by‐election's effect on turnout.27 If the electorate was, indeed, aware of the peculiar incentives and reacted rationally, there should have been lower turnout in the by‐election – after all the distribution of seats in parliament had, for the most part, been already determined. Table 6 Strategic Voting? . Turnout . FDP share of list vote . CDU share of list vote . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . District 160 × 2005 −3.857 −16.499 6.500 20.918 −3.459 −14.735 −9.275 −11.155 (0.610) (1.270) (0.378) (0.514) (0.288) (1.225) (1.244) (1.540) SPD share of candidate vote −0.133 −0.224 −0.726 (0.044) (0.014) (0.052) District 160 × 2005 × SPD share of candidate vote 0.430 −0.395 0.527 (0.044) (0.014) (0.052) Green Party share of candidate vote −0.457 (0.098) District 160 × 2005 × Green Party share of candidate vote 1.044 (0.098) The Left share of candidate vote −0.648 (0.062) District 160 × 2005 × The Left share of candidate vote 0.309 (0.062) Green Party share of list vote District 160 × 2005 × Green Party share of list vote CDU share of list vote District 160 × 2005 × CDU share of list vote Year fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Yes Yes Included voters All All All All All All All All R2 0.507 0.510 0.634 0.680 0.616 0.717 0.641 0.737 Number of observations 11,512 11,512 13,107 13,107 13,107 13,107 13,107 13,107 . Turnout . FDP share of list vote . CDU share of list vote . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . District 160 × 2005 −3.857 −16.499 6.500 20.918 −3.459 −14.735 −9.275 −11.155 (0.610) (1.270) (0.378) (0.514) (0.288) (1.225) (1.244) (1.540) SPD share of candidate vote −0.133 −0.224 −0.726 (0.044) (0.014) (0.052) District 160 × 2005 × SPD share of candidate vote 0.430 −0.395 0.527 (0.044) (0.014) (0.052) Green Party share of candidate vote −0.457 (0.098) District 160 × 2005 × Green Party share of candidate vote 1.044 (0.098) The Left share of candidate vote −0.648 (0.062) District 160 × 2005 × The Left share of candidate vote 0.309 (0.062) Green Party share of list vote District 160 × 2005 × Green Party share of list vote CDU share of list vote District 160 × 2005 × CDU share of list vote Year fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Yes Yes Included voters All All All All All All All All R2 0.507 0.510 0.634 0.680 0.616 0.717 0.641 0.737 Number of observations 11,512 11,512 13,107 13,107 13,107 13,107 13,107 13,107 . SPD share of candidate vote . CDU share of candidate vote . CDU share of candidate vote . Difference in CDU share of candidate and list vote . (9) . (10) . (11) . (12) . (13) . (14) . District 160 × 2005 7.773 7.266 4.479 21.858 3.991 2.192 (0.378) (0.661) (0.430) (0.376) (0.293) (0.492) SPD share of candidate vote District 160 × 2005 × SPD share of candidate vote Green Party share of candidate vote District 160 × 2005 × Green Party share of candidate vote The Left share of candidate vote District 160 × 2005 × The Left share of candidate vote Green Party share of list vote 0.170 (0.031) District 160 × 2005 × Green Party share of list vote 0.141 (0.031) CDU share of list vote 1.014 (0.012) District 160 × 2005 × CDU share of list vote −0.569 (0.012) Year fixed effects Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Included voters All All All All Absentee Absentee R2 0.872 0.876 0.621 0.940 0.663 0.582 Number of observations 13,107 13,107 13,107 13,107 1,595 1,595 . SPD share of candidate vote . CDU share of candidate vote . CDU share of candidate vote . Difference in CDU share of candidate and list vote . (9) . (10) . (11) . (12) . (13) . (14) . District 160 × 2005 7.773 7.266 4.479 21.858 3.991 2.192 (0.378) (0.661) (0.430) (0.376) (0.293) (0.492) SPD share of candidate vote District 160 × 2005 × SPD share of candidate vote Green Party share of candidate vote District 160 × 2005 × Green Party share of candidate vote The Left share of candidate vote District 160 × 2005 × The Left share of candidate vote Green Party share of list vote 0.170 (0.031) District 160 × 2005 × Green Party share of list vote 0.141 (0.031) CDU share of list vote 1.014 (0.012) District 160 × 2005 × CDU share of list vote −0.569 (0.012) Year fixed effects Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Included voters All All All All Absentee Absentee R2 0.872 0.876 0.621 0.940 0.663 0.582 Number of observations 13,107 13,107 13,107 13,107 1,595 1,595 Notes Entries are coefficients and standard errors from estimating models analogous to equation (4) using weighted least squares with weights corresponding to the number of voters. The respective dependent variables are listed at the top of each column. Individual polling precincts are the level of observation. Columns (1)–(12) use information on all voters, whereas columns (13) and (14) restrict attention to absentee voters. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab Table 6 Strategic Voting? . Turnout . FDP share of list vote . CDU share of list vote . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . District 160 × 2005 −3.857 −16.499 6.500 20.918 −3.459 −14.735 −9.275 −11.155 (0.610) (1.270) (0.378) (0.514) (0.288) (1.225) (1.244) (1.540) SPD share of candidate vote −0.133 −0.224 −0.726 (0.044) (0.014) (0.052) District 160 × 2005 × SPD share of candidate vote 0.430 −0.395 0.527 (0.044) (0.014) (0.052) Green Party share of candidate vote −0.457 (0.098) District 160 × 2005 × Green Party share of candidate vote 1.044 (0.098) The Left share of candidate vote −0.648 (0.062) District 160 × 2005 × The Left share of candidate vote 0.309 (0.062) Green Party share of list vote District 160 × 2005 × Green Party share of list vote CDU share of list vote District 160 × 2005 × CDU share of list vote Year fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Yes Yes Included voters All All All All All All All All R2 0.507 0.510 0.634 0.680 0.616 0.717 0.641 0.737 Number of observations 11,512 11,512 13,107 13,107 13,107 13,107 13,107 13,107 . Turnout . FDP share of list vote . CDU share of list vote . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . District 160 × 2005 −3.857 −16.499 6.500 20.918 −3.459 −14.735 −9.275 −11.155 (0.610) (1.270) (0.378) (0.514) (0.288) (1.225) (1.244) (1.540) SPD share of candidate vote −0.133 −0.224 −0.726 (0.044) (0.014) (0.052) District 160 × 2005 × SPD share of candidate vote 0.430 −0.395 0.527 (0.044) (0.014) (0.052) Green Party share of candidate vote −0.457 (0.098) District 160 × 2005 × Green Party share of candidate vote 1.044 (0.098) The Left share of candidate vote −0.648 (0.062) District 160 × 2005 × The Left share of candidate vote 0.309 (0.062) Green Party share of list vote District 160 × 2005 × Green Party share of list vote CDU share of list vote District 160 × 2005 × CDU share of list vote Year fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Yes Yes Included voters All All All All All All All All R2 0.507 0.510 0.634 0.680 0.616 0.717 0.641 0.737 Number of observations 11,512 11,512 13,107 13,107 13,107 13,107 13,107 13,107 . SPD share of candidate vote . CDU share of candidate vote . CDU share of candidate vote . Difference in CDU share of candidate and list vote . (9) . (10) . (11) . (12) . (13) . (14) . District 160 × 2005 7.773 7.266 4.479 21.858 3.991 2.192 (0.378) (0.661) (0.430) (0.376) (0.293) (0.492) SPD share of candidate vote District 160 × 2005 × SPD share of candidate vote Green Party share of candidate vote District 160 × 2005 × Green Party share of candidate vote The Left share of candidate vote District 160 × 2005 × The Left share of candidate vote Green Party share of list vote 0.170 (0.031) District 160 × 2005 × Green Party share of list vote 0.141 (0.031) CDU share of list vote 1.014 (0.012) District 160 × 2005 × CDU share of list vote −0.569 (0.012) Year fixed effects Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Included voters All All All All Absentee Absentee R2 0.872 0.876 0.621 0.940 0.663 0.582 Number of observations 13,107 13,107 13,107 13,107 1,595 1,595 . SPD share of candidate vote . CDU share of candidate vote . CDU share of candidate vote . Difference in CDU share of candidate and list vote . (9) . (10) . (11) . (12) . (13) . (14) . District 160 × 2005 7.773 7.266 4.479 21.858 3.991 2.192 (0.378) (0.661) (0.430) (0.376) (0.293) (0.492) SPD share of candidate vote District 160 × 2005 × SPD share of candidate vote Green Party share of candidate vote District 160 × 2005 × Green Party share of candidate vote The Left share of candidate vote District 160 × 2005 × The Left share of candidate vote Green Party share of list vote 0.170 (0.031) District 160 × 2005 × Green Party share of list vote 0.141 (0.031) CDU share of list vote 1.014 (0.012) District 160 × 2005 × CDU share of list vote −0.569 (0.012) Year fixed effects Yes Yes Yes Yes Yes Yes Municipality fixed effects Yes Yes Yes Yes Yes Yes Municipality‐specific trends Yes Yes Yes Yes Yes Yes Included voters All All All All Absentee Absentee R2 0.872 0.876 0.621 0.940 0.663 0.582 Number of observations 13,107 13,107 13,107 13,107 1,595 1,595 Notes Entries are coefficients and standard errors from estimating models analogous to equation (4) using weighted least squares with weights corresponding to the number of voters. The respective dependent variables are listed at the top of each column. Individual polling precincts are the level of observation. Columns (1)–(12) use information on all voters, whereas columns (13) and (14) restrict attention to absentee voters. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab In line with this prediction, overall turnout is estimated to be 3.9 percentage points lower. More importantly, the additional interaction term in column (2) indicates a larger effect for CDU partisans. To see this, note that voters who wish the CDU to gain an additional mandate should never cast their candidate vote for the rival SPD. Thus, the share of a precinct's candidate vote accruing to the SPD can be interpreted as a proxy for the fraction of voters who do not support the CDU. By this measure, turnout is estimated to be higher among those who view the CDU unfavourably. Recall that CDU supporters could potentially cost their most preferred party a seat by voting for it, whereas supporters of other parties were not able to influence their parties’ seat totals. The results in columns (1) and (2) are, therefore, consistent with the view that some voters abstained for strategic reasons.28 Columns (3) and (4) demonstrate that the FDP, the CDU's traditional coalition partner, benefited from the by‐election. Taking the coefficient in column (3) at face value, the FDP received an additional 6.5% of the list vote. By a similar argument as above, column (4) shows that the FDP's gain was much lower, even negative, in precincts more critical of the CDU. This suggests that CDU partisans who did go to the polls cast their list votes for the FDP. If supporters of the SPD fully grasped the situation and behaved strategically, then one might even expect them to attempt to hurt the CDU by voting for its list. Although in regular election years there is a very strong negative correlation between a precinct's SPD share of the candidate vote and its CDU share of the list vote, as evidenced by the large negative coefficient in the second row of column (6), this was not the case during the by‐election in District 160. The positive interaction term in the third row of column (6) indicates that a significant number of individuals split their tickets between SPD candidate and CDU list. As neither sincere, nor strategic, CDU supporters would ever vote for the SPD candidate, these must have been agents who viewed the CDU unfavourably. Along the same lines, the coefficients in column (7) show a strong negative correlation between CDU list and Green Party candidate votes in regular election years. But this relationship becomes positive and large in 2005 – exactly when an additional list vote might cost the CDU a seat in parliament. The next column shows that a similar phenomenon cannot only be observed among adherents of SPD and Green Party but also among those of The Left. As explained in subsection 2, The Left's representatives explicitly instructed its supporters to refrain from casting tactical list votes, asking them to vote for The Left instead – just as in any other election year. Yet, the results in column (8) show that some of The Left's followers chose to ignore their party's request, and voted for the CDU instead. The evidence, therefore, suggests that adherents of opposition parties attempted to hurt the CDU by voting for it – despite the fact that none of these rival parties had instructed their followers to do so. Moreover, the results in columns (9) and (10) show that the SPD received a higher than usual fraction of the candidate vote. As the Green Party's own candidate had essentially no chance of winning the direct mandate and, given that the Green Party is much closer in policy space to the SPD than the CDU, strategic Green Party supporters should choose the SPD candidate with their first vote. Consistent with this prediction, the interaction term in the seventh row of column (10) indicates that the relationship between the Green Party's share of the list vote and the SPD's share of the candidate vote was substantially stronger during the by‐election than in regular election years. It therefore appears that supporters of the Green Party did not only attempt to hurt the CDU by casting tactical list votes but that they also behaved strategically with their candiate vote. Note well that this comes despite the Green Party's refusal to ask its supporters to cast tactical ballots. Taken together, these results suggest that agents did not merely follow official party positions as to how to cast their vote. At least some voters behaved strategically in the narrower sense of the word and cast tactical ballots out of their own volition. That is, they strategically misrepresented their preferences. Additional support in favour of this conclusion comes from absentee voters.29 After the death of the NPD direct candidate, new ballots were mailed to registered absentee voters in District 160 to ensure that they would be able to participate in the by‐election. Critically, one would expect absentee voters to be less exposed and, therefore, less affected by parties’ rallies, posters and other campaign activities, all of which might affect the behaviour of ‘regular’ voters. Thus, looking at absentee voters provides another way to disentangle the impact of parties’ activities from individuals themselves being strategic. Interestingly, estimating (4) on the set of absentee voters yields results that are qualitatively similar to those above. For instance, in the by‐election absentee voters in District 160 were 3.99 percentage points more likely to vote for the CDU candidate than absentee voters in the control group. Not only is this estimate economically large (about 89% the size of its counterpart in Table 5) but, given a standard error of 0.293, it is also statistically highly significant. The empirical approach relying on the weakest identifying assumptions is the triple differencing specification in (5). The corresponding estimate for absentee voters is 2.19 percentage points (with a standard error of 0.492). Although this estimate is considerably smaller than that in Table 5, it implies that parties adapting their campaign strategies cannot be the whole story. Instead, the evidence suggests that at least some voters internalised the by‐election's strange incentives and strategically misrepresented their preferences. 3.4. Sensitivity and Robustness Analysis Broadly summarising, the results presented above indicate that parts of the electorate reacted to the perverse incentives in the by‐election by casting misaligned votes. Supporters of the CDU either abstained or cast their list vote for the FDP, whereas adherents of the rival SPD appear to have voted for the CDU. This subsection explores the sensitivity and robustness of these results. Table 7 Sensitivity and Robustness Analysis Sample/specification . CDU share of list vote . CDU share of candidate vote . Turnout . FDP share of list vote . SPD share of candidate vote . (a) Main effects Baseline −3.459 4.479 −3.857 6.500 7.773 (0.288) (0.430) (0.610) (0.378) (0.378) Unweighted −3.982 4.384 −3.830 6.888 8.076 (0.298) (0.537) (0.555) (0.382) (0.401) Controlling for CDU share of list votes in previous election −3.584 4.873 −3.748 7.005 5.887 (0.404) (0.426) (0.690) (0.340) (0.402) Controlling for district‐level covariates −4.335 4.702 −3.141 7.418 5.174 (0.710) (0.788) (0.476) (0.316) (0.744) Geographically constant districts −4.768 4.347 −4.199 7.581 6.108 (1.097) (0.970) (0.827) (0.420) (0.586) Control group Other districts in East Germany −4.931 3.543 −2.830 7.374 5.439 (0.202) (0.293) (0.221) (0.090) (0.433) All other districts in Germany −6.720 1.173 −0.594 7.654 4.437 (0.101) (0.140) (0.081) (0.060) (0.126) As percentage of all eligible voters −5.801 0.765 – 5.108 3.657 (0.171) (0.200) (0.243) (0.332) Sample/specification . CDU share of list vote . CDU share of candidate vote . Turnout . FDP share of list vote . SPD share of candidate vote . (a) Main effects Baseline −3.459 4.479 −3.857 6.500 7.773 (0.288) (0.430) (0.610) (0.378) (0.378) Unweighted −3.982 4.384 −3.830 6.888 8.076 (0.298) (0.537) (0.555) (0.382) (0.401) Controlling for CDU share of list votes in previous election −3.584 4.873 −3.748 7.005 5.887 (0.404) (0.426) (0.690) (0.340) (0.402) Controlling for district‐level covariates −4.335 4.702 −3.141 7.418 5.174 (0.710) (0.788) (0.476) (0.316) (0.744) Geographically constant districts −4.768 4.347 −4.199 7.581 6.108 (1.097) (0.970) (0.827) (0.420) (0.586) Control group Other districts in East Germany −4.931 3.543 −2.830 7.374 5.439 (0.202) (0.293) (0.221) (0.090) (0.433) All other districts in Germany −6.720 1.173 −0.594 7.654 4.437 (0.101) (0.140) (0.081) (0.060) (0.126) As percentage of all eligible voters −5.801 0.765 – 5.108 3.657 (0.171) (0.200) (0.243) (0.332) Sample/specification . Column (2) . Column (4) . Column (6) . Column (7) . Column (10) . (b) Additional interaction terms Baseline 0.430 −0.395 0.527 1.044 0.141 (0.044) (0.014) (0.052) (0.098) (0.031) Unweighted 0.426 −0.419 0.521 1.038 0.138 (0.033) (0.014) (0.038) (0.087) (0.026) Controlling for CDU share of list votes in previous election 0.432 −0.397 0.574 1.044 0.137 (0.044) (0.016) (0.050) (0.098) (0.031) Controlling for district‐level covariates 0.430 −0.393 0.559 1.053 0.135 (0.047) (0.015) (0.050) (0.103) (0.031) Geographically constant districts 0.267 −0.416 0.479 1.001 0.133 (0.113) (0.027) (0.030) (0.069) (0.039) Control group Other districts in East Germany 0.507 −0.468 0.442 0.984 0.174 (0.027) (0.012) (0.050) (0.103) (0.054) All other districts in Germany 0.523 −0.472 0.635 1.294 0.079 (0.012) (0.007) (0.011) (0.059) (0.030) As percentage of all eligible voters – −0.120 0.439 0.390 0.320 (0.011) (0.040) (0.096) (0.021) Sample/specification . Column (2) . Column (4) . Column (6) . Column (7) . Column (10) . (b) Additional interaction terms Baseline 0.430 −0.395 0.527 1.044 0.141 (0.044) (0.014) (0.052) (0.098) (0.031) Unweighted 0.426 −0.419 0.521 1.038 0.138 (0.033) (0.014) (0.038) (0.087) (0.026) Controlling for CDU share of list votes in previous election 0.432 −0.397 0.574 1.044 0.137 (0.044) (0.016) (0.050) (0.098) (0.031) Controlling for district‐level covariates 0.430 −0.393 0.559 1.053 0.135 (0.047) (0.015) (0.050) (0.103) (0.031) Geographically constant districts 0.267 −0.416 0.479 1.001 0.133 (0.113) (0.027) (0.030) (0.069) (0.039) Control group Other districts in East Germany 0.507 −0.468 0.442 0.984 0.174 (0.027) (0.012) (0.050) (0.103) (0.054) All other districts in Germany 0.523 −0.472 0.635 1.294 0.079 (0.012) (0.007) (0.011) (0.059) (0.030) As percentage of all eligible voters – −0.120 0.439 0.390 0.320 (0.011) (0.040) (0.096) (0.021) Notes Entries in the upper panel are coefficients and standard errors on δ, obtained by estimating equation (4) using least squares. Entries in the lower panel correspond to the additional interaction terms added in columns (2), (4), (6), (7) and (10) in Table 6. The respective dependent variables are listed at the top of each column, and the relevant sample restriction or change in the set of controls is denoted on the left of each row. All specifications, except for those including districts outside the state of Saxony, contain year fixed effects, municipality fixed effects, municipality‐specific linear trends and, if applicable, indicator variables for missing covariates. Regressions including districts outside the state of Saxony contain year fixed effects, district fixed effects and district‐specific linear trends instead. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab Table 7 Sensitivity and Robustness Analysis Sample/specification . CDU share of list vote . CDU share of candidate vote . Turnout . FDP share of list vote . SPD share of candidate vote . (a) Main effects Baseline −3.459 4.479 −3.857 6.500 7.773 (0.288) (0.430) (0.610) (0.378) (0.378) Unweighted −3.982 4.384 −3.830 6.888 8.076 (0.298) (0.537) (0.555) (0.382) (0.401) Controlling for CDU share of list votes in previous election −3.584 4.873 −3.748 7.005 5.887 (0.404) (0.426) (0.690) (0.340) (0.402) Controlling for district‐level covariates −4.335 4.702 −3.141 7.418 5.174 (0.710) (0.788) (0.476) (0.316) (0.744) Geographically constant districts −4.768 4.347 −4.199 7.581 6.108 (1.097) (0.970) (0.827) (0.420) (0.586) Control group Other districts in East Germany −4.931 3.543 −2.830 7.374 5.439 (0.202) (0.293) (0.221) (0.090) (0.433) All other districts in Germany −6.720 1.173 −0.594 7.654 4.437 (0.101) (0.140) (0.081) (0.060) (0.126) As percentage of all eligible voters −5.801 0.765 – 5.108 3.657 (0.171) (0.200) (0.243) (0.332) Sample/specification . CDU share of list vote . CDU share of candidate vote . Turnout . FDP share of list vote . SPD share of candidate vote . (a) Main effects Baseline −3.459 4.479 −3.857 6.500 7.773 (0.288) (0.430) (0.610) (0.378) (0.378) Unweighted −3.982 4.384 −3.830 6.888 8.076 (0.298) (0.537) (0.555) (0.382) (0.401) Controlling for CDU share of list votes in previous election −3.584 4.873 −3.748 7.005 5.887 (0.404) (0.426) (0.690) (0.340) (0.402) Controlling for district‐level covariates −4.335 4.702 −3.141 7.418 5.174 (0.710) (0.788) (0.476) (0.316) (0.744) Geographically constant districts −4.768 4.347 −4.199 7.581 6.108 (1.097) (0.970) (0.827) (0.420) (0.586) Control group Other districts in East Germany −4.931 3.543 −2.830 7.374 5.439 (0.202) (0.293) (0.221) (0.090) (0.433) All other districts in Germany −6.720 1.173 −0.594 7.654 4.437 (0.101) (0.140) (0.081) (0.060) (0.126) As percentage of all eligible voters −5.801 0.765 – 5.108 3.657 (0.171) (0.200) (0.243) (0.332) Sample/specification . Column (2) . Column (4) . Column (6) . Column (7) . Column (10) . (b) Additional interaction terms Baseline 0.430 −0.395 0.527 1.044 0.141 (0.044) (0.014) (0.052) (0.098) (0.031) Unweighted 0.426 −0.419 0.521 1.038 0.138 (0.033) (0.014) (0.038) (0.087) (0.026) Controlling for CDU share of list votes in previous election 0.432 −0.397 0.574 1.044 0.137 (0.044) (0.016) (0.050) (0.098) (0.031) Controlling for district‐level covariates 0.430 −0.393 0.559 1.053 0.135 (0.047) (0.015) (0.050) (0.103) (0.031) Geographically constant districts 0.267 −0.416 0.479 1.001 0.133 (0.113) (0.027) (0.030) (0.069) (0.039) Control group Other districts in East Germany 0.507 −0.468 0.442 0.984 0.174 (0.027) (0.012) (0.050) (0.103) (0.054) All other districts in Germany 0.523 −0.472 0.635 1.294 0.079 (0.012) (0.007) (0.011) (0.059) (0.030) As percentage of all eligible voters – −0.120 0.439 0.390 0.320 (0.011) (0.040) (0.096) (0.021) Sample/specification . Column (2) . Column (4) . Column (6) . Column (7) . Column (10) . (b) Additional interaction terms Baseline 0.430 −0.395 0.527 1.044 0.141 (0.044) (0.014) (0.052) (0.098) (0.031) Unweighted 0.426 −0.419 0.521 1.038 0.138 (0.033) (0.014) (0.038) (0.087) (0.026) Controlling for CDU share of list votes in previous election 0.432 −0.397 0.574 1.044 0.137 (0.044) (0.016) (0.050) (0.098) (0.031) Controlling for district‐level covariates 0.430 −0.393 0.559 1.053 0.135 (0.047) (0.015) (0.050) (0.103) (0.031) Geographically constant districts 0.267 −0.416 0.479 1.001 0.133 (0.113) (0.027) (0.030) (0.069) (0.039) Control group Other districts in East Germany 0.507 −0.468 0.442 0.984 0.174 (0.027) (0.012) (0.050) (0.103) (0.054) All other districts in Germany 0.523 −0.472 0.635 1.294 0.079 (0.012) (0.007) (0.011) (0.059) (0.030) As percentage of all eligible voters – −0.120 0.439 0.390 0.320 (0.011) (0.040) (0.096) (0.021) Notes Entries in the upper panel are coefficients and standard errors on δ, obtained by estimating equation (4) using least squares. Entries in the lower panel correspond to the additional interaction terms added in columns (2), (4), (6), (7) and (10) in Table 6. The respective dependent variables are listed at the top of each column, and the relevant sample restriction or change in the set of controls is denoted on the left of each row. All specifications, except for those including districts outside the state of Saxony, contain year fixed effects, municipality fixed effects, municipality‐specific linear trends and, if applicable, indicator variables for missing covariates. Regressions including districts outside the state of Saxony contain year fixed effects, district fixed effects and district‐specific linear trends instead. See the Data Appendix for the precise definitions and sources of all variables. Open in new tab To this end, the upper panel in Table 7 probes the sensitivity of δ^ with respect to different specifications. For comparison, the first row displays the baseline results, that is, those from columns (1), (3), (5), (9) and (11) in Table 6. Successive rows vary the weighting scheme, the set of included controls, as well as the control group. The last row estimates the baseline model using vote shares calculated as the percentage of all eligible voters. As turnout is itself affected by the reversal of incentives and the distribution of seats depends on the number of list votes, these estimates may be more informative for answering certain questions than ones based on ordinary vote shares. Although individual point estimates do, of course, vary, δ^ is qualitatively quite robust. Neither including different covariates, nor changing the control group, would alter the conclusions drawn above.30 At first glance, there appear to be two exceptions, however. The CDU's share of the candidate vote increases by only 0.8 percentage points when calculated as a fraction of all eligible voters and the SPD's gain is estimated to be much smaller than at baseline. Given that turnout is itself endogenous, this is hardly surprising. If supporters of all parties were more likely to abstain than in ordinary years but adherents of the CDU did so disproportionally, then this alone may explain the differences. Compositional effects, that is, endogenous turnout, are an especially important concern as the by‐election in District 160 was held on the Sunday of a three‐day weekend, whereas the election in all other districts took place on a ‘regular’ Sunday. Yet, compositional effects cannot explain why the CDU's share of the list vote measured as percentage of all eligible voters declined by almost 6 percentage points when turnout itself declined by only 3.9 percentage points and why the share of agents casting list votes for the FDP actually increased by more than 5 percentage points. Reconciling these estimates requires a theory of why CDU supporters would substitute towards the FDP.31 The lower panel in Table 7 performs robustness checks on the additional interaction terms included in columns (2), (4), (6), (7) and (10) of Table 6. Again, the first row shows the baseline results. As was the case for the main effects, the estimates appear to be quite robust. 4. Concluding Remarks Whether and, if so, to what extent, individuals cast strategic ballots is one of the most important questions at the intersection of economics and political science. This article provides empirical evidence from a natural experiment in which a party benefited by receiving fewer votes. Comparing the behaviour of constituencies facing different incentives, identification in this article is not subject to the problems that plague most of the existing literature. The evidence presented above points clearly towards a non‐trivial incidence of preference misrepresentation, despite the fact that the by‐election could at most have had a small effect on the overall distribution of seats. Although agents might not vote according to their preferences for non‐instrumental reasons, the evidence suggests that at least some voters did not merely follow official party recommendations but misrepresented their preferences strategically. Taking the estimates at face value, one can derive a lower bound on the fraction of agents who did not vote ideologically in the sense of Degan and Merlo (2009), that is, who voted for a candidate or party other than the ideologically closest one. To do so, add the change in the FDP's share of the list vote and the change in the SPD's share of the candidate vote (both calculated as percentage of all eligible voters, see Table 7). Although this number includes CDU supporters substituting towards the FDP and adherents of other parties supporting the SPD candidate, it does not capture SPD supporters casting their list votes for the CDU, nor does it encompass FDP adherents voting for the CDU candidate. Therefore, it is likely to understate the true extent of misaligned voting in the by‐election. Nevertheless, according to this measure (at least) 8.8% of the electorate misrepresented their preferences when faced with new, paradoxical incentives. Thus, the lower bound identified under relatively weak assumptions in this article is among the largest of findings in the literature (most of which come from elections under plurality rule or from run‐off elections). Especially compared to Kawai and Watanabe's (2013)1.2–2.7% estimate of misaligned voting, the lower bound derived for the by‐election in District 160 appears to be very large. This may be surprising as electoral systems with (approximately) proportional representation are often believed to be relatively immune to strategic voting. In sum, the evidence in this article suggests that the share of agents who do not vote ideogolically may be much higher than previously thought.32 Of course, it is difficult to say what the results from this, admittedly, unusual by‐election imply for strategic voting under ordinary circumstances, or even in other electoral systems. One may, however, be willing to draw somewhat more general conclusions by focusing on the race for the outstanding direct mandate. As in elections under plurality rule elsewhere, voters in District 160 knew that the winner of the district‐level race would represent them in parliament and that only two or three candidates were in contention for victory. Faced with this particular set of incentives, voters chose to misrepresent their preferences. As this is essentially the same situation as in many countries around the world, it may not be unreasonable to expect similar behaviour elsewhere. Moreover, the results of this article have potentially important implications for the design of electoral systems. If a significant number of voters do, indeed, misrepresent their true preferences, then the issue of strategy‐proofness in social choice becomes more than a theoretical curiosity. With a non‐trivial number of non‐ideological voters, susceptibility to strategic manipulation should be an important criterion by which to judge the performance of different mechanisms. Footnotes 1 " Throughout the article, the terms ‘strategic’ and ‘tactical’ are used interchangeably. 2 " Small‐scale laboratory experiments relying on induced preferences are an important exception. These studies typically find convincing evidence of strategic voting, which increases with the availability of co‐ordination devices such as polls or access to voting histories (Forsythe et al.,). Palfrey (2009) provides a useful review. 3 " Alvarez et al. (2006) as well as Kawai and Watanabe (2013) point out that some of the literature mistakes the share of voters who actually voted for someone other than their most preferred candidate as equal to the fraction of voters who would do so if their preferred candidate was believed to be unlikely to win the race. Not surprisingly, both papers estimate the former number to be significantly smaller than the latter. 4 " Throughout most of the article, the identifying assumption is that the one‐time reversal in incentives did not coincide with larger deviations from trend for affected voters than for unaffected ones. 5 " Given past election results in District 160, only the CDU had a realistic chance of gaining or losing a seat. Although other parties stood to gain a mandate in the state of Saxony, this would have come at the expense of a mandate in some other state. 6 " The inherent uncertainty in a party's number of votes in each state makes it almost impossible to predict similar situations in advance, although ex post one can in most election years find instances in which a party may have benefited by receiving fewer votes in some state. 7 " Although winning the plurality of the candidate vote may not necessarily increase a party's seat total in regular election years (Section 1), in the 2005 by‐election it was known that the CDU would gain a seat in the Bundestag by securing the direct mandate in District 160. 8 " As the political landscape in East Germany continues to be quite different from that in the West, electoral districts in West Germany may be a relatively poor comparison group. Nevertheless, the results in this article are qualitatively robust to including those districts as well (Table 7). 9 " In doing so, it borrows from Korte (2010). 10 " Single‐member districts are represented by exactly one delegate in parliament, and plurality rule refers to an electoral system in which whoever receives the most votes is declared the winner of the election. 11 " A party list is a pre‐determined ranking of candidates based on which list mandates are awarded. By law, parties must post different lists in each state, and a candidate can appear on only one list (§27 BWG). Proportionality rule means that a party's vote share equals its share of seats in parliament. 12 " In this context, mandate stands for a parliamentary seat won in an election. 13 " Although it had been known for a long time that a voting system which provides constituencies with the opportunity to elect some (but not all) members of parliament directly might give rise to perverse incentives, until the by‐election in District 160 the German Constitutional Court had judged this to be inconsequential. 14 " In 2009, the Sainte‐Laguë method was used instead. 15 " Generally, they accrue only to the two major factions, that is, the CDU/CSU or the SPD. 16 " In Germany, elections are almost always held on Sundays. It is important to point out that 3 October 2005, was a national holiday. Although the fact that the by‐election was held on a three‐day weekend may explain the lower turnout, it cannot account for the CDU receiving a lower share of the list vote while simultaneously garnering a higher fraction of the candidate vote (see Section 3). 17 " Traditionally, the president (Bundespräsident) charges the leader of the largest faction with forming a new government. 18 " Unfortunately, there are no geographically disaggregated campaign spending data. 19 " Both FDP and CDU officials denied allegations of collusion. In the past, the FDP had often campaigned for the list votes of CDU supporters to pass the national 5% threshold. But it had not asked its own supporters to abandon the FDP direct candidate in favour of that of the CDU. In 2005, the FDP gained 9.8% of the national list vote, and was thus far from failing to clear the 5% threshold. By winning enough list votes in District 160, the FDP could only receive an additional list mandate in Saxony at the expense of one in North Rhine‐Westphalia or Saxony‐Anhalt. 20 " Note that, compared to the preliminary results, the final list vote count in other districts changes slightly. 21 " See the Data Appendix for additional information on the construction of the mapping. As individual precincts change considerably between 2002 and 2009 and there are no data on the sub‐precinct level, it is not possible to link precincts over time. Reassuringly, the results in this article are robust to excluding all electoral districts that did not remain geographically constant over time (Table 7). 22 " Districts in the same state are likely subject to similar overall trends and may thus be a better comparison group than districts in East Germany as a whole, or even West Germany. The robustness checks in Table 7 show that the main results are qualitatively robust to the choice of control group. 23 " Other parties’ gains and losses are estimated as follows: +3.23% for the SPD, +7.02% for the FDP, −1.55% for the Green Party and −2.81% for The Left, and −2.31% for all remaining parties. 24 " Note well that the DD estimates indicate the difference in the extent of non‐ideological voting between the by‐election and the ‘regular’ election on September 18. This does not require the assumption that all voters in the ‘regular’ election behaved sincerely. For this reason, one may want to regard the DD estimates as lower bound (see also, the discussion in Section 4) 25 " At least in principle, voters could have changed their rankings of parties not because of a change in preferences but due to the arrival of new information between 18 September and the by‐election. Although this may cause problems for standard DD estimates, the fact that CDU received more list votes but fewer candidate votes is at odds with such an explanation. 26 " The municipality‐level data even allow for quadratic or cubic trends. The point estimates, however, remain almost unaffected. 27 " The number of observations in columns (1) and (2) of Table 6 is lower than in the remaining columns, because turnout cannot be calculated for precincts that handle only absentee ballots. Absentee voters are included in the turnout figure of the precinct in which they reside. 28 " Of course, there exist alternative explanations for why overall turnout might have been lower during the by‐election. For instance, individuals might derive less utility from the act of voting when the results of the election are already known. However, such ‘behavioural’ explanations are incompatible with there being large partisan differences in the by‐election's effect on turnout. 29 " Absentee voters are individuals who were not physically present in District 160 on election day, or who were not physically able to appear at a polling station in person, that is, elderly or handicapped citizens. Absentee ballots are generally cast by mail and are reported separately in the official data. 30 " At first, it may seem preferable to include covariates even in the baseline specifications. Yet, this is not necessarily the case. Since the baseline specifications already account for municipality‐specific trends, there is very little variation left to identify the coefficients on these controls. In fact, for most of the control variables, the estimates are statistically indistinguishable from zero. Moreover, given that very little variation is used to identify the model's coefficients, measurement error in the covariates becomes a first‐order concern. Unfortunately, without imposing strong assumptions, it is not possible to sign the bias in δ resulting from measurement error in the covariates. 31 " In Germany, it is also possible to abstain with one vote without invalidating the other. However, there is no evidence that this kind of selective abstention was quantitatively important during the by‐election. 32 " Kawai and Watanabe (2013) estimate a 63–85% extent of strategic voting. In the context of the by‐election in District 160 one can derive an upper bound on the share of strategic voters, although it is unlikely to be tight. To do so note that individuals voting for the direct candidates of the FDP, Green Party, The Left or the NPD cannot have cast strategic ballots, as these candidates were known to have had virtually no chance of winning the district, or being tied for first (see, for instance, the representative poll published nine days before the by‐election in the Sächsische Zeitung, 2005b). By this measure, at most 69.2% of voters could have behaved strategically. Interestingly, this upper bound is somewhat smaller than the lower bound in Kawai and Watanabe (2013). References Abramson , P.R. , Aldrich , J.H., Paolino , P. and Rohde , D.W. ( 1992 ). ‘ Sophisticated voting in the 1988 presidential primaries ’, American Political Science Review , vol. 86 ( 1 ), pp. 55 – 69 . Google Scholar Crossref Search ADS WorldCat Alvarez , R.M. , Boehmke , F.J. and Nagler , J. ( 2006 ). ‘ Strategic voting in British elections ’, Electoral Studies , vol. 25 ( 2 ), pp. 1 – 19 . 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( 2013 ). ‘ Inferring strategic voting ’, American Economic Review , vol. 103 ( 2 ), pp. 624 – 62 . Google Scholar Crossref Search ADS WorldCat Korte , K.‐R . ( 2010 ). Wahlen in Deutschland. Bonn: Bundeszentrale für Politische Bildung. Landesvermessungsamt Sachsen . ( 2006 ). Übersichtskarte Freistaat Sachsen 1:200 000 , Dresden : Landesvermessungsamt Sachsen . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Myerson , R.B . and Weber , R.J. ( 1993 ). ‘ A theory of voting equilibria ’, American Political Science Review , vol. 87 ( 1 ), pp. 102 – 14 . Google Scholar Crossref Search ADS WorldCat Niemi , R.G. , Whitten , G. and Franklin , M.N. ( 1992 ). ‘ Constituency characteristics, individual characteristics and tactical voting in the 1987 British general election ’, British Journal of Political Science , vol. 22 ( 2 ), pp. 229 – 40 . Google Scholar Crossref Search ADS WorldCat Palfrey , T.R . 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Google Scholar Crossref Search ADS WorldCat Author notes " This article is based on the first chapter of my dissertation at the University of Chicago. I would like to thank the editor, Andrea Galeotti, and three anonymous referees for helpful comments and suggestions. Moreover, I have benefited from conversations with Gary Becker, Eric Budish, Dana Chandler, Tony Cookson, Roland Fryer, Steven Levitt, Roger Myerson, Elisa Olivieri, Philipp Tillmann and David Toniatti. I am also indebted to Gabriele Schömel at the office of the Bundeswahlleiter for help in acquiring the data used in this article. Steven Castongia and Sathish Kumar provided excellent research assistance. Financial support from the Beryl W. Sprinkel Fund at the University of Chicago is gratefully acknowledged. All views expressed in this article as well as any remaining errors are solely my responsibility. © 2014 Royal Economic Society

Smith,, Sarah;Windmeijer,, Frank;Wright,, Edmund

doi: 10.1111/ecoj.12114pmid: N/A

Abstract There is a widespread belief that peer effects are important in charitable giving but little evidence on how donors respond to their peers. Analysing a unique data set of donations to online fund‐raising pages, we find positive and sizeable peer effects: a £10 increase in the mean of past donations increases giving by £2.50, on average. Donations respond to both very large and very small amounts and to changes in the mode. We find little evidence that donations signal charity quality – our preferred explanation is that donors use information on earlier donations to decide what is appropriate for them to give. The size of your gift can persuade your peer to make a contribution as significant as yours. ‘How to succeed in fundraising by really trying’ by Lewis B. Cullman. This study is concerned with peer effects in charitable giving – specifically the way in which the amount that donors give responds to donations made by others in their peer group. There is a widespread belief that such peer effects are important but there is surprisingly little direct evidence. Early studies used cross‐section data to define generic reference groups in terms of income (Feldstein and Clotfelter, 1976) and other socio‐demographic characteristics such as age and education (Andreoni and Scholz, 1998). More recent experimental studies have looked at the effect of ‘social cues’ – i.e. single pieces of information about how much has been given by other people, unknown to the donor, such as a previous cohort or a typical donor (Frey and Meier, 2004; Alpizar et al., 2008; Shang and Croson, 2009). There are two studies that have looked directly at peer effects in giving. Meer (2011) focused on peer effects in solicitation, looking at whether people give more if the ask comes from someone they know. Carman (2004) studied peer effects among workplace teams but, in this case, the peer group included the team captain who played a role in encouraging and motivating giving among team members. Ours is the first study we are aware of to look at purely horizontal (donor‐to‐donor) peer effects in giving. We empirically investigate how donors are influenced by the donations of their peers in the context of individual online fund‐raising. In the UK, this is a major source of income for many charities. Since 2001, more than two million individual fund‐raisers have raised more than £1 billion for a wide range of different charities through the biggest individual online fund‐raising website, and this has been growing over time.1 The way that individual online fund‐raising typically works is as follows: Individual fund‐raisers decide on a fund‐raising activity to raise money for their chosen charity (these activities often involve a sporting event such as running a marathon or swimming the English Channel but novelty activities such as head shaving are also popular). The fund‐raisers then set‐up personalised web pages on a fund‐raising website and invite people to make donations to their chosen charities. Most of the donations come from the fund‐raiser's friends, family and colleagues.2 Almost all are made online via the fund‐raising page and are passed directly by the fund‐raising website to the charity. The online donations are listed on the fund‐raising page, with the most recent first.3 Information on how much has been given and by whom,4 is then visible to each donor who arrives at the fund‐raising page. When donors go to the page to make a donation they can see all the previous online donations that have been made; we exploit this set‐up to look at whether donors are influenced by how much other people have given. Of course, donations made to the same page will be correlated because of the common characteristics of the peer group – the fund‐raiser's friends, family and work colleagues. Our identification strategy relies on the within‐page variation in the observed history of donations that arises as a result of donors arriving at the website at different times.5 In essence, we argue that there is plausibly exogenous variation in the set of donations observed by each donor because exactly when donors make their donation is subject to random factors, such as when they turn on their computer and find time to log on to the fund‐raising website to make a donation. We further discuss our identification strategy in Sections 3 and 4. We provide direct evidence on the direction and magnitude of peer effects in giving. In principle, it is possible that other people's donations could ‘crowd out’ giving (Warr, 1982; Roberts, 1984) but we show that higher (average) donations cause people to increase the amount that they give – a £10 increase in the mean of past donations causes people to give £2.50 more on average. One potential criticism of a simple ‘linear‐in‐means’ specification is that it can mask the potentially diverse ways in which peer effects can work (Sacerdote, 2011). We are able to shed light on the nature of peer effects in giving and show that the amount given is affected both by ‘shining knights’ (very large donations) and by ‘widows’ mites’ (very small donations), as well as there being ‘herd behaviour’ (donations following the mode). We also exploit the richness of our data to explore some of the underlying mechanisms that might explain why donors respond positively to how much their peers have given. We find no evidence that peer donations provide a signal about the quality of a charity (Vesterlund, 2003), or that peer effects are only related to fund‐raising targets (Andreoni, 1998). The explanation that is most consistent with observed behaviour is that donors use information on (the distribution of) past donations as a benchmark in deciding how much it is appropriate for them to give. The plan of the remainder of the study is as follows. The next Section provides information on our data – a subset of fund‐raising pages set‐up by runners in the 2010 London marathon. Section 2 explores the effect of other donations and the nature of the peer effects by looking at the effect of large and small donations and changes in the mode, whereas Section 4 contains our main econometric analysis. Section 5 explores alternative explanations of why donors might respond to their peers and Section 6 concludes. 1. The Setting – Online Fund‐raising In this study, we focus on fund‐raising pages set‐up by people who raised money for charity by running in the 2010 London marathon and who fund raised via the two largest fund‐raising websites in the UK – Justgiving (www.justgiving.co.uk) and Virgin Money Giving (http://uk.virginmoneygiving.com/giving/). The London marathon claims to be the biggest single fund‐raising event in the world and, of the approximately 35,000 runners who line up each year, an estimated 20,000 are raising money for charity. Our initial sample contained information from more than 12,000 fund‐raising pages. The data were captured on 30 April 2010, five days after the marathon took place. For each page we have all the information that is publicly available (examples of fund‐raising pages are shown in online Appendix B). This includes the fund‐raiser's name, the charity they were fund‐raising for, their target amount (if they had one), the total amount raised offline at the time the data were captured, the full history of donations to the website, the donors’ names (where available) and the amount given. Table 1 provides a basic sample summary. Each fund‐raiser gets an average of 34.5 donations and raises an average of £1,093 in online donations and £335 in reported offline donations.6 Donations are spread over time. The typical page is set‐up just over two months before the marathon. Some fund‐raisers create pages up to six months before the event. Over this period, fund‐raisers may sequentially target different sets of people within their wider peer group. In this case, any observed change in donation amounts (e.g. following a large or small donation or a change in the mode) may simply reflect the arrival of a new donor group. When we look at amounts donated in Section 4, we test for changes in arrival rates; we also carry out an additional robustness check focusing only on donations made within the same day. Table 1 Sample Summary Statistics . Mean . SD . Minimum . 1st percentile . Median . 99th percentile . Maximum . Full sample Number of donations per page 34.5 25.4 1 1 29 114 370 Number of days 74.8 50.7 0 0 67 204 225 Online donations – all £30.31 £66.02 £1 £5 £20 £200 £10,000 Total raised online per page £1,093 £1,401 £1 £20 £778 £5,710 £40,326 Total raised offline per page £335 £1,115 £0 £0 £0 £3,077 £53,000 Proportion of pages with target 0.803 Proportion of pages with target achieved 0.395 Target amounts £99,985 £9.9 m £0.01 £200 £1,500 £9,000 £1 bn Number of fund‐raisers 12,750 Estimation sample Number of donations per page 36.7 19.7 10 10 33 91 100 Number of days 79.5 49.5 2 6 73 205 225 Online donations £29.81 £46.58 £1 £5 £20 £200 £1,000 Total raised online per page £1,115 £916 £53 £136 £892 £4,458 £12,260 Total raised offline per page £310 £827 £0 £0 £0 £2,725 £43,897 Proportion of pages with target 0.823 Proportion of pages with target achieved 0.420 Target amounts £1,511 £832 £200 £200 £1,500 £5,000 £7,000 Number of fund‐raisers 10,597 . Mean . SD . Minimum . 1st percentile . Median . 99th percentile . Maximum . Full sample Number of donations per page 34.5 25.4 1 1 29 114 370 Number of days 74.8 50.7 0 0 67 204 225 Online donations – all £30.31 £66.02 £1 £5 £20 £200 £10,000 Total raised online per page £1,093 £1,401 £1 £20 £778 £5,710 £40,326 Total raised offline per page £335 £1,115 £0 £0 £0 £3,077 £53,000 Proportion of pages with target 0.803 Proportion of pages with target achieved 0.395 Target amounts £99,985 £9.9 m £0.01 £200 £1,500 £9,000 £1 bn Number of fund‐raisers 12,750 Estimation sample Number of donations per page 36.7 19.7 10 10 33 91 100 Number of days 79.5 49.5 2 6 73 205 225 Online donations £29.81 £46.58 £1 £5 £20 £200 £1,000 Total raised online per page £1,115 £916 £53 £136 £892 £4,458 £12,260 Total raised offline per page £310 £827 £0 £0 £0 £2,725 £43,897 Proportion of pages with target 0.823 Proportion of pages with target achieved 0.420 Target amounts £1,511 £832 £200 £200 £1,500 £5,000 £7,000 Number of fund‐raisers 10,597 Open in new tab Table 1 Sample Summary Statistics . Mean . SD . Minimum . 1st percentile . Median . 99th percentile . Maximum . Full sample Number of donations per page 34.5 25.4 1 1 29 114 370 Number of days 74.8 50.7 0 0 67 204 225 Online donations – all £30.31 £66.02 £1 £5 £20 £200 £10,000 Total raised online per page £1,093 £1,401 £1 £20 £778 £5,710 £40,326 Total raised offline per page £335 £1,115 £0 £0 £0 £3,077 £53,000 Proportion of pages with target 0.803 Proportion of pages with target achieved 0.395 Target amounts £99,985 £9.9 m £0.01 £200 £1,500 £9,000 £1 bn Number of fund‐raisers 12,750 Estimation sample Number of donations per page 36.7 19.7 10 10 33 91 100 Number of days 79.5 49.5 2 6 73 205 225 Online donations £29.81 £46.58 £1 £5 £20 £200 £1,000 Total raised online per page £1,115 £916 £53 £136 £892 £4,458 £12,260 Total raised offline per page £310 £827 £0 £0 £0 £2,725 £43,897 Proportion of pages with target 0.823 Proportion of pages with target achieved 0.420 Target amounts £1,511 £832 £200 £200 £1,500 £5,000 £7,000 Number of fund‐raisers 10,597 . Mean . SD . Minimum . 1st percentile . Median . 99th percentile . Maximum . Full sample Number of donations per page 34.5 25.4 1 1 29 114 370 Number of days 74.8 50.7 0 0 67 204 225 Online donations – all £30.31 £66.02 £1 £5 £20 £200 £10,000 Total raised online per page £1,093 £1,401 £1 £20 £778 £5,710 £40,326 Total raised offline per page £335 £1,115 £0 £0 £0 £3,077 £53,000 Proportion of pages with target 0.803 Proportion of pages with target achieved 0.395 Target amounts £99,985 £9.9 m £0.01 £200 £1,500 £9,000 £1 bn Number of fund‐raisers 12,750 Estimation sample Number of donations per page 36.7 19.7 10 10 33 91 100 Number of days 79.5 49.5 2 6 73 205 225 Online donations £29.81 £46.58 £1 £5 £20 £200 £1,000 Total raised online per page £1,115 £916 £53 £136 £892 £4,458 £12,260 Total raised offline per page £310 £827 £0 £0 £0 £2,725 £43,897 Proportion of pages with target 0.823 Proportion of pages with target achieved 0.420 Target amounts £1,511 £832 £200 £200 £1,500 £5,000 £7,000 Number of fund‐raisers 10,597 Open in new tab The mean online donation is £30.31. The distribution of donations is heavily concentrated with spikes at £10 and £20 (and to a lesser extent other rounded amounts) with just over half of all donations at exactly £10 or £20. The distributions of donation amounts and the number of donations per page are skewed by the presence of a few very successful fund‐raisers7 and generous donors. In our analysis, we exclude pages which have single donations of more than £1,000. We also exclude pages with fewer than 10 donations (1,783 pages) or more than 100 donations (212 pages). With these exclusions, our sample is 10,597 pages. 2. Empirical Strategy A commonly estimated model in the peer effects literature is a linear‐in‐means model. In our case, this can be written as: din=α+γd¯i,n−1+uin, where the donation amount, d, given by donor n to page i is estimated as a function of the mean of all past donations to the same page up to that point d¯i,n−1 ⁠. There are well‐known problems in identifying peer effects (Brock and Durlauf, 2001; Manski, 2003). In our case we can rule out the reflection problem as the amount given by the nth donor will not affect the donations made by previous donors. Correlated effects are a clear concern. Donors to a page will share socio‐economic and demographic characteristics because they are likely to be drawn from a fund‐raiser's network of friends, family and work colleagues. They will also be subject to the common influence of the same fund‐raiser who may be more or less effective at encouraging people to give. Our identification strategy therefore relies on within‐page variation in observed past donations arising as a result of donors arriving at a page at different times to make their donation. Of course there is likely to be some endogenous sorting within a page: close family and friends will be among the first to give, as well as people with a strong connection to the cause – and both these groups are likely to give more. This is clear from the observed decline in mean donation size over the first few donations to a page (see Figure 1a). In our analysis, we run regressions excluding the first three donations to a page – this is both to allow for some donation history for subsequent donors to respond to and also because the first three donations are systematically higher than the rest and may possibly behave differently to those that follow (e.g. because they are from the donor's closest friends/family). Our main findings are not sensitive to this sample selection. It is clear from a randomly selected sub‐sample of pages (Figure 1b) that there is also non‐systematic variation in the size of donations within a page that causes the within‐page mean to vary. We exploit this variation to identify peer effects. As a number of studies have pointed out (see discussion in Sacerdote, 2011), a limitation of the linear‐in‐means model is that it may over‐simplify – and potentially obscure – the many different ways in which peer effects work in practice. Following Sacerdote (2011), who presents a typology of potential peer effects in relation to education, we can distinguish a number of different ways in which peer effects might affect giving. First, donations may be affected by ‘shining knights’, i.e. by large donations to a page. A large donation is likely to place upwards pressure on amounts given among donors who want to signal their wealth (Glazer and Konrad, 1996) or generosity (Harbaugh, 1998) or the closeness of their relationship with the fund‐raiser by being among the biggest donors. This would be likely only to affect the upper end of the distribution among donors competing to give the most. Large donations may, however, have a wider effect on all donors to the extent that they crowd out other giving, assuming standard public good giving (Warr, 1982; Roberts, 1984) or crowd it in if there is a threshold for the provision of the public good (Andreoni, 1998). Large donations may also provide a signal about the quality of the charity (Vesterlund, 2003) or affect individuals’ beliefs about how much it is appropriate to give, assuming such beliefs are based on the observed distribution of amounts given. Second, donations may be affected by ‘widows’ mites’, i.e. by small donations to a page. Becker (1974) emphasised that donations might be motivated by the desire to avoid social stigma as well as to gain social prestige. Some donors will want to get away with giving as little as possible and a small donation will allow them to reduce how much they give. This is likely to affect donations at the lower end of the distribution. More generally, a small donation may also affect all others in ways similar to a large donation – i.e. through crowd out/crowd in, signalling effects or benchmarking. Fig. 1. Open in new tabDownload slide (a) Mean Amount by Order of Donation on Page (full sample). (b) Within‐page Variation in Past Mean (randomly selected sub‐sample) Fig. 1. Open in new tabDownload slide (a) Mean Amount by Order of Donation on Page (full sample). (b) Within‐page Variation in Past Mean (randomly selected sub‐sample) Table 2 Nature of Peer Effects in Giving . Donation will have an effect on ··· . Type of donation . … only some donors if there is: . … all donors if there is: . Large donations (‘Shining knights’) Competition to be the top donor Crowding in/out Signalling quality Benchmark for appropriate amount Small donations (‘Widows’ mites’) Desire to avoid being the bottom donor Crowding in/out Signalling quality Benchmark for appropriate amount Modal donations (‘The herd’) Following the herd; giving what other people give Crowding in/out Signalling quality Benchmark for appropriate amount . Donation will have an effect on ··· . Type of donation . … only some donors if there is: . … all donors if there is: . Large donations (‘Shining knights’) Competition to be the top donor Crowding in/out Signalling quality Benchmark for appropriate amount Small donations (‘Widows’ mites’) Desire to avoid being the bottom donor Crowding in/out Signalling quality Benchmark for appropriate amount Modal donations (‘The herd’) Following the herd; giving what other people give Crowding in/out Signalling quality Benchmark for appropriate amount Open in new tab Table 2 Nature of Peer Effects in Giving . Donation will have an effect on ··· . Type of donation . … only some donors if there is: . … all donors if there is: . Large donations (‘Shining knights’) Competition to be the top donor Crowding in/out Signalling quality Benchmark for appropriate amount Small donations (‘Widows’ mites’) Desire to avoid being the bottom donor Crowding in/out Signalling quality Benchmark for appropriate amount Modal donations (‘The herd’) Following the herd; giving what other people give Crowding in/out Signalling quality Benchmark for appropriate amount . Donation will have an effect on ··· . Type of donation . … only some donors if there is: . … all donors if there is: . Large donations (‘Shining knights’) Competition to be the top donor Crowding in/out Signalling quality Benchmark for appropriate amount Small donations (‘Widows’ mites’) Desire to avoid being the bottom donor Crowding in/out Signalling quality Benchmark for appropriate amount Modal donations (‘The herd’) Following the herd; giving what other people give Crowding in/out Signalling quality Benchmark for appropriate amount Open in new tab Third, there may be ‘herd behaviour’. Donors with a desire to conform may try to target how much they give on the modal amount (Bernheim, 1994). In this case, the amount given may be affected by (changes in) the mode of donations to a page. As with small and large donations, a change in the mode may affect only some donors or all other donors to a page. Table 2 summarises the different ways in which donations by peers might influence giving. The online fund‐raising data allow us to explore these different types of peer effects. In particular, we can look directly at the effect of ‘shining knights’ and ‘widows’ mites’ and of changes in the mode on amounts given. We also look at whether large and small donations affect only some donors (in the upper/lower end of the distribution) and/or whether the effects appear to be more general. 3. Estimates of Peer Effects – A Natural Experiment Approach To look at the effects of ‘large’ and ‘small’ donations and changes in the modal amount we estimate the following specification: din=α+βTin+zin′δ+uin, where din refers to the nth donation to fund‐raising page i (in pounds) and Tin is a ‘treatment’ indicator equal to 1 if the donation follows a large/small donation or a change in the mode, or equal to 0 otherwise. We define a ‘large’ donation as being at least twice the page mean (and more than £50). The mean ‘large’ donation is £102. A ‘small’ donation is defined as half the page mean. The mean ‘small’ donation is £8.61. We look separately at increases and decreases in the mode.8zin is a vector of controls for the systematic component of the timing of donations – the order on the page and the date of donation respectively. The error term is decomposed into a constant page‐specific effect that will pick up common differences in donations across pages and a pure random error term: uin = ηi + vin. We estimate this model using a fixed‐effects regression that removes the effect on donations of the page‐specific unobservable factors. We drop pages where a large or small donation occurs within the first three donations; we also restrict the first change in the mode to occur after the first three donations. Our identifying assumption is that there is random variation in the timing of donations, after controlling for systematic within‐page variation, such that the random error term, vin, is uncorrelated with the ‘treatment’ variable, Tin. We would argue that this assumption is plausible, at least within a narrow window of donations, given that the exact timing of when people make an online donation will be subject to a number of exogenous factors. Exactly when donors arrive at the page – and hence whether they arrive just before or just after a large/small donation – will be influenced by a number of random factors such as when they turn on their computer and when they find a moment to log on to the fund‐raising website to make an online donation. Under our identifying assumption, the coefficient β will identify the average causal effect of a large/small donation on the amount subsequently given. There are two possible violations of this identifying assumption. One is if large/small donations affect the extensive margin – i.e. the probability that donors make a donation. In this case, the observed donations before and after would be subject to a differential selection process. A second is if fund‐raisers sequentially target different groups of donors – in which case the first large/small donation would herald the arrival of a new group of donors. We have no information on visits to the websites, or on donor characteristics that allow us to test for these effects directly. However, we can look at the arrival rate of donations (i.e. the number of donations made to a page per day) to give some indication of whether either of these is likely to be material. Both a change in the extensive margin and the arrival of a new group of donors would be associated with a change in the arrival rate. Figure 2 plots the distributions of the arrival rates (i.e. the number of donations per day) on the days before and after each of the four treatments we look at. There is little obvious change in the distributions and this is confirmed by Kolgomorov–Smirnov tests. The p‐values for the equality of distributions before/after large and small donations are 0.219 and 0.352, respectively, whereas the p‐values for the equality of distributions before/after increases and decreases in the mode are, respectively, 0.094 and 0.668. In all four cases we fail to reject that the distributions of arrival rates are the same. By contrast, Figure 3 provides clear evidence of effects on amounts given after each of the four ‘treatments’. Donations increase after both a large donation and an increase in the mode, whereas donations fall after both a small donation and a decrease in the mode. These findings are confirmed by regression results, summarised in Table 3. We vary the size of the window before and after – looking at a very narrow window of one donation before/after and also five donations before/after and five before and ten after. We do a further robustness check where we restrict the before and after donations to lie within the same day, making it less likely that they have been made by different groups of (sequentially targeted) donors. Fig. 2. Open in new tabDownload slide Distributions of Arrivals. Before/After: (a) A ‘Large’ Donation; (b) A ‘Small’ Donation; (c) An Increase in Mode; and (d) A Decrease in Mode Notes. A large donation is defined as twice the page mean and at least £50. A small donation is half the page mean. We focus on the first large/small donation or change in mode to occur on a page, excluding those within the first three donations. p‐value is for test of equality of distributions (Kolgomorov–Smirnov). Fig. 2. Open in new tabDownload slide Distributions of Arrivals. Before/After: (a) A ‘Large’ Donation; (b) A ‘Small’ Donation; (c) An Increase in Mode; and (d) A Decrease in Mode Notes. A large donation is defined as twice the page mean and at least £50. A small donation is half the page mean. We focus on the first large/small donation or change in mode to occur on a page, excluding those within the first three donations. p‐value is for test of equality of distributions (Kolgomorov–Smirnov). The results in panels (a)–(d) confirm that there is a change in how much subsequent donors give following each of the four treatments. The coefficients indicate fairly sizeable effects. Within a narrow window of one donation either side, large donations are associated with a £12.49 increase in donation size, compared to a previous donation level around £20, whereas a small donation reduces donation size by a similar magnitude. The effects also appear to be fairly persistent affecting at least 10 donations that follow; this is likely to work not just through the first large/small donation or change in mode but also through changes in subsequent donations. As discussed in the previous Section, large and small donations may affect amounts given either by triggering competition among some donors (other large/small donations) or, more generally, by influencing all other donors through crowd out/in, signalling or benchmarking effects. We shed light on this by looking at the effect on subsequent amounts given, excluding other large and small donations. This will tell us whether the effect is (just) to trigger other large/small donations or whether it goes wider than this. The results, shown in panels (e) and ( f ), indicate that large and small donations do indeed trigger other similar‐sized donations (the coefficients are smaller than in panels (a) and (b)) but that there are effects even on ‘regular‐sized’ donations. Fig. 3. Open in new tabDownload slide Mean Amounts Given Before/After: (a) A ‘Large’ Donation; (b) A ‘Small’ Donation; (c) An Increase in Mode; and (d) A Decrease in Mode (all donations measured in £) Notes. A large donation is defined as twice the page mean and at least £50. A small donation is half the page mean. We focus on the first large/small donation or change in mode to occur on a page, excluding those within the first three donations. Fig. 3. Open in new tabDownload slide Mean Amounts Given Before/After: (a) A ‘Large’ Donation; (b) A ‘Small’ Donation; (c) An Increase in Mode; and (d) A Decrease in Mode (all donations measured in £) Notes. A large donation is defined as twice the page mean and at least £50. A small donation is half the page mean. We focus on the first large/small donation or change in mode to occur on a page, excluding those within the first three donations. The coefficients in panels (a)–(d) indicate that the peer effects are increasing in amount size – a large donation is associated with a bigger effect than an increase in the mode. We explore this further by looking at the effects of different‐sized large donations (twice previous mean, three times previous mean, five times previous mean and more than ten times previous mean). As in previous studies (Shang and Croson, 2009) we find that larger donations produce a greater response from subsequent donors, at least up to very large donations of ten or more times the page mean. Combined with our results on the effects of large/small donations and changes in the mode, this supports our use of a linear‐in‐means model in the next Section. Finally, we look at whether there is evidence of spillover effects from donors giving more in response to a large donation on one fund‐raising page to how much they give on other fund‐raising pages. We do this by exploiting the fact that, within the Justgiving sample, we can identify donors who give to more than one fund‐raising page. We construct a donor‐level panel of amounts given sequentially across different pages.9 We estimate an equation of the following form: dsj=α+β1Tsj+β2T(s−1)j+ηj+ωsj, where dsj refers to the sth donation of donor j. Tsj is an indicator equal to 1 if there has been a large donation (within ten donations) to the page currently visited, whereas T(s−1)j is an indicator equal to 1 if the previous page visited by the donor had a large donation. β1 captures the own‐page effect and β2 any spillover effect of a large donation on a previously visited page. We estimate this equation on the full sample of ( Justgiving) donors but the own‐page and spillover effects are identified from donors who give to multiple pages. We include a trend to allow for the fact that donors may reduce their donations as they are asked to sponsor more people. Table 3 Effect of Large/Small Donation and Change In Mode . One before/one after . One before/one after (same day) . Five before/five after . Five before/ten after . (a) Effect of a ‘large’ donation After 12.458** 13.392** 12.611** 12.134** (0.789) (2.609) (0.661) (0.496) N 15,508 6,464 68,926 102,492 (b) Effect of a ‘small’ donation After −11.411** −9.493** −11.169** −10.232** (0.911) (2.090) (0.770) (0.550) N 14,499 6,600 58,858 91,422 (c) Effect of an increase in the mode After −0.424 1.755 0.887 1.137* (1.211) (2.818) (0.961) (0.671) N 11,394 7,137 55,272 80,104 (d) Effect of a decrease in the mode After −3.250* −2.195 −2.732** −4.142** (1.290) (3.772) (0.959) (0.666) N 12,665 8,754 55,114 87,904 (e) Effect of a large donation – excluding other large donations After 2.541** 2.724** 3.051** 2.793** (0.348) (1.001) (0.278) (0.208) N 14,690 6,079 6,5386 9,6125 (f) Effect of a small donation – excluding other small donations After −2.050* −1.214 −0.610 −1.014* (1.124) (2.751) (0.830) (0.606) N 12,399 5,546 4,8705 7,2011 Twice mean Three times mean Five times mean Ten times mean (g) Effect of different‐sized large donations (five donations before/five after) After 11.154** 10.663** 17.396** 20.327** (1.043) (0.973) (1.825) (3.155) N 27,647 24,585 12,285 4,409 . One before/one after . One before/one after (same day) . Five before/five after . Five before/ten after . (a) Effect of a ‘large’ donation After 12.458** 13.392** 12.611** 12.134** (0.789) (2.609) (0.661) (0.496) N 15,508 6,464 68,926 102,492 (b) Effect of a ‘small’ donation After −11.411** −9.493** −11.169** −10.232** (0.911) (2.090) (0.770) (0.550) N 14,499 6,600 58,858 91,422 (c) Effect of an increase in the mode After −0.424 1.755 0.887 1.137* (1.211) (2.818) (0.961) (0.671) N 11,394 7,137 55,272 80,104 (d) Effect of a decrease in the mode After −3.250* −2.195 −2.732** −4.142** (1.290) (3.772) (0.959) (0.666) N 12,665 8,754 55,114 87,904 (e) Effect of a large donation – excluding other large donations After 2.541** 2.724** 3.051** 2.793** (0.348) (1.001) (0.278) (0.208) N 14,690 6,079 6,5386 9,6125 (f) Effect of a small donation – excluding other small donations After −2.050* −1.214 −0.610 −1.014* (1.124) (2.751) (0.830) (0.606) N 12,399 5,546 4,8705 7,2011 Twice mean Three times mean Five times mean Ten times mean (g) Effect of different‐sized large donations (five donations before/five after) After 11.154** 10.663** 17.396** 20.327** (1.043) (0.973) (1.825) (3.155) N 27,647 24,585 12,285 4,409 Notes Dependent variable = £ amount given. A large donation is twice the page mean and at least £50. A small donation is half the page mean. All regressions include fund‐raising page fixed effects. Columns (III) and (IV) in panels (a)–( f ) and all columns in panel (g) include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. *p < 0.10; **p < 0.05. Open in new tab Table 3 Effect of Large/Small Donation and Change In Mode . One before/one after . One before/one after (same day) . Five before/five after . Five before/ten after . (a) Effect of a ‘large’ donation After 12.458** 13.392** 12.611** 12.134** (0.789) (2.609) (0.661) (0.496) N 15,508 6,464 68,926 102,492 (b) Effect of a ‘small’ donation After −11.411** −9.493** −11.169** −10.232** (0.911) (2.090) (0.770) (0.550) N 14,499 6,600 58,858 91,422 (c) Effect of an increase in the mode After −0.424 1.755 0.887 1.137* (1.211) (2.818) (0.961) (0.671) N 11,394 7,137 55,272 80,104 (d) Effect of a decrease in the mode After −3.250* −2.195 −2.732** −4.142** (1.290) (3.772) (0.959) (0.666) N 12,665 8,754 55,114 87,904 (e) Effect of a large donation – excluding other large donations After 2.541** 2.724** 3.051** 2.793** (0.348) (1.001) (0.278) (0.208) N 14,690 6,079 6,5386 9,6125 (f) Effect of a small donation – excluding other small donations After −2.050* −1.214 −0.610 −1.014* (1.124) (2.751) (0.830) (0.606) N 12,399 5,546 4,8705 7,2011 Twice mean Three times mean Five times mean Ten times mean (g) Effect of different‐sized large donations (five donations before/five after) After 11.154** 10.663** 17.396** 20.327** (1.043) (0.973) (1.825) (3.155) N 27,647 24,585 12,285 4,409 . One before/one after . One before/one after (same day) . Five before/five after . Five before/ten after . (a) Effect of a ‘large’ donation After 12.458** 13.392** 12.611** 12.134** (0.789) (2.609) (0.661) (0.496) N 15,508 6,464 68,926 102,492 (b) Effect of a ‘small’ donation After −11.411** −9.493** −11.169** −10.232** (0.911) (2.090) (0.770) (0.550) N 14,499 6,600 58,858 91,422 (c) Effect of an increase in the mode After −0.424 1.755 0.887 1.137* (1.211) (2.818) (0.961) (0.671) N 11,394 7,137 55,272 80,104 (d) Effect of a decrease in the mode After −3.250* −2.195 −2.732** −4.142** (1.290) (3.772) (0.959) (0.666) N 12,665 8,754 55,114 87,904 (e) Effect of a large donation – excluding other large donations After 2.541** 2.724** 3.051** 2.793** (0.348) (1.001) (0.278) (0.208) N 14,690 6,079 6,5386 9,6125 (f) Effect of a small donation – excluding other small donations After −2.050* −1.214 −0.610 −1.014* (1.124) (2.751) (0.830) (0.606) N 12,399 5,546 4,8705 7,2011 Twice mean Three times mean Five times mean Ten times mean (g) Effect of different‐sized large donations (five donations before/five after) After 11.154** 10.663** 17.396** 20.327** (1.043) (0.973) (1.825) (3.155) N 27,647 24,585 12,285 4,409 Notes Dependent variable = £ amount given. A large donation is twice the page mean and at least £50. A small donation is half the page mean. All regressions include fund‐raising page fixed effects. Columns (III) and (IV) in panels (a)–( f ) and all columns in panel (g) include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. *p < 0.10; **p < 0.05. Open in new tab Our results confirm the own‐page crowd‐in effect. Our estimate is 5.91 (SE 3.11) which is significant at the 10% level. The estimated spillover effect is also positive (5.60), but insignificant (SE 1.80), suggesting that there is no crowd out of a large donation to one page on donations to other fund‐raising pages. 4. Econometric Analysis In this Section we present estimates from a linear‐in‐means model. The attraction of the mean is that it provides a simple summary statistic of the distribution of donations that donors appear to be responding to. We have shown in the previous Section that donors respond to large and small donations and to the mode. The linear‐in‐means model provides a parsimonious specification to capture these behaviours, particularly when we come to test for heterogeneity of effects in the next Section. We estimate the following specification: din=α+γd¯i,n−1+zin′δ+uin, where din refers to the nth donation to fund‐raising page i and d¯i,n−1 is the mean of all donations made online to the fund‐raising page up to the point at which the nth donor arrives at the page.10 As before, zin is a set of indicators for the order in which the donation occurs on the page and date controls, including indicators for the days since the page was set‐up (capped at 100) and also for the days in the immediate run up to the day of the marathon. We are interested in the coefficient γ which measures the extent to which a higher level of past donations across the page is associated with people giving more or less. The OLS estimate of γ is likely to be biased upwards by unobservable factors that affect all donations to a page that can be captured in a page‐specific error term, i.e. uin = ηi + vin These factors will include both shared (unobserved) characteristics of the donors to a page, such as their income, as well as (unobserved) characteristics of the fund‐raiser, such as their persuasive power or their personal connection to a particular cause.11 Because of the latter factor, we cannot identify the effect of past donations from within‐donor variation across pages but only from variation within pages over time. Estimating a fixed‐effects model using a within‐groups specification, however, will lead to a downwards‐biased estimate of γ because the mean‐differenced error term, uin−(1/N−1)∑j=2Nuij ⁠, will be negatively correlated with the mean‐differenced lagged dependent variable, d¯i,n−1−(1/N−1)∑j=1N−1d¯ij ⁠. In the case of estimating the effect of the past mean of all donations, this bias will not be negligible even though we have a long panel (the average number of donations per page in our analysis is 37 and we observe many pages with 50 or more donations), unlike the standard case of ‘Nickell bias’ (Nickell, 1981). We show this formally in online Appendix C. Our preferred approach, therefore, is to estimate γ using the Arellano and Bond (1991) GMM estimator.12 First, the page‐specific effect ηi is eliminated by first differencing: Δdin=γΔd¯i,n−1+Δzin′δ+Δvin. In this first‐differenced model there is, however, an endogeneity problem due to the correlation between d¯i,n−1and vi,n−1. As shown in online Appendix C, the bias of the OLS estimator in this first‐differenced model does not decrease with N. In our main specification we use the two‐period lag and the three‐period lag of the page mean as instruments for the (change in) mean of past donations, with different reduced form coefficients per donation order. The Arellano–Bond test for serial correlation does not reject the null of no second‐order serial correlation, implying that the two‐period lag is valid as an instrument. The Hansen test does not indicate that the instrument set is not valid. Our main results are presented in Table 4. For comparison, we show both the upwards‐biased OLS and the downwards‐biased fixed‐effects results for all specifications. Our preferred GMM results lie between these two for all specifications. We also present results for the effect of the last donation and the effect of the mean of the past five and ten donations. As demonstrated in online Appendix C, the extent of downwards bias to the fixed‐effects estimator is greater when looking at the past mean of all donations to a page than for the simple lagged dependent variable. In our main specification, the GMM estimate of γ is positive and significant, implying positive peer effects. This finding is robust to a number of robustness checks, presented in Appendix A. In our main specification we drop the first three observations from each page – we also show results dropping no and five observations. We also vary the instrument set, using different lags as instruments for the past mean. The estimated coefficient indicates that a £10 increase in the mean of past donations leads to people giving £2.50 more on average. To illustrate the magnitude of this, the effect of a £150 donation following three donations of £20 would be to increase giving by £8.13, whereas the effect of a £150 donation following six donations of £20 would be to increase giving by £4.64 (in both cases, the effect on giving in the case of the lagged dependent variable would be to increase giving by £2.86). This highlights an important feature of estimating the effect of the past mean – that the effect of a single donation diminishes, the later it occurs on a page. This is intuitively plausible as a donor may give less weight to a single large donation if there are more other donations on the page. We also find further empirical support for this finding by repeating the analysis from the previous Section and looking at the effect of a single ‘large’ donation made after 10 donations and after 15 donations to a page (compared with a large donation that occurs between five and ten donations). The estimated effect of a large donation is reduced by £1.19 when it occurs after 10 or more donations and by £2.50 when it occurs after 15 or more donations. This lends further support to including the past mean of all donations as the preferred empirical specification and we focus on this specification in the next Section. Table 4 Main Regression Results . (I) . (II) . (III) . OLS . Page fixed effects . Difference GMM . (a) Past_mean (£) 0.525** −0.359** 0.250** (0.013) (0.023) (0.028) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.322 Hansen test, p‐value (217 over‐ID restrictions) 0.864 (b) Mean, last ten (£) 0.458** −0.114** 0.202** (0.012) (0.012) (0.019) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.313 Hansen test, p‐value (217 over‐ID restrictions) 0.397 (c) Mean, last five (£) 0.361** −0.047** 0.116** (0.011) (0.007) (0.010) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.348 Hansen test, p‐value (217 over‐ID restrictions) 0.771 (d ) Past_donation (£) 0.125** 0.003 0.022** (0.005) (0.003) (0.001) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.051 Hansen test, p‐value (217 over‐ID restrictions) 0.630 . (I) . (II) . (III) . OLS . Page fixed effects . Difference GMM . (a) Past_mean (£) 0.525** −0.359** 0.250** (0.013) (0.023) (0.028) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.322 Hansen test, p‐value (217 over‐ID restrictions) 0.864 (b) Mean, last ten (£) 0.458** −0.114** 0.202** (0.012) (0.012) (0.019) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.313 Hansen test, p‐value (217 over‐ID restrictions) 0.397 (c) Mean, last five (£) 0.361** −0.047** 0.116** (0.011) (0.007) (0.010) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.348 Hansen test, p‐value (217 over‐ID restrictions) 0.771 (d ) Past_donation (£) 0.125** 0.003 0.022** (0.005) (0.003) (0.001) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.051 Hansen test, p‐value (217 over‐ID restrictions) 0.630 Notes Dependent variable: donation amount (£). Sample size: I = 10,597, NI = 364,286. Instruments are the second and third‐period lag of the (level) independent variable. All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. **p < 0.01. Open in new tab Table 4 Main Regression Results . (I) . (II) . (III) . OLS . Page fixed effects . Difference GMM . (a) Past_mean (£) 0.525** −0.359** 0.250** (0.013) (0.023) (0.028) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.322 Hansen test, p‐value (217 over‐ID restrictions) 0.864 (b) Mean, last ten (£) 0.458** −0.114** 0.202** (0.012) (0.012) (0.019) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.313 Hansen test, p‐value (217 over‐ID restrictions) 0.397 (c) Mean, last five (£) 0.361** −0.047** 0.116** (0.011) (0.007) (0.010) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.348 Hansen test, p‐value (217 over‐ID restrictions) 0.771 (d ) Past_donation (£) 0.125** 0.003 0.022** (0.005) (0.003) (0.001) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.051 Hansen test, p‐value (217 over‐ID restrictions) 0.630 . (I) . (II) . (III) . OLS . Page fixed effects . Difference GMM . (a) Past_mean (£) 0.525** −0.359** 0.250** (0.013) (0.023) (0.028) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.322 Hansen test, p‐value (217 over‐ID restrictions) 0.864 (b) Mean, last ten (£) 0.458** −0.114** 0.202** (0.012) (0.012) (0.019) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.313 Hansen test, p‐value (217 over‐ID restrictions) 0.397 (c) Mean, last five (£) 0.361** −0.047** 0.116** (0.011) (0.007) (0.010) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.348 Hansen test, p‐value (217 over‐ID restrictions) 0.771 (d ) Past_donation (£) 0.125** 0.003 0.022** (0.005) (0.003) (0.001) Arellano–Bond test for AR(1), p‐value 0.000 Arellano–Bond test for AR(2), p‐value 0.051 Hansen test, p‐value (217 over‐ID restrictions) 0.630 Notes Dependent variable: donation amount (£). Sample size: I = 10,597, NI = 364,286. Instruments are the second and third‐period lag of the (level) independent variable. All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. **p < 0.01. Open in new tab 5. Inside the Black Box – Exploring Why Peers Matter We would like to understand why peers matter. As discussed in Section 3, there are several potential explanations. On the basis of our findings so far, we can rule out that donors are (just) aiming to be the most generous donor to a page as both small donations and changes in the mode matter; large donations also have a wider effect than simply triggering similar‐sized donations. By analogous reasoning, we can also rule out that donors are (just) trying to avoid being the least generous donor to a page. The observed effects of large and small donations also imply that donors do not just try to follow the herd and match the mode. Table 2 summarised a number of potential explanations of why large/small donations and changes in the mode may affect all donations that follow. Our estimates of peer effects are positive, ruling out classic crowd out. Andreoni (1998) discusses the case in which threshold contribution levels, such as a minimum level of funding required before the public good can be produced, can result in crowd in – essentially large donations make it more likely that the threshold will be reached, which can encourage other donations. The potential effects of thresholds are relevant to the London marathon fund‐raising pages, the majority of which have fund‐raising targets. We find that peer effects are stronger for pages with a target but we also find a positive and significant effect for pages without a target (Table 5, column (I)). This indicates that targets do not provide the full explanation of the observed peer effects. There are further interesting differences in behaviour around the target. Regression analysis, summarised in Table 6, columns (I) and (II) shows, first, that the size of the first donation to take the total over the target donation is significantly higher and second that donations are lower on average after the target than before. Assuming as before that there is some random variation in exactly when donors arrive at a page (and that they are equally likely to arrive before or after the target, within a narrow window), this could be interpreted as a negative effect of hitting the target on donations. One important caveat to this is that it is possible for fund‐raisers to change their target (e.g. to increase the target amount once it has been reached). We have no evidence on the extent to which this happens in practice. Finally, column (III) of Table 6 provides the results from a further GMM regression in which the past mean of donations is interacted with an indicator for the donor arriving after the target has been reached. This tests whether the crowd‐in effect of past donations is the same on either side of the target. We find that the coefficient on the interaction term is negative and similar in magnitude to the coefficient on the past mean implying that there is no crowd‐in effect of past donations once the target has been reached. Table 5 Testing for Heterogeneous Effects . (I) . (II) . (III) . (IV) . (V) . Past_mean (£) 0.104** 0.160** 0.098** 0.264** 0.214** (0.042) (0.041) (0.031) (0.032) (0.034) Past_mean × page with target 0.158** (0.050) Past_mean × medium charity −0.043 (0.057) Past_mean × large charity 0.078 (0.056) Past_mean × major charity 0.085 (0.054) Past_mean × charity age > 10 years 0.127** (0.047) Past_mean × charity age > 20 years 0.018 (0.047) Past_mean × overseas charity −0.079 (0.045) Past_mean × young donors 0.020 (0.046) Number of observation = NI 364,286 183,619 280,660 260,362 364,286 Number of pages = I 10,597 5,248 8,208 8,194 10,597 . (I) . (II) . (III) . (IV) . (V) . Past_mean (£) 0.104** 0.160** 0.098** 0.264** 0.214** (0.042) (0.041) (0.031) (0.032) (0.034) Past_mean × page with target 0.158** (0.050) Past_mean × medium charity −0.043 (0.057) Past_mean × large charity 0.078 (0.056) Past_mean × major charity 0.085 (0.054) Past_mean × charity age > 10 years 0.127** (0.047) Past_mean × charity age > 20 years 0.018 (0.047) Past_mean × overseas charity −0.079 (0.045) Past_mean × young donors 0.020 (0.046) Number of observation = NI 364,286 183,619 280,660 260,362 364,286 Number of pages = I 10,597 5,248 8,208 8,194 10,597 Notes Difference GMM: Dependent variable = donation amount (£). All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. Instruments are the two‐period and three‐period lag of the past mean. **p < 0.01. Medium, large and major charities have incomes of £1 m–£5 m, £5 m–£50 m and £50 m+ respectively. Young donors are defined by the fund‐raiser being <40. Open in new tab Table 5 Testing for Heterogeneous Effects . (I) . (II) . (III) . (IV) . (V) . Past_mean (£) 0.104** 0.160** 0.098** 0.264** 0.214** (0.042) (0.041) (0.031) (0.032) (0.034) Past_mean × page with target 0.158** (0.050) Past_mean × medium charity −0.043 (0.057) Past_mean × large charity 0.078 (0.056) Past_mean × major charity 0.085 (0.054) Past_mean × charity age > 10 years 0.127** (0.047) Past_mean × charity age > 20 years 0.018 (0.047) Past_mean × overseas charity −0.079 (0.045) Past_mean × young donors 0.020 (0.046) Number of observation = NI 364,286 183,619 280,660 260,362 364,286 Number of pages = I 10,597 5,248 8,208 8,194 10,597 . (I) . (II) . (III) . (IV) . (V) . Past_mean (£) 0.104** 0.160** 0.098** 0.264** 0.214** (0.042) (0.041) (0.031) (0.032) (0.034) Past_mean × page with target 0.158** (0.050) Past_mean × medium charity −0.043 (0.057) Past_mean × large charity 0.078 (0.056) Past_mean × major charity 0.085 (0.054) Past_mean × charity age > 10 years 0.127** (0.047) Past_mean × charity age > 20 years 0.018 (0.047) Past_mean × overseas charity −0.079 (0.045) Past_mean × young donors 0.020 (0.046) Number of observation = NI 364,286 183,619 280,660 260,362 364,286 Number of pages = I 10,597 5,248 8,208 8,194 10,597 Notes Difference GMM: Dependent variable = donation amount (£). All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. Instruments are the two‐period and three‐period lag of the past mean. **p < 0.01. Medium, large and major charities have incomes of £1 m–£5 m, £5 m–£50 m and £50 m+ respectively. Young donors are defined by the fund‐raiser being <40. Open in new tab Table 6 Donations Around the Target Amount . (I) . (II) . (III) . Fixed effects . Difference GMM . Difference GMM . Target donation 53.988** 47.506** 50.554** (3.957) (3.455) (1.490) Reached target −3.517** −2.563 3.588** (0.564) (1.482) (1.398) Past_mean (£) 0.262** 0.268** (0.040) (0.030) Past_mean × reached target −0.191** (0.030) Arellano–Bond test for AR(1), p‐value 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.940 0.943 Hansen test, p‐value (over‐ID restrictions) 0.669 0.898 (205) (395) Number of observation = NI 139,201 135,308 135,308 Number of pages = I 3,893 3,893 3,839 . (I) . (II) . (III) . Fixed effects . Difference GMM . Difference GMM . Target donation 53.988** 47.506** 50.554** (3.957) (3.455) (1.490) Reached target −3.517** −2.563 3.588** (0.564) (1.482) (1.398) Past_mean (£) 0.262** 0.268** (0.040) (0.030) Past_mean × reached target −0.191** (0.030) Arellano–Bond test for AR(1), p‐value 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.940 0.943 Hansen test, p‐value (over‐ID restrictions) 0.669 0.898 (205) (395) Number of observation = NI 139,201 135,308 135,308 Number of pages = I 3,893 3,893 3,839 Notes Dependent variable = donation amount (£). All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. Target donation is the first donation to take the total over the target amount. Reached target is an indicator variable if the total is greater than the target. Instruments are the two‐period and three‐period lag of the past mean. **p < 0.01. Open in new tab Table 6 Donations Around the Target Amount . (I) . (II) . (III) . Fixed effects . Difference GMM . Difference GMM . Target donation 53.988** 47.506** 50.554** (3.957) (3.455) (1.490) Reached target −3.517** −2.563 3.588** (0.564) (1.482) (1.398) Past_mean (£) 0.262** 0.268** (0.040) (0.030) Past_mean × reached target −0.191** (0.030) Arellano–Bond test for AR(1), p‐value 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.940 0.943 Hansen test, p‐value (over‐ID restrictions) 0.669 0.898 (205) (395) Number of observation = NI 139,201 135,308 135,308 Number of pages = I 3,893 3,893 3,839 . (I) . (II) . (III) . Fixed effects . Difference GMM . Difference GMM . Target donation 53.988** 47.506** 50.554** (3.957) (3.455) (1.490) Reached target −3.517** −2.563 3.588** (0.564) (1.482) (1.398) Past_mean (£) 0.262** 0.268** (0.040) (0.030) Past_mean × reached target −0.191** (0.030) Arellano–Bond test for AR(1), p‐value 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.940 0.943 Hansen test, p‐value (over‐ID restrictions) 0.669 0.898 (205) (395) Number of observation = NI 139,201 135,308 135,308 Number of pages = I 3,893 3,893 3,839 Notes Dependent variable = donation amount (£). All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. Target donation is the first donation to take the total over the target amount. Reached target is an indicator variable if the total is greater than the target. Instruments are the two‐period and three‐period lag of the past mean. **p < 0.01. Open in new tab Another possibility is that donations may be important as a signal of the quality of the charity, with higher (lower) donations indicating that the particular cause is more (less) worthy of support (Vesterlund, 2003). To explore this empirically we adopt an idea, proposed by Heutel (2013), that the information content of past donations should be more important for smaller charities and for younger charities, for charities operating overseas whose activities are less easy to observe directly and for younger donors. To implement this we match data from the Charity Commission Register, comprising all registered charities in England and Wales. We are able to find a match in the case of 78% of fund‐raising pages (some of those we cannot match are Scottish and Irish charities), although information is not always available for all charities even where a match is made. Table 5 summarises the results from a set of regressions that include interaction terms, allowing the effect of the past mean to vary by, respectively – the size of the charity, the age of the charity, the location of charitable activity (UK or overseas) and the age of the fund‐raiser (which proxies for the age of donors, defined by a cut‐off of 40). The results provide little support for this particular signalling story. The effect of past donations is actually stronger for larger charities and for older charities, although the differences are not statistically significant. We find no difference in peer effects between overseas and UK‐based charities. We find no evidence of statistically significant differences by age.13 Instead of signalling charity quality, past donations may alternatively signal to donors how much it is socially appropriate for them to give. This is our preferred explanation of why past donations affect the amount given. When they arrive at a page, donors observe the distribution of past donations and use this to form – or update – their beliefs about how much they should give. These beliefs are likely to be donor (and possibly fund‐raiser) specific; donors will have some idea of where they should locate within the distribution depending on their characteristics relative to those of other donors, including the proximity of their relationship with the fund‐raiser, their support for a particular charity and their income (and possibly what their peers know about their income). Large/small donations and the mode will all affect amounts donated subsequently because they will be used to inform donors’ beliefs. We cannot test this benchmarking story explicitly but it is consistent with the observed pattern of behaviour, including both donor responses to past donations and the fact that, individually, past donations have less effect if they occur later on in the page. 6. Discussion This study adds to the empirical literature on what Andreoni has referred to as ‘the inherent sociality of giving’ by providing new evidence on the importance of peer effects in charitable giving in the context of online individual fund‐raising. Online fund‐raising is important to look at in its own right as a sizeable – and growing – channel for raising money for charities in the UK and elsewhere. It also provides an excellent setting to look at peer effects as it offers an environment in which donors observe donations from people within their naturally occurring peer groups (i.e. their friends, family and colleagues). There is an inevitable issue about the extent to which our findings can be generalised beyond this particular setting. The online fund‐raising context in which donors can see all other donations – and know that their donations will be seen – is arguably quite distinctive. However, it is one that is potentially relevant to practitioners and policy makers interested in whether they can exploit the power of peer effects by providing similar levels of publicity to donations in other settings. Furthermore, by looking at data that span more than 1,000 different charities, we have been able to demonstrate that peer effects are not limited to particular charities or groups of donors, suggesting that the effects are likely to be more broadly generalisable. The richness of the data also allows us to explore potential explanations of why peers matter. We can reject that donors systematically compete to be the top, or strive to avoid being the bottom or align themselves with the mode or median. Our preferred explanation, which is consistent with the empirical findings, is that donors give what they think that they personally are expected to give where the distribution of the donations of their peers (along with other factors, such as income and specific cause) feed into the formation of that expectation. In this study we have analysed only a small sub‐sample of the population of online fund‐raising pages that are potentially available. Going forward, information from online fund‐raising pages, particularly matched with social network data, has the potential to yield even further insights into how donors behave in social settings. Appendix A. Robustness Checks . (I) . (II) . (III) . (IV) . (V) . (VI) . (VII) . (VIII) . . Excluding first 3 observations . Excluding no observation . Excluding first 5 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Past_mean (£) 0.250** 0.143** 0.358** 0.283** 0.188** 0.151** 0.216** 0.256** (0.028) (0.015) (0.043) (0.078) (0.031) (0.049) (0.025) (0.039) Instruments d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3d¯i,n−4 d¯i,n−2,d¯i,n−3 Collapsed Collapsed One‐step Arellano–Bond test for AR(1), p‐value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.322 0.275 0.539 0.325 0.327 0.332 0.325 0.326 Hansen test, p‐value (over‐ID restrictions) 0.864 0.768 0.865 0.021 0.811 0.209 0.547 0.864 (217) (217) (217) (1) (217) (1) (323) (218) . (I) . (II) . (III) . (IV) . (V) . (VI) . (VII) . (VIII) . . Excluding first 3 observations . Excluding no observation . Excluding first 5 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Past_mean (£) 0.250** 0.143** 0.358** 0.283** 0.188** 0.151** 0.216** 0.256** (0.028) (0.015) (0.043) (0.078) (0.031) (0.049) (0.025) (0.039) Instruments d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3d¯i,n−4 d¯i,n−2,d¯i,n−3 Collapsed Collapsed One‐step Arellano–Bond test for AR(1), p‐value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.322 0.275 0.539 0.325 0.327 0.332 0.325 0.326 Hansen test, p‐value (over‐ID restrictions) 0.864 0.768 0.865 0.021 0.811 0.209 0.547 0.864 (217) (217) (217) (1) (217) (1) (323) (218) Notes Difference GMM: Dependent variable = donation amount (£). All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. **p < 0.01. Open in new tab . (I) . (II) . (III) . (IV) . (V) . (VI) . (VII) . (VIII) . . Excluding first 3 observations . Excluding no observation . Excluding first 5 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Past_mean (£) 0.250** 0.143** 0.358** 0.283** 0.188** 0.151** 0.216** 0.256** (0.028) (0.015) (0.043) (0.078) (0.031) (0.049) (0.025) (0.039) Instruments d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3d¯i,n−4 d¯i,n−2,d¯i,n−3 Collapsed Collapsed One‐step Arellano–Bond test for AR(1), p‐value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.322 0.275 0.539 0.325 0.327 0.332 0.325 0.326 Hansen test, p‐value (over‐ID restrictions) 0.864 0.768 0.865 0.021 0.811 0.209 0.547 0.864 (217) (217) (217) (1) (217) (1) (323) (218) . (I) . (II) . (III) . (IV) . (V) . (VI) . (VII) . (VIII) . . Excluding first 3 observations . Excluding no observation . Excluding first 5 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Excluding first 3 observations . Past_mean (£) 0.250** 0.143** 0.358** 0.283** 0.188** 0.151** 0.216** 0.256** (0.028) (0.015) (0.043) (0.078) (0.031) (0.049) (0.025) (0.039) Instruments d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3 d¯i,n−2,d¯i,n−3d¯i,n−4 d¯i,n−2,d¯i,n−3 Collapsed Collapsed One‐step Arellano–Bond test for AR(1), p‐value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Arellano–Bond test for AR(2), p‐value 0.322 0.275 0.539 0.325 0.327 0.332 0.325 0.326 Hansen test, p‐value (over‐ID restrictions) 0.864 0.768 0.865 0.021 0.811 0.209 0.547 0.864 (217) (217) (217) (1) (217) (1) (323) (218) Notes Difference GMM: Dependent variable = donation amount (£). All regressions include additional controls for place within page (linear trend), indicators for days since page was set‐up (capped at 100) and indicator variables for two days and one day before the marathon, the day of the marathon and (any) days after the marathon. **p < 0.01. Open in new tab Footnotes 1 " For comparison, total donations from individuals in the UK were estimated to be £13 billion in 2010–11. 2 " We do not have direct information on the identity of the donors or their relationship with the fund‐raiser. However, we have supporting evidence that they are mainly friends, family and colleagues from a separate survey of approximately 19,000 Justgiving donors (Payne et al., 2011). Of those who had been asked to give to a fund‐raising page, 84% had been asked by a family member (of whom 87% said that they always gave when asked); 96% had been asked by a friend (67% always gave); 89% had been asked by a colleague (48% always gave); 70% had been asked by a charity representative (only 9% always gave). 3 " Donors can see up to 30 or 50 past donations by scrolling down without having to click through. As the median number of donations is 33, this means that most donors can see all previous donations in one go. 4 " Donors can choose to donate anonymously. Unfortunately, whether or not a donation was given anonymously was miscoded for more than half our sample, which means that we cannot do a full analysis on the effects of anonymity. Where we do have information, we find that 11% of donations are made anonymously. Large and small donations are more likely to be made anonymously as might be expected. We find that the effect of large and small donations is not affected by whether or not the donation was made anonymously. We also find that the probability of giving anonymously does not change after a large or small donation. 5 " Mas and Moretti (2009) provide perhaps the closest study to our study in terms of identification. They look at the effect of peers’ productivity in the context of supermarket checkouts, exploiting randomness arising from the scheduling of checkout operatives. They estimate individual‐specific fixed effects; we do not have sufficient observations to allow us to do this. 6 " These totals exclude the value of UK Gift Aid tax relief, which is additionally passed to the charity by the tax authorities. 7 " The biggest individual fund‐raisers include Richard Branson who raised more than £35,000 for Virgin Unite, including a single donation of £6,550, and popstar Natalie Imbruglia, also running for Virgin Unite who raised more than £32,000, including a single donation of £10,000. 8 " Where there is more than one mode, we look at increases in the maximum of the modes and decreases in the minimum of the modes. 9 " We drop 4% of donations which were made on the same day as we cannot identify donation order. 10 " The donor will also see the amount raised offline up to the point at which they arrive at the website, whereas we only know the total amount raised offline at the time the data were captured. 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( 2011 ) ‘Peer effects in education: How might they work, how big are they and how much do we know thus far?’ in ( E. Hanushek, S. Machin and L. Woessmann, eds.), Handbook of the Economics of Education , vol. 3, pp. 249 – 77 , Amsterdam : Elsevier Science . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Shang , J. and Croson , R. ( 2009 ) ‘Field experiments in charitable contribution: the impact of social influence on the voluntary provision of public goods’, Economic Journal , vol. 119 ( 54 ), pp. 1422 – 39 . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Vesterlund , L. ( 2003 ). ‘ The informational value of sequential fundraising ’, Journal of Public Economics , vol. 87 ( 3–4 ), pp. 627 – 58 . Google Scholar Crossref Search ADS WorldCat Warr , P. ( 1982 ) ‘ Pareto optimal redistribution and private charity ’, Journal of Public Economics , vol. 19 ( 1 ), pp. 131 – 8 . Google Scholar Crossref Search ADS WorldCat Author notes " This research was funded by the Leverhulme Foundation (Grant F/00 182/A2) and the Economic and Social Research Council (Grant RES‐343‐28‐0001). Thanks to Justgiving for providing us with their data and to two anonymous referees, Rachel Griffith, Clare Leaver, Abigail Payne and Kimberley Scharf and participants at seminars in Essex, Mannheim, Rotterdam, Southampton, Stirling, the workshop on the social dimension of organisations at the CUI in Budapest and the CEPR conference on charitable giving and altruism, Paris, for helpful comments and discussions. Any views – and in particular any errors – are our own. © 2013 The Author(s). The Economic Journal Published by John Wiley & Sons Ltd on behalf of the Royal Economic Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Aikman,, David;Haldane, Andrew, G.;Nelson, Benjamin, D.

doi: 10.1111/ecoj.12113pmid: N/A

Abstract Credit cycles have been a characteristic of advanced economies for over 100 years. On average, a sustained pick‐up in the ratio of credit to GDP has been highly correlated with banking crises. The boom phases of the cycle are characterised by large deviations in credit from trend. A range of mechanisms can generate these effects, each of which has strategic complementarity between banks at its core. Macro‐prudential policy could curb these credit cycles, both through raising the cost of maintaining risky portfolios and through an expectations channel that operates via banks' perceptions of other banks' actions. Credit lies at the heart of crises. Credit booms sow the seeds of subsequent credit crunches. This is a key lesson of past financial crashes, manias and panics (Kindleberger, 1978; Minsky, 1986; Reinhart and Rogoff, 2009). It was a lesson re‐taught painfully to policy makers during the most recent financial crisis. This time's credit cycle has been particularly severe and synchronous. In 2006, private sector credit across the UK, US and euro area rose by around 10%. During 2009, private credit in these countries fell by around 2%.1 The knock‐on consequences for real growth were severe and synchronous equally. Peak‐to‐trough, G7 real output fell by 3.6% during the Great Recession. In response, there have been widespread calls for remedial policy action. These proposals come in various stripes. Some have proposed a more active role for monetary policy in addressing financial imbalances (White, 2009; Taylor, 2010). Others have suggested using new macro‐prudential tools to rein in credit excesses (Borio and Lowe, 2002; Bank of England, 2009, 2011, 2013; Kashyap et al., 2011). Others still have proposed a radical root‐and‐branch reform of the structure of banking (Kay, 2010; Kotlikoff, 2010). Evaluating the merits of these proposals requires a conceptual understanding of the causes of the credit cycle and an empirical quantification of its dynamic behaviour. What is the underlying friction generating credit booms and busts? Are credit cycles distinct from cycles in the real economy? And how have they evolved, both over time and across countries? Answers to these questions should help frame public policy choices for curbing the credit cycle. Our first contribution is to present some empirical evidence on the credit cycle. Across countries and across a sweep of history, credit cycles (measured by variation in the ratio of bank lending to GDP) are both clearly identifiable and regular. Typically, they presage banking crises. Importantly, however, credit cycles are distinct from the business cycle in their frequency and amplitude. It is fluctuations in credit to output operating over the medium term – that is, beyond business cycle frequency, with peak‐to‐trough cycles completed over the course of a decade or more – that, over the last century, have been so damaging. There is also evidence of increased synchronisation of credit cycles across countries over the second half of the last century. Second, we present a simple theoretical framework within which to understand how large fluctuations in credit relative to real activity can occur. We show that a number of frictions studied in the literature can be understood to generate strategic complementarity between banks – including systemic risk shifting, reputational concerns and so‐called ‘moral hazard’. Each of these generates a tendency for banks collectively to take on more risk. We use global games techniques to pin down unique equilibria in these games, which are characterised by threshold strategies. These threshold effects open the door for small changes in fundamentals to generate large swings in aggregate credit. We argue that this is a central feature of the credit cycles that have operated for over a century across the industrialised world. Drawing on this, we offer some implications for the design of public policy. In particular, new policy apparatus may be needed that targets bank balance sheets directly but, unlike micro‐prudential policy, does so by looking across the system systematically. This is one key dimension of so‐called macro‐prudential policy.2 Various international macro‐prudential policy committees are now in place – in the US the Financial Stability Oversight Committee, in the euro area the European Systemic Risk Board and in the UK the Financial Policy Committee. These provide one element of a macro‐prudential policy framework. Other elements remain to be put in place. Knowledge of the sources and dynamics of the credit cycle will be important in assembling those missing pieces. This study is intended to be a contribution towards that goal. 0.1 Related Literature The idea that credit booms sow the seeds of subsequent credit crunches goes back many centuries. Kindleberger (1978) and Minsky (1986) were both attempts to make sense of these regular patterns in the relationship between credit growth and crises, drawing on detailed case studies of past financial crises. In documenting some of the empirical facts about financial cycles, our work is closely related to a number of more recent studies examining the properties of credit and leverage over a relatively large sweep of history. We use data collected by Schularick and Taylor (2012), who study credit booms and leverage cycles over 1870–2008. We document the presence of a medium‐term credit cycle in these data and, like Schularick and Taylor, relate the large swings in credit observed at this frequency to banking crises. The properties of financial cycles have also been studied by Aikman et al. (2010), Claessens et al. (2011a,b) and Drehmann et al. (2012). Claessens et al. (2011a) analyse cycles in credit, housing and equity prices over the period 1960–2007, emphasising that financial cycles tend to be long and severe and, like us, Drehmann et al. (2012) emphasise the importance of lower frequency financial cycles. Comin and Gertler (2006) study lower frequency cycles but limit their attention to explaining fluctuations in real, rather than financial activity. In related work, Bordo et al. (2001) examine whether financial crises have become more frequent and severe over the last century. Jorda et al. (2011a, b) extend Schularick and Taylor (2012) by documenting that more credit‐intensive booms tend to be followed by deeper recessions and slower recoveries. This complements recent work by Claessens et al. (2008, 2011,b) who find that recessions associated with credit crunches and financial disruption tend to be deeper and longer than other recessions. Adrian et al. (2010) explore the links between monetary, financial and business cycles by linking changes in monetary policy to the profitability and risk‐taking capacity of financial intermediaries, which shifts credit supply. The simple theoretical framework we develop to understand the empirics of credit and crises uses global games techniques pioneered in Carlsson and van Damme (1993), as developed in Morris and Shin (2003).3 We use global games methods to shed light on a range of theoretical mechanisms that we identify as giving rise to strategic complementarities in risk taking between financial intermediaries. Sources of such complementarities include: government bailout policies, as studied in Acharya and Yorulmazer (2007) and Farhi and Tirole (2012); concerns for reputation, as in Rajan (1994) and Aikman et al. (2012) and what we identify as ‘systemic’ risk shifting, which has some similarities to Allen and Gale (2000) but which is also related to Merton (1977), Stiglitz and Weiss (1981) and Keeley (1990), among others. Other strands of the literature in macroeconomics seek to explain swings in leverage and credit by appeal to the ‘financial accelerator’ mechanisms of Bernanke et al. (1999) or Kiyotaki and Moore (1997), among others. Recent work on this front has put the balance sheets of financial intermediaries at centre stage in explaining movements in credit, leverage and asset prices (Geanakoplos, 2010; Gertler and Kiyotaki, 2011; Adrian and Shin, 2013; He and Krishnamurthy, 2013). Our focus is on particular sources of strategic complementarity operating at the micro level, although clearly there is scope for strategic complementarities and financial accelerator mechanisms to interact in general equilibrium settings. In contrast to the studies mentioned above, we examine how a policy tool that affects system‐wide bank capitalisation can alter the likelihood that intermediaries coordinate on equilibria with excessive risk taking. We also discuss how macro‐prudential policy differs from micro‐prudential approaches to bank supervision as a tool for enhancing macroeconomic stability. For related discussions of policy implications, see inter alia Bank of England (2009, 2011, 2013) and Dell'Ariccia et al. (2012). 1. Credit Cycles and Business Cycles, 1880–2008 Irving Fisher noted almost 80 years ago that: [t]he old and apparently still persistent notion of ‘the’ business cycle, as a single, simple, self‐generating cycle … is a myth. Instead of one cycle, there are many co‐existing cycles, constantly aggravating or neutralising each other, as well as co‐existing with many non‐cyclical forces (Fisher, 1933, p. 338). Historically, the credit cycle appears to have been just such a phenomenon. To investigate, we draw on a data set recently collected by Schularick and Taylor (2012).4 This covers a lengthy time series (often more than a century of data) across 14 developed countries5 and includes data on bank loans and bank assets, together with GDP and money aggregates.6 We combine these data with those of Bordo et al. (2001), which provide dates for banking crises and other systemic risk episodes. 1.1. Descriptive Statistics We begin with a simple description of the data. Tables Table Tabel 1 and Table Tabel 2 present some summary statistics (mean and standard deviation) of real GDP growth, real bank loan growth, real bank asset growth and real money aggregates in the full sample of countries since 1880. Table 2 focuses on the UK and the US and splits the sample into four periods: 1880–1913; 1914–45; 1945–79 and 1980–2008. Figure 1 shows the volatility of real bank loan growth and real bank asset growth relative to real GDP growth. Figure 2 plots bank loan and asset‐to‐GDP ratios for all countries over the sample. Several features are clear: (i) For the UK and the US, average real GDP growth is little changed either side of the wars (Table 2). But for the UK, real credit has grown around twice as quickly since 1945 relative to pre‐First World War. In consequence, loan‐to‐GDP ratios trend upwards in the post‐war era, consistent with financial liberalisation and deepening – a pattern observed across all countries in the sample (Figure 2). (ii) The same relative pattern is evident in the volatilities of output and credit. The variability in real GDP growth has fallen in both countries since 1945 relative to its pre‐First World War level. By contrast, the standard deviation of real credit growth has either increased – in the case of the UK – or stayed broadly unchanged – in the case of the US. Another notable feature of these data is a reduction in the volatility of bank credit and assets in both countries between 1980 and 2008. Table Tabel 1. Summary Statistics, Real Growth Rates (Means and Standard Deviations), 1880–2008 (%) Country . GDP . Loans . Assets . Narrow money . Broad money . AUS 3.48 4.18 4.48 3.62 4.22 5.52 7.48 6.28 9.02 5.53 CAN 3.51 4.67 4.76 3.59 4.40 4.86 7.92 5.58 7.00 4.87 CHE 2.74 2.90 4.29 4.01 3.30 4.22 5.84 6.06 11.40 4.86 DEU 3.40 6.61 6.72 4.38 5.73 4.41 10.47 8.39 4.87 5.35 DNK 2.78 4.55 4.63 5.00 4.07 3.31 7.15 7.06 11.10 6.96 ESP 2.73 6.82 6.46 3.17 3.88 5.79 12.12 7.65 10.25 6.29 FRA 3.08 6.43 5.99 2.76 2.66 5.07 13.67 7.10 7.82 7.58 GBR 2.16 4.08 4.47 0.94 2.90 3.97 8.18 8.09 5.07 5.13 ITA 2.82 3.86 4.25 3.68 4.12 5.33 16.38 12.39 6.18 6.04 JPN 3.87 6.27 7.33 5.41 5.49 4.66 9.43 9.15 10.56 7.87 NLD 2.82 4.39 4.38 4.43 4.92 3.76 12.41 8.52 9.18 10.08 NOR 3.26 4.93 4.78 2.05 3.45 4.06 6.46 5.83 7.92 5.36 SWE 2.80 4.16 4.83 2.26 3.55 3.23 7.64 6.70 6.72 4.54 USA 3.34 3.96 4.65 3.58 3.81 5.18 5.75 4.66 8.37 4.96 Country . GDP . Loans . Assets . Narrow money . Broad money . AUS 3.48 4.18 4.48 3.62 4.22 5.52 7.48 6.28 9.02 5.53 CAN 3.51 4.67 4.76 3.59 4.40 4.86 7.92 5.58 7.00 4.87 CHE 2.74 2.90 4.29 4.01 3.30 4.22 5.84 6.06 11.40 4.86 DEU 3.40 6.61 6.72 4.38 5.73 4.41 10.47 8.39 4.87 5.35 DNK 2.78 4.55 4.63 5.00 4.07 3.31 7.15 7.06 11.10 6.96 ESP 2.73 6.82 6.46 3.17 3.88 5.79 12.12 7.65 10.25 6.29 FRA 3.08 6.43 5.99 2.76 2.66 5.07 13.67 7.10 7.82 7.58 GBR 2.16 4.08 4.47 0.94 2.90 3.97 8.18 8.09 5.07 5.13 ITA 2.82 3.86 4.25 3.68 4.12 5.33 16.38 12.39 6.18 6.04 JPN 3.87 6.27 7.33 5.41 5.49 4.66 9.43 9.15 10.56 7.87 NLD 2.82 4.39 4.38 4.43 4.92 3.76 12.41 8.52 9.18 10.08 NOR 3.26 4.93 4.78 2.05 3.45 4.06 6.46 5.83 7.92 5.36 SWE 2.80 4.16 4.83 2.26 3.55 3.23 7.64 6.70 6.72 4.54 USA 3.34 3.96 4.65 3.58 3.81 5.18 5.75 4.66 8.37 4.96 Notes All variables are real. Standard deviations in italics. Open in new tab Table Tabel 1. Summary Statistics, Real Growth Rates (Means and Standard Deviations), 1880–2008 (%) Country . GDP . Loans . Assets . Narrow money . Broad money . AUS 3.48 4.18 4.48 3.62 4.22 5.52 7.48 6.28 9.02 5.53 CAN 3.51 4.67 4.76 3.59 4.40 4.86 7.92 5.58 7.00 4.87 CHE 2.74 2.90 4.29 4.01 3.30 4.22 5.84 6.06 11.40 4.86 DEU 3.40 6.61 6.72 4.38 5.73 4.41 10.47 8.39 4.87 5.35 DNK 2.78 4.55 4.63 5.00 4.07 3.31 7.15 7.06 11.10 6.96 ESP 2.73 6.82 6.46 3.17 3.88 5.79 12.12 7.65 10.25 6.29 FRA 3.08 6.43 5.99 2.76 2.66 5.07 13.67 7.10 7.82 7.58 GBR 2.16 4.08 4.47 0.94 2.90 3.97 8.18 8.09 5.07 5.13 ITA 2.82 3.86 4.25 3.68 4.12 5.33 16.38 12.39 6.18 6.04 JPN 3.87 6.27 7.33 5.41 5.49 4.66 9.43 9.15 10.56 7.87 NLD 2.82 4.39 4.38 4.43 4.92 3.76 12.41 8.52 9.18 10.08 NOR 3.26 4.93 4.78 2.05 3.45 4.06 6.46 5.83 7.92 5.36 SWE 2.80 4.16 4.83 2.26 3.55 3.23 7.64 6.70 6.72 4.54 USA 3.34 3.96 4.65 3.58 3.81 5.18 5.75 4.66 8.37 4.96 Country . GDP . Loans . Assets . Narrow money . Broad money . AUS 3.48 4.18 4.48 3.62 4.22 5.52 7.48 6.28 9.02 5.53 CAN 3.51 4.67 4.76 3.59 4.40 4.86 7.92 5.58 7.00 4.87 CHE 2.74 2.90 4.29 4.01 3.30 4.22 5.84 6.06 11.40 4.86 DEU 3.40 6.61 6.72 4.38 5.73 4.41 10.47 8.39 4.87 5.35 DNK 2.78 4.55 4.63 5.00 4.07 3.31 7.15 7.06 11.10 6.96 ESP 2.73 6.82 6.46 3.17 3.88 5.79 12.12 7.65 10.25 6.29 FRA 3.08 6.43 5.99 2.76 2.66 5.07 13.67 7.10 7.82 7.58 GBR 2.16 4.08 4.47 0.94 2.90 3.97 8.18 8.09 5.07 5.13 ITA 2.82 3.86 4.25 3.68 4.12 5.33 16.38 12.39 6.18 6.04 JPN 3.87 6.27 7.33 5.41 5.49 4.66 9.43 9.15 10.56 7.87 NLD 2.82 4.39 4.38 4.43 4.92 3.76 12.41 8.52 9.18 10.08 NOR 3.26 4.93 4.78 2.05 3.45 4.06 6.46 5.83 7.92 5.36 SWE 2.80 4.16 4.83 2.26 3.55 3.23 7.64 6.70 6.72 4.54 USA 3.34 3.96 4.65 3.58 3.81 5.18 5.75 4.66 8.37 4.96 Notes All variables are real. Standard deviations in italics. Open in new tab Table Tabel 2. Summary Statistics, Real Growth Rates (Means and Standard Deviations), UK and US (%) . . GDP . Loans . Assets . UK 1880–1913 2.01 2.89 3.20 4.46 4.32 4.02 1914–45 0.82 1.03 1.99 6.01 12.49 6.63 1945–79 2.98 5.07 4.91 1.82 8.63 12.27 1980–2008 2.68 7.02 7.73 2.11 4.52 6.85 US 1880–1913 3.97 5.85 6.67 4.57 4.16 5.22 1914–45 3.78 0.23 2.85 8.24 6.50 4.92 1945–79 3.05 6.41 4.70 3.93 5.72 3.18 1980–2008 2.25 3.69 3.26 2.09 3.92 3.58 . . GDP . Loans . Assets . UK 1880–1913 2.01 2.89 3.20 4.46 4.32 4.02 1914–45 0.82 1.03 1.99 6.01 12.49 6.63 1945–79 2.98 5.07 4.91 1.82 8.63 12.27 1980–2008 2.68 7.02 7.73 2.11 4.52 6.85 US 1880–1913 3.97 5.85 6.67 4.57 4.16 5.22 1914–45 3.78 0.23 2.85 8.24 6.50 4.92 1945–79 3.05 6.41 4.70 3.93 5.72 3.18 1980–2008 2.25 3.69 3.26 2.09 3.92 3.58 Note Standard deviations appear in italics. Open in new tab Table Tabel 2. Summary Statistics, Real Growth Rates (Means and Standard Deviations), UK and US (%) . . GDP . Loans . Assets . UK 1880–1913 2.01 2.89 3.20 4.46 4.32 4.02 1914–45 0.82 1.03 1.99 6.01 12.49 6.63 1945–79 2.98 5.07 4.91 1.82 8.63 12.27 1980–2008 2.68 7.02 7.73 2.11 4.52 6.85 US 1880–1913 3.97 5.85 6.67 4.57 4.16 5.22 1914–45 3.78 0.23 2.85 8.24 6.50 4.92 1945–79 3.05 6.41 4.70 3.93 5.72 3.18 1980–2008 2.25 3.69 3.26 2.09 3.92 3.58 . . GDP . Loans . Assets . UK 1880–1913 2.01 2.89 3.20 4.46 4.32 4.02 1914–45 0.82 1.03 1.99 6.01 12.49 6.63 1945–79 2.98 5.07 4.91 1.82 8.63 12.27 1980–2008 2.68 7.02 7.73 2.11 4.52 6.85 US 1880–1913 3.97 5.85 6.67 4.57 4.16 5.22 1914–45 3.78 0.23 2.85 8.24 6.50 4.92 1945–79 3.05 6.41 4.70 3.93 5.72 3.18 1980–2008 2.25 3.69 3.26 2.09 3.92 3.58 Note Standard deviations appear in italics. Open in new tab Fig. 1. Open in new tabDownload slide Volatility of Real Bank Loan Growth and Real Bank Asset Growth Relative to Real GDP Growth, 1880–2008, All Countries Fig. 1. Open in new tabDownload slide Volatility of Real Bank Loan Growth and Real Bank Asset Growth Relative to Real GDP Growth, 1880–2008, All Countries (iii) The standard deviation of real credit growth in the UK was 4.7 times that of real activity during 1945–79 and 2.1 times that of real activity during 1980–2008. From Figure 1, across all countries and all eras, real credit growth was between 1.5–3 times as volatile as growth in real activity. (iv) Growth in real bank assets has tended to outstrip growth in broad money across most countries, on average, across the whole sample. We turn next to a more detailed exploration of the cyclical components of the data. 1.2. Cycles Figure 3 plots an estimate of the spectral density of real loan growth for the UK, along with approximate 90% confidence bands.7 The peak in the density at a (normalised) frequency of around 0.15 suggests a cycle with duration of around 13 years – a medium‐term cycle. The smaller peak at around 0.45 corresponds to a business cycle frequency of around 4.5 years. The relative size of these two peaks suggests that medium‐term fluctuations are an important source of overall variation in real loan growth. Fig. 2. Open in new tabDownload slide Ratio of Loans to GDP and Assets to GDP Fig. 2. Open in new tabDownload slide Ratio of Loans to GDP and Assets to GDP Fig. 3. Open in new tabDownload slide Estimated Spectral Density for UK Real Loan Growth, 1880–2008 Notes. Approximate 90% confidence bands shown, using the χ2 approximation discussed in Preistley (1999). A Parzen window with M = 35 was used to smooth the periodogram. Fig. 3. Open in new tabDownload slide Estimated Spectral Density for UK Real Loan Growth, 1880–2008 Notes. Approximate 90% confidence bands shown, using the χ2 approximation discussed in Preistley (1999). A Parzen window with M = 35 was used to smooth the periodogram. The limited sample size means that the confidence bands around our estimated density are large. Nonetheless, the spectral density is suggestive of the empirical relevance of medium‐term variations in credit. To focus on variation within a particular frequency range, we apply a band‐pass filter to the data.8 This isolates the component of a series operating in a frequency range specified by the user.9 Figure 4 panels (a) and (b) show cyclical components of fluctuations in real loan growth and real GDP growth in the UK and US. The Figures plot the medium‐term component of loans and output (solid line), which, as the estimated density is subject to uncertainty, we take to comprise fluctuations within the 8 to 20‐year range, together with all the components in the 2 to 20‐year range (dashed line). The difference between the solid and dashed series is then the business cycle component. Consistent with the estimated density, shorter term, business cycle frequency fluctuations between 2 and 8 years typically do not account for much of the overall variation in credit in the 2 to 20‐year range.10 We estimate the standard deviation of the 8 to 20‐year cycle to be around 8%, with a 95% confidence interval of [6.6%, 9.8%].11 In short, the credit cycle appears to be a well‐defined empirical regularity. It also appears to have been operating for well over a century. And its frequency suggests that factors other than the business cycle may be responsible for driving it. Fig. 4. Open in new tabDownload slide Medium‐term Cycle in Real GDP and Real Bank Loans, 1880–2008. (a) UK, (b) US Note. Vertical lines denote banking crises. Fig. 4. Open in new tabDownload slide Medium‐term Cycle in Real GDP and Real Bank Loans, 1880–2008. (a) UK, (b) US Note. Vertical lines denote banking crises. Medium‐term fluctuations in credit appear distinct from fluctuations in GDP at both business cycle and medium‐term cycle frequencies. To show this, Figure 4(a) and (b) also plot both business cycles and medium‐term cycles in real GDP for the UK and US. The amplitude of the credit cycle is roughly four times that of fluctuations in GDP over the medium term. And it is roughly five times that of fluctuations in GDP at conventional business cycle frequencies. The peak‐to‐trough variation in the typical credit cycle has been around 40 percentage points in the UK. For real GDP, it has been around 10 percentage points. As a result, ratios of credit to GDP themselves exhibit a distinct cyclical pattern in the UK and US, as illustrated in Figure 5 panels (a) and (b). One striking implication of these cycles is that small changes in real activity (output) have historically been accompanied by large changes in credit quantities. The ‘great moderation’ is a recent example of a general historical phenomenon that seems to have involved decoupling of financial and real activity over the medium term. Claessens et al. (2011a) and Drehmann et al. (2012) make similar observations for the post‐war era. Figure 5(c) shows results for the full set of countries, applying the same techniques to the ratio of credit to GDP. Although the cycles are sometimes not as regular, the same general patterns in output and credit are present using the wider panel of countries. 1.3. Cycles and Crises What are the economic consequences of such large fluctuations in credit? Are credit booms and busts systematically related to the incidence of crises, with their associated social costs? Using the crisis dates from Bordo et al. (2001) and Schularick and Taylor (2012), it is possible to test this hypothesis. Bordo et al. take banking crises to be episodes where we ‘observe financial distress resulting in the erosion of most of all of aggregate banking system capital’, as in Caprio and Klingebiel (1999).12 Over the sample period, the unconditional probability of such a banking crisis starting in any given year was around 4%. Table 3 reports the results from logit regressions relating the probability of being in a state of banking crisis to lagged growth in the ratio of bank loans to GDP. Given the persistence of the estimated credit cycle, we follow Schularick and Taylor by including five lags of the growth in the ratio of bank credit to GDP. We also include time‐invariant country‐specific effects.13 Column (1) in Table 3 reports the results for the whole sample across all 14 countries.14 The coefficient estimates suggest a strongly significant relationship between the growth in the credit‐to‐GDP ratio and banking distress; a joint test of the lagged coefficients strongly rejects the null of no relationship. Columns (2) and (3) run the same regression for pre and post‐1945 subsamples. The finding is relatively robust across the pre and post‐war eras, though, if anything, the relationship is statistically stronger in the pre‐war period. These results are consistent with Jorda et al. (2011a, b) and Schularick and Taylor (2012). Table 4 repeats this exercise with the addition of broad money growth as an explanatory variable. Across the whole sample, broad money growth appears to have only a weak relation to subsequent crises. This is also the message of Schularick and Taylor (2012). We conclude from this that credit dominates broad money as an indicator of financial distress. For completeness, we report an additional set of regressions in the online Appendix to this article that control for real GDP growth and inflation, in addition to credit and money. We also present results that exclude war periods from the regressions. Our headline finding that booms in the credit‐to‐GDP ratio are a systematic indicator of banking crises is not altered by these robustness checks. Fig. 5. Open in new tabDownload slide Medium‐term Cycle in Bank Loan‐to‐GDP Ratio, 1880–2008. (a) UK, (b) US and (c) All Countries Note. Vertical lines denote banking crises. Fig. 5. Open in new tabDownload slide Medium‐term Cycle in Bank Loan‐to‐GDP Ratio, 1880–2008. (a) UK, (b) US and (c) All Countries Note. Vertical lines denote banking crises. Table Tabel 3. The Credit Cycle and Subsequent Crises. Logit Models (Dependent Variable: Banking Crisis) Sample . (1) . (2) . (3) . Whole sample . Pre‐1945 . Post‐1945 . D ln(Loans/GDP)−1 1.02214 1.09391 2.13687 (0.64) (0.68) (0.57) D ln(Loans/GDP)−2 5.33510*** 4.22891*** 8.34439** (4.05) (3.09) (2.33) D ln(Loans/GDP)−3 4.44875*** 3.70178*** 6.34084* (3.52) (2.79) (1.80) D ln(Loans/GDP)−4 1.51519 1.58624 2.12933 (1.07) (1.10) (0.55) D ln(Loans/GDP)−5 2.72905** 2.61701** 1.56466 (2.16) (2.09) (0.41) Constant −3.60827*** −3.08920*** −4.46720*** (−19.31) (−14.91) (−11.21) Observations 1,437 682 755 Countries 14 14 14 Degrees of freedom 5 5 5 χ2 33.14 20.85 16.08 Model p‐value 0.00 0.00 0.01 Sample . (1) . (2) . (3) . Whole sample . Pre‐1945 . Post‐1945 . D ln(Loans/GDP)−1 1.02214 1.09391 2.13687 (0.64) (0.68) (0.57) D ln(Loans/GDP)−2 5.33510*** 4.22891*** 8.34439** (4.05) (3.09) (2.33) D ln(Loans/GDP)−3 4.44875*** 3.70178*** 6.34084* (3.52) (2.79) (1.80) D ln(Loans/GDP)−4 1.51519 1.58624 2.12933 (1.07) (1.10) (0.55) D ln(Loans/GDP)−5 2.72905** 2.61701** 1.56466 (2.16) (2.09) (0.41) Constant −3.60827*** −3.08920*** −4.46720*** (−19.31) (−14.91) (−11.21) Observations 1,437 682 755 Countries 14 14 14 Degrees of freedom 5 5 5 χ2 33.14 20.85 16.08 Model p‐value 0.00 0.00 0.01 Marginal effects for model (1), evaluated at sample means: . x . dPr(crisis)/dx . SE . D ln(Loans/GDP)−1 0.0348 0.0541 D ln(Loans/GDP)−2 0.1817*** 0.0433 D ln(Loans/GDP)−3 0.1515*** 0.0414 D ln(Loans/GDP)−4 0.0516 0.0480 D ln(Loans/GDP)−5 0.0929** 0.0426 Marginal effects for model (1), evaluated at sample means: . x . dPr(crisis)/dx . SE . D ln(Loans/GDP)−1 0.0348 0.0541 D ln(Loans/GDP)−2 0.1817*** 0.0433 D ln(Loans/GDP)−3 0.1515*** 0.0414 D ln(Loans/GDP)−4 0.0516 0.0480 D ln(Loans/GDP)−5 0.0929** 0.0426 Notes ***, ** and * denote significance at 1%, 5% and 10% levels respectively. Includes country effects. Open in new tab Table Tabel 3. The Credit Cycle and Subsequent Crises. Logit Models (Dependent Variable: Banking Crisis) Sample . (1) . (2) . (3) . Whole sample . Pre‐1945 . Post‐1945 . D ln(Loans/GDP)−1 1.02214 1.09391 2.13687 (0.64) (0.68) (0.57) D ln(Loans/GDP)−2 5.33510*** 4.22891*** 8.34439** (4.05) (3.09) (2.33) D ln(Loans/GDP)−3 4.44875*** 3.70178*** 6.34084* (3.52) (2.79) (1.80) D ln(Loans/GDP)−4 1.51519 1.58624 2.12933 (1.07) (1.10) (0.55) D ln(Loans/GDP)−5 2.72905** 2.61701** 1.56466 (2.16) (2.09) (0.41) Constant −3.60827*** −3.08920*** −4.46720*** (−19.31) (−14.91) (−11.21) Observations 1,437 682 755 Countries 14 14 14 Degrees of freedom 5 5 5 χ2 33.14 20.85 16.08 Model p‐value 0.00 0.00 0.01 Sample . (1) . (2) . (3) . Whole sample . Pre‐1945 . Post‐1945 . D ln(Loans/GDP)−1 1.02214 1.09391 2.13687 (0.64) (0.68) (0.57) D ln(Loans/GDP)−2 5.33510*** 4.22891*** 8.34439** (4.05) (3.09) (2.33) D ln(Loans/GDP)−3 4.44875*** 3.70178*** 6.34084* (3.52) (2.79) (1.80) D ln(Loans/GDP)−4 1.51519 1.58624 2.12933 (1.07) (1.10) (0.55) D ln(Loans/GDP)−5 2.72905** 2.61701** 1.56466 (2.16) (2.09) (0.41) Constant −3.60827*** −3.08920*** −4.46720*** (−19.31) (−14.91) (−11.21) Observations 1,437 682 755 Countries 14 14 14 Degrees of freedom 5 5 5 χ2 33.14 20.85 16.08 Model p‐value 0.00 0.00 0.01 Marginal effects for model (1), evaluated at sample means: . x . dPr(crisis)/dx . SE . D ln(Loans/GDP)−1 0.0348 0.0541 D ln(Loans/GDP)−2 0.1817*** 0.0433 D ln(Loans/GDP)−3 0.1515*** 0.0414 D ln(Loans/GDP)−4 0.0516 0.0480 D ln(Loans/GDP)−5 0.0929** 0.0426 Marginal effects for model (1), evaluated at sample means: . x . dPr(crisis)/dx . SE . D ln(Loans/GDP)−1 0.0348 0.0541 D ln(Loans/GDP)−2 0.1817*** 0.0433 D ln(Loans/GDP)−3 0.1515*** 0.0414 D ln(Loans/GDP)−4 0.0516 0.0480 D ln(Loans/GDP)−5 0.0929** 0.0426 Notes ***, ** and * denote significance at 1%, 5% and 10% levels respectively. Includes country effects. Open in new tab Model (1) in Table 3 can be used to quantify the expected marginal effect of higher growth in the credit‐to‐GDP ratio on the likelihood of a banking crisis. As a simple rule of thumb, a 1 percentage point increase in the growth rate of credit to GDP for one year has on average been associated with an increase in the probability of a banking crisis of around 0.18 percentage points two years hence. A one‐year 1 standard deviation increase in the growth of credit to GDP, which is around 9% in the sample, has been associated with around a 1.6 percentage point increase in the probability of a banking crisis two years hence on average over the last century. Of course, the regressions paint a probabilistic picture. The upwards march of the credit‐to‐GDP ratios shown in Figure 2 would have entailed periods of financial deepening that did not lead to banking crises (Dell'Ariccia et al., 2012). To isolate these episodes, consider intervals in which the cumulative growth rate of bank loans over a consecutive five‐year period exceeded the 75th percentile for the whole sample. For the UK, this 75th percentile is around 24% cumulative growth. We then look for years in which the ratio of loans to GDP grew at least as fast as this five‐year rate, but which were not followed by banking crises in the five years thereafter.15 Around one in ten years satisfy these criteria for the UK. Intervals when these criteria are satisfied comprise periods of reconstruction following major conflicts or recoveries following recessions. They are identified as 1921–3, 1960–7, 1984–5 and 1992. Over the period 1921–3, the median five‐year growth rate in bank loans to GDP was of the order of 40%. Similarly, the median five‐year growth rate in bank loans to GDP exceeded 30% over 1960–3. Neither of these periods was followed immediately by a banking crisis, although there were, however, currency crises in the UK in 1931, 1961 and 1964–7.16 The remaining periods (1984–5 and 1992) were both periods of recession; that credit grew rapidly relative to GDP in these periods mechanically reflects weak growth in real activity in these years. Table Tabel 4. Credit, Money and Crises. Logit Models (Dependent Variable: Banking Crisis) . (1) . (2) . (3) . (4) . Whole sample . Whole sample . Pre‐1945 . Post‐1945 . Δ ln(Loans/GDP)−1 1.02214 −0.18370 0.31815 0.47120 (0.64) (−0.10) (0.17) (0.11) Δ ln(Loans/GDP)−2 5.33510*** 5.17842*** 4.73293*** 8.14783** (4.05) (3.48) (2.97) (2.17) Δ ln(Loans/GDP)−3 4.44875*** 5.02197*** 4.59481*** 5.73375 (3.52) (3.76) (3.21) (1.57) Δ ln(Loans/GDP) −4 1.51519 0.73451 0.98475 2.63007 (1.07) (0.45) (0.58) (0.65) Δ ln(Loans/GDP)−5 2.72905** 1.88247 2.01939 −0.11617 (2.16) (1.27) (1.35) (−0.03) Δ ln(bMoney/GDP)−1 2.64921 1.26805 3.05611 (1.15) (0.45) (0.72) Δ ln(bMoney/GDP)−2 1.48451 −0.57333 4.44342 (0.64) (−0.22) (1.30) Δ ln(bMoney/GDP)−3 2.01371 0.25485 4.20789 (0.87) (0.10) (1.12) Δ ln(bMoney/GDP)−4 1.25464 1.08866 −2.44164 (0.50) (0.38) (−0.44) Δ ln(bMoney/GDP)−5 3.96269* 2.84409 4.14682 (1.85) (1.04) (1.15) Constant −3.60827*** −3.72621*** −3.18652*** −4.53349*** (−19.31) (−18.52) (−13.67) (−11.00) Observations 1,437 1,411 658 753 Countries 14 14 14 14 Degrees of freedom 5 10 10 10 χ2 33.14 40.43 23.67 19.55 Model p‐value 0.000 0.000 0.009 0.034 Joint significance of credit/GDP lags (p‐value) 0.000 0.000 0.002 0.028 . (1) . (2) . (3) . (4) . Whole sample . Whole sample . Pre‐1945 . Post‐1945 . Δ ln(Loans/GDP)−1 1.02214 −0.18370 0.31815 0.47120 (0.64) (−0.10) (0.17) (0.11) Δ ln(Loans/GDP)−2 5.33510*** 5.17842*** 4.73293*** 8.14783** (4.05) (3.48) (2.97) (2.17) Δ ln(Loans/GDP)−3 4.44875*** 5.02197*** 4.59481*** 5.73375 (3.52) (3.76) (3.21) (1.57) Δ ln(Loans/GDP) −4 1.51519 0.73451 0.98475 2.63007 (1.07) (0.45) (0.58) (0.65) Δ ln(Loans/GDP)−5 2.72905** 1.88247 2.01939 −0.11617 (2.16) (1.27) (1.35) (−0.03) Δ ln(bMoney/GDP)−1 2.64921 1.26805 3.05611 (1.15) (0.45) (0.72) Δ ln(bMoney/GDP)−2 1.48451 −0.57333 4.44342 (0.64) (−0.22) (1.30) Δ ln(bMoney/GDP)−3 2.01371 0.25485 4.20789 (0.87) (0.10) (1.12) Δ ln(bMoney/GDP)−4 1.25464 1.08866 −2.44164 (0.50) (0.38) (−0.44) Δ ln(bMoney/GDP)−5 3.96269* 2.84409 4.14682 (1.85) (1.04) (1.15) Constant −3.60827*** −3.72621*** −3.18652*** −4.53349*** (−19.31) (−18.52) (−13.67) (−11.00) Observations 1,437 1,411 658 753 Countries 14 14 14 14 Degrees of freedom 5 10 10 10 χ2 33.14 40.43 23.67 19.55 Model p‐value 0.000 0.000 0.009 0.034 Joint significance of credit/GDP lags (p‐value) 0.000 0.000 0.002 0.028 Notes ***, ** and * denote significance at 1%, 5% and 10% levels respectively. Includes country effects. Open in new tab Table Tabel 4. Credit, Money and Crises. Logit Models (Dependent Variable: Banking Crisis) . (1) . (2) . (3) . (4) . Whole sample . Whole sample . Pre‐1945 . Post‐1945 . Δ ln(Loans/GDP)−1 1.02214 −0.18370 0.31815 0.47120 (0.64) (−0.10) (0.17) (0.11) Δ ln(Loans/GDP)−2 5.33510*** 5.17842*** 4.73293*** 8.14783** (4.05) (3.48) (2.97) (2.17) Δ ln(Loans/GDP)−3 4.44875*** 5.02197*** 4.59481*** 5.73375 (3.52) (3.76) (3.21) (1.57) Δ ln(Loans/GDP) −4 1.51519 0.73451 0.98475 2.63007 (1.07) (0.45) (0.58) (0.65) Δ ln(Loans/GDP)−5 2.72905** 1.88247 2.01939 −0.11617 (2.16) (1.27) (1.35) (−0.03) Δ ln(bMoney/GDP)−1 2.64921 1.26805 3.05611 (1.15) (0.45) (0.72) Δ ln(bMoney/GDP)−2 1.48451 −0.57333 4.44342 (0.64) (−0.22) (1.30) Δ ln(bMoney/GDP)−3 2.01371 0.25485 4.20789 (0.87) (0.10) (1.12) Δ ln(bMoney/GDP)−4 1.25464 1.08866 −2.44164 (0.50) (0.38) (−0.44) Δ ln(bMoney/GDP)−5 3.96269* 2.84409 4.14682 (1.85) (1.04) (1.15) Constant −3.60827*** −3.72621*** −3.18652*** −4.53349*** (−19.31) (−18.52) (−13.67) (−11.00) Observations 1,437 1,411 658 753 Countries 14 14 14 14 Degrees of freedom 5 10 10 10 χ2 33.14 40.43 23.67 19.55 Model p‐value 0.000 0.000 0.009 0.034 Joint significance of credit/GDP lags (p‐value) 0.000 0.000 0.002 0.028 . (1) . (2) . (3) . (4) . Whole sample . Whole sample . Pre‐1945 . Post‐1945 . Δ ln(Loans/GDP)−1 1.02214 −0.18370 0.31815 0.47120 (0.64) (−0.10) (0.17) (0.11) Δ ln(Loans/GDP)−2 5.33510*** 5.17842*** 4.73293*** 8.14783** (4.05) (3.48) (2.97) (2.17) Δ ln(Loans/GDP)−3 4.44875*** 5.02197*** 4.59481*** 5.73375 (3.52) (3.76) (3.21) (1.57) Δ ln(Loans/GDP) −4 1.51519 0.73451 0.98475 2.63007 (1.07) (0.45) (0.58) (0.65) Δ ln(Loans/GDP)−5 2.72905** 1.88247 2.01939 −0.11617 (2.16) (1.27) (1.35) (−0.03) Δ ln(bMoney/GDP)−1 2.64921 1.26805 3.05611 (1.15) (0.45) (0.72) Δ ln(bMoney/GDP)−2 1.48451 −0.57333 4.44342 (0.64) (−0.22) (1.30) Δ ln(bMoney/GDP)−3 2.01371 0.25485 4.20789 (0.87) (0.10) (1.12) Δ ln(bMoney/GDP)−4 1.25464 1.08866 −2.44164 (0.50) (0.38) (−0.44) Δ ln(bMoney/GDP)−5 3.96269* 2.84409 4.14682 (1.85) (1.04) (1.15) Constant −3.60827*** −3.72621*** −3.18652*** −4.53349*** (−19.31) (−18.52) (−13.67) (−11.00) Observations 1,437 1,411 658 753 Countries 14 14 14 14 Degrees of freedom 5 10 10 10 χ2 33.14 40.43 23.67 19.55 Model p‐value 0.000 0.000 0.009 0.034 Joint significance of credit/GDP lags (p‐value) 0.000 0.000 0.002 0.028 Notes ***, ** and * denote significance at 1%, 5% and 10% levels respectively. Includes country effects. Open in new tab Fig. 6. Open in new tabDownload slide GDP Growth in Year of Banking Crisis Against Growth in Bank Loans‐to‐GDP Ratio in Five Years Preceding the Crisis Fig. 6. Open in new tabDownload slide GDP Growth in Year of Banking Crisis Against Growth in Bank Loans‐to‐GDP Ratio in Five Years Preceding the Crisis In addition to their relation to the probability of banking crises, are credit booms, and their attendant downswings also related to the severity of crises? As a simple measure of severity, first consider real GDP growth in the year a banking crisis strikes. Figure 6 plots a simple relationship between this measure and the size of the preceding credit boom, as measured by the growth in the bank credit‐to‐GDP ratio in the five years before the crisis. This provides tentative evidence of a negative (unconditional) relationship between growth in the first year of a crisis and the preceding credit boom (pair‐wise correlation: −0.20, p‐value 0.10). These findings are consistent with the analysis of Jorda et al. (2011a), who show that more credit‐intensive booms tend to be followed by deeper recessions and slower recoveries. They are complementary to Claessens et al. (2008, 2011b), who find that recessions associated with credit crunches and financial disruption tend to be deeper and longer than other recessions. We next examine the question of how the GDP gains enjoyed during the upwards phase of the cycle compare with the GDP losses suffered in the bust. A full‐blown analysis of this question is beyond the scope of this study. But to get a sense of the likely magnitudes involved, we first compute country‐specific full sample trend growth rates and use these to compute no‐crisis counterfactual trends for the level of real GDP.17 We then calculate the gap between this counterfactual trend and the actual path followed by GDP in a fixed 10‐year window either side of each crisis. Figure 7 summarises the results. The median cumulative gain in GDP in the decade running up to a crisis is around 21.5%. But the median cumulative loss over the subsequent decade is over twice as large at around 48%. Even allowing for discounting, this provides some speculative evidence that the net impact on GDP of a credit boom that eventually results in a banking crisis is negative. Fig. 7. Open in new tabDownload slide Level of Real GDP Relative to No‐crisis Counterfactual Level of Real GDP (Normalised to Zero in the Year of the Crisis) Fig. 7. Open in new tabDownload slide Level of Real GDP Relative to No‐crisis Counterfactual Level of Real GDP (Normalised to Zero in the Year of the Crisis) 1.4. Cross‐country Correlations Finally, we investigate the extent to which the credit cycle has developed an international dimension. To do this, we calculate the correlation between credit cycles across countries for two different sample windows. The first is 1945–79, covering the post‐war era but prior to major financial liberalisation. The second period covers 1980–2008, which includes, among other things, the ‘big bang’ financial liberalisation in the UK and other advanced countries. Figures 8(a) and (b) show cumulative distribution functions (cdfs) for the pair‐wise correlations across countries’ cycles in real bank loans in the [2, 20] and [8, 20] year windows respectively. We use these cdfs to describe the range of pair‐wise correlations observed. Each shows a marked change, with a tendency for the pair‐wise correlations to increase in the second half of the sample. This is particularly marked for the medium‐term [8, 20] year credit cycle, where all pair‐wise correlations rise as indicated by the rightward shift in the cdf in panel (b). More formally, for the medium‐term cycle both Wilcoxon's rank‐sum test rejects the null that the two samples are drawn from the same distribution at the 1% level, and Jenrich's test rejects the null that the correlation matrices are equal at the 1% level across the two samples. Fig. 8. Open in new tabDownload slide Cumulative Distribution Functions for Cross‐country Correlations in Cycles in Real Bank Loans. (a) Both Medium‐term and Business Cycle Components ([2, 20] Years). (b) Medium‐term Components Only ([8, 20] Years) Fig. 8. Open in new tabDownload slide Cumulative Distribution Functions for Cross‐country Correlations in Cycles in Real Bank Loans. (a) Both Medium‐term and Business Cycle Components ([2, 20] Years). (b) Medium‐term Components Only ([8, 20] Years) Table 5 summarises the findings by reporting the average (mean and median) cross‐country correlations for the two subperiods for cycles in credit in both the full [2, 20] and medium‐term [8, 20] year ranges. Consistent with Figure 8(a) and (b), the average cross‐country correlation between credit cycles has risen. But the absolute level of correlation remains quite low on average – at between 0.3 and 4 – suggesting that, although synchronisation has increased, significant cross‐country asymmetries have remained, even in the era of financial globalisation. Table Tabel 5. Average Cross‐country Cycle Correlations, 1945–79 versus 1980–2008 . . [2, 20] year full cycle in real bank loans . [8, 20] year medium‐term cycle in real bank loans . 1945–79 Mean 0.152 0.159 Median 0.168 0.143 1980–2008 Mean 0.305 0.371 Median 0.345 0.431 . . [2, 20] year full cycle in real bank loans . [8, 20] year medium‐term cycle in real bank loans . 1945–79 Mean 0.152 0.159 Median 0.168 0.143 1980–2008 Mean 0.305 0.371 Median 0.345 0.431 Notes Reports mean and median cross‐country correlations in cycles in real bank credit for the two windows 1945–79 and 1980–2008 for cycles in the [2, 20] (‘full’) and [8, 20] (‘medium‐term’) frequency ranges. Open in new tab Table Tabel 5. Average Cross‐country Cycle Correlations, 1945–79 versus 1980–2008 . . [2, 20] year full cycle in real bank loans . [8, 20] year medium‐term cycle in real bank loans . 1945–79 Mean 0.152 0.159 Median 0.168 0.143 1980–2008 Mean 0.305 0.371 Median 0.345 0.431 . . [2, 20] year full cycle in real bank loans . [8, 20] year medium‐term cycle in real bank loans . 1945–79 Mean 0.152 0.159 Median 0.168 0.143 1980–2008 Mean 0.305 0.371 Median 0.345 0.431 Notes Reports mean and median cross‐country correlations in cycles in real bank credit for the two windows 1945–79 and 1980–2008 for cycles in the [2, 20] (‘full’) and [8, 20] (‘medium‐term’) frequency ranges. Open in new tab 2. A Simple Conceptual Framework These empirics suggest that large swings in credit relative to real activity have been the rule, rather than the exception, over the past century. Credit cycles have been accompanied by banking crises and widespread financial distress. How can small changes in fundamentals be associated with large swings in credit? And what does this imply for macroeconomic policy in general, and macro‐prudential policy in particular? In this Section we set out a simple framework that can accommodate a number of mechanisms developed in the literature that could explain these large swings in finance relative to real activity. What these mechanisms have in common is some form of underlying strategic complementarity. In the presence of imperfect information, financial institutions may adopt strategies whereby the optimal action varies either side of a threshold level of fundamentals. This has been well understood since at least Morris and Shin (1998, 2003).18 Our objective here is to show how this framework could underpin large fluctuations in credit relative to real activity. A key challenge for general equilibrium macroeconomic model builders is to accommodate the sort of discontinuity implied by banking crises.19 Our goal is more modest than this. The framework we present is deliberately simple and is intended to capture the essential features of various explanations of credit booms. Our intention is to develop a simple conceptual model of excessive risk taking and then to examine the role of macro‐prudential regulation in curbing this risk taking. A fully articulated general equilibrium framework is beyond the scope of this study, as is an empirical analysis of the relative importance of different structural causes of excessive risk taking, although each of the mechanisms we point to has some empirical support. We abstract from the role that monetary policy might play in triggering credit booms (Christiano et al., 2007), which is an important topic but one which we leave for future work. 2.1. Environment We begin by setting out the generic environment. There is a continuum of banks, indexed by i ∈ [0, 1]. The economy evolves over two periods, denoted as 0 and 1.20 The state of fundamentals in period 1 is denoted by θ. There is imperfect information about this state: each bank i receives a private signal about fundamentals, ui = θ + ɛi, where σ > 0, where ε∼N(0,1) and where εi is independently drawn across banks. Each bank is endowed with fixed equity capital at time 0, the book value of which we normalise to unity for all i. Each bank raises insured debt from depositors d which pays fixed return Rd. The bank invests its funds in assets x (which could be thought of as loans), such that the balance sheet reads: x=1+d. At time 1, the bank realises the return on its assets, and pays depositors. We allow for two types of asset portfolio: risky (r) and safe (s). If the expected gross return on asset portfolio j = r, s is E(Rj) and its expected cost of debt on funding a portfolio of type j is E(Rdj) ⁠,21 the bank's expected going concern value Vj under portfolio j at time 1 – equivalently, the market value of its equity at time 1 – is as follows: Vj=E(Rj)xj−E(Rdj)dj.(1) The bank is subject to regulatory oversight. We will use this later to talk about macro‐prudential policy. The regulator imposes a constraint by requiring that the bank's asset portfolio is such that expected firm value does not fall below some fraction λ ∈ [0, 1] of the bank's assets22 : Vj≥λxj. Using (1) and the balance sheet of the bank, bank value will be increasing in its asset portfolio xj whenever expected returns exceed the cost of debt. To determine the size of this portfolio, we make: Assumption 1. E(Rj)>λ>E(Rj)−E(Rdj)>0 ⁠. Under this Assumption, the bank's privately optimal scale is given by the value of xj at which the regulator's constraint binds with equality (see Appendix A.1): xj=E(Rdj)λ−[E(Rj)−E(Rdj)]. Note that, by the normalisation that time‐0 book equity capital equals unity, this is also the bank's leverage ratio. The first inequality in Assumption 1 guarantees that leverage exceeds unity. The chosen leverage ratio rises in the expected excess return of assets over debt E(Rj)−E(Rdj) (since then the bank's expected value is higher), and is decreasing in the tightness of the regulator's constraint, λ. We will refer later to λ as the policy maker's instrument that restricts leverage. As this regulatory constraint binds under Assumption 1, the bank's going concern value is Vj = λxj on portfolio j. We place the following generic structure on asset pay‐offs and explain below how each of the games we study relates to this generic structure. First, the safe asset returns E(Rs)=R>Rd with probability 1. Second, we let the risky asset yield a high return (indicated by an upper bar) R¯(θ,b)>R with probability b ∈ [0, 1], where ∂R¯(θ,b)/∂θ≥0 and ∂R¯(θ,b)/∂b≤0 ⁠. The first of these derivatives indicates that the high return on the risky portfolio is (weakly) increasing in fundamentals, whereas the second indicates that the high return on the risky portfolio is (weakly) increasing in the riskiness of the asset (i.e. 1 − b). With probability 1 − b, the risky asset yields a low return (indicated by a lower bar), R̲<Rd ⁠. If R̲ is sufficiently low, the bank's equity will be wiped out; in this case, 1 − b is the probability of bank failure conditional on choosing the risky portfolio. Third, throughout we use l ∈ [0, 1] to denote the proportion of banks choosing the risky portfolio, and allow b potentially to vary with this proportion, where ∂b/ ∂l ≤ 0. We discuss this further below. Finally, we assume that the bank is subject to limited liability. Each of the games we consider uses specific variants of these conditions.23 We turn next to the question of how the bank chooses between the two portfolios. We outline a set of different ways to think about this, beginning with risk shifting, before describing equilibrium in each of the games in subsection 2.5. 2.2. Game 1: Systemic Risk Shifting In this game, we model the effects of a limit in the supply of ‘good’ risky projects – projects that are risky but yield high returns with a relatively high probability. To relate this to reality, consider that as more risky credit is extended, the marginal project becomes riskier and so more likely to default.24 By contrast, we assume that safe returns will not be affected as more banks hold safe portfolios. To incorporate this into the model, make: Assumption 2. ∂b(l)/ ∂l < 0, b(0)=b¯<1,b(1)=b̲>0. An increase in the proportion of banks lending to risky borrowers l depletes the remaining quantity of high‐quality borrowers, such that the marginal lender faces a higher probability of receiving low returns. Banks would tend to price this risk. By appeal to risk neutrality, suppose they do so such that the expected loan return is proportional to fundamentals, bR¯(θ,b)=θ ⁠, which would be the case under: Assumption 3. R¯(θ,b)=θ/b ⁠. Second, let the bad return on the risky portfolio R̲ satisfy: Assumption 4. R̲<Rd(xr−1)/xr ⁠. This says that the low return is insufficient for the bank to pay back depositors, so the bank goes bust. Assumptions 2, 3 and 4 have two effects. First, as more banks make risky loans, the marginal lender will face a higher probability of asset default, which it will account for when it sets its loan rate. The higher likelihood of default will, however, mean that the effective cost of debt is lower due to limited liability. Banks become more likely to use the limited liability shield as riskiness increases. Due to the call‐like pay‐off structure this introduces, it has value to the bank – which will be increasing in the riskiness of the underlying pool of borrowers – e.g. Merton (1977); see also Stiglitz and Weiss (1981), Keeley (1990) and Hellmann et al. (2000). To see this more clearly, under Assumptions 2–4, the bank's expected value under the risky portfolio is as follows: Vr=[θxr−b(l)Rd(xr−1)]+[1−b(l)]×0=θxr−b(l)Rd(xr−1), where, from the regulator's constraint and Assumption 1, xr, the bank's leverage, satisfies: xr=b(l)Rdλ−[θ−b(l)Rd]. This is sufficient to generate systemic risk shifting. To see this, write the marginal return to choosing the risky rather than the safe portfolio V(θ,l)≡Vr(θ,l)−Vs as: V(θ,l)=λ(xr−xs)=λb(l)Rdλ−[θ−b(l)Rd]−Rdλ−(R−Rd), which satisfies: ∂V(θ,l)∂θ>0,∂V(θ,l)∂l>0(2) (see Appendix A.2). That is, the marginal return to choosing the risky rather than the safe portfolio is (a) increasing in fundamentals, and (b) increasing in the number of other banks that choose the risky portfolio. Part (b) arises because the value of limited liability is higher when the pool of remaining borrowers is riskier. This enhances the incentive to risk shift en masse. In Appendix A.2, we show that when fundamentals are neither so strong that the risky portfolio dominates nor so weak that the safe portfolio dominates, there are multiple equilibria such that each bank's privately optimal portfolio choice depends on its beliefs about what other banks are likely to choose. 2.3. Game 2: Assets Under Management and ‘Keeping up with the Goldmans’ In this game, we return to the generic setting and impose: Assumption 5. ∂b(l)/ ∂l = 0, so b is constant. Assumption 6. ∂R¯(θ,b)/∂θ>0 and ∂R¯(θ,b)/∂b=0 ⁠. Now suppose each bank is run by a banker with career concerns (Rajan, 1994). Let the banker obtain a pecuniary benefit proportional to the going concern value of the bank (e.g. a bonus), together with a non‐pecuniary benefit derived from managing a relatively large, risky banking operation. In particular, the larger is the banker's stock of assets under management relative to his peers, the larger is his sense of prestige, the better is his market reputation and the brighter are his future job prospects. Remuneration might also depend on relative assets under management, for example, through fee income targets.25 To capture this, suppose the banker's pay‐off for portfolio j is as follows: γVj+(1−γ)ρj, where γ ∈ [0, 1] captures the banker's relative concern for firm value and ρj captures the effect on reputation of managing portfolio j. The marginal return to choosing the risky rather than the safe portfolio is then: V(θ,l)=γ(Vr−Vs)+(1−γ)ρ(l),(3) where ρ(l) ≡ ρr − ρs captures the marginal effect of holding the risky portfolio on reputation. Suppose: Assumption 7. The marginal gain to reputation ρ(l) to managing the risky rather than the safe portfolio is increasing in the proportion of other banks that also manage risky portfolios, ∂ρ(l)/ ∂l > 0. In a model where bankers have heterogeneous underlying abilities, there are natural cases in which ∂ρ(l)/ ∂l > 0. For example, suppose bankers are high ability with probability α and low ability with probability 1 − α, where bankers' types are unobservable to the market. Low and high types differ in their ability to access investment opportunities. In particular, low types are constrained to choose only safe portfolios with probability 1 − σ, so only with probability σ do they face the same portfolio choice as high types. Of those banks that face a choice, let l denote the fraction that choose the risky portfolio. If the market is using Bayes' rule to update its beliefs on bankers' types, it will assess the probability of being low ability conditional on choosing a safe portfolio as increasing in l.26 Intuitively, as more banks choose the risky portfolio, those that do not are more likely to be drawn from the pool of low‐ability banks which face limited investment choices. If banks' market reputations depend on avoiding being assessed as low ability, then as more banks choose risky portfolios the remainder will be more exposed to bad reputation.27 In this case, the marginal gain to choosing the risky portfolio (3) can be written as: V(θ,l)=γλbRdλ−b[R¯(θ,b)−Rd]−Rdλ−(R−Rd)+(1−λ)ρ(l), which satisfies the properties in (2) (see Appendix A.3). As a result, being small and safe when everyone else is large and risky could be pernicious for prestige. That is, there is a desire to ‘keep up with the Goldmans’ as bankers impose reputational externalities on one another. As in game 1, we show in Appendix A.3 that when fundamentals are neither so strong that the risky portfolio dominates nor so weak that the safe portfolio dominates, the privately optimal action depends on what other banks are doing, as this determines the size of the non‐pecuniary reputation effect. 2.4. Game 3: Moral Hazard In our final example, if a sufficient number of banks fail, we assume that the authorities may be faced with the uncomfortable choice of covering private losses with public funds to maintain the flow of essential financial services. Acharya and Yorulmazer (2007) analyse such a situation; see also Farhi and Tirole (2012). In both contexts, a no bailout policy turns out to be time inconsistent, generating strategic complementarities in risk taking ex ante. To capture this in our benchmark model – suppose that under the risky portfolio – limited liability only comes into play when a sufficient number of banks fail. Only then are the authorities confronted with the uncomfortable choice of stepping in. In particular, suppose: Assumption 8. ∂b(l)/ ∂l = 0, so b constant.28 Assumption 9. ∂R¯(θ,b)/∂θ=0 and ∂R¯(θ,b)/∂b=0 ⁠, so R¯ constant. Assumption 10. For a bailout to occur, a sufficient number t(θ) ∈ [0, 1] of banks must fail, where ∂t(θ)/ ∂θ < 0. Else there is no limited liability. This is sufficient to generate moral hazard. To see this, note that under these assumptions the return to the risky portfolio is as follows: Vr=bR¯xr+(1−b)R̲xr−Rddrifl<t(θ)/(1−b)b(R¯xr−Rddr)ifl≥t(θ)/(1−b), as, by the law of large numbers l(1 − b) banks fail when proportion l choose the risky portfolio. When the number of gamblers l falls below some critical threshold t(θ)/(1 − b), we assume that it is not credible that the authorities will cover losses with public funds. Therefore, when gambling increases above this level, a bailout will occur. Assumption 10 makes the threshold for a bailout a function of fundamentals. In particular, as fundamentals improve, committing not to bailout failed banks becomes less credible: for example, it is during these times that tax receipts are buoyant and there are fewer pressures on the public purse.29 This implies ∂t(θ)/ ∂θ < 0. The risky portfolios under ‘no bailout’ (subscript n) and ‘bailout’ (xnr,xr) must satisfy, respectively: xnr=Rdλ−[bR¯+(1−b)R̲−Rd],xr=bRdλ−(bR¯−Rd), such that xr>xnr if the risky portfolio yields a sufficiently low return conditional on failure, R̲ (see Appendix A.4). When this is the case, an improvement in fundamentals (implying a higher likelihood of bailout) increases expected returns: for all l ∈ [0, 1], the bank will be more likely to enjoy the large ‘bailout’ portfolio xr rather than the smaller ‘no bailout’ portfolio xnr as fundamentals improve. As such, the marginal return to the risky portfolio will be non‐decreasing in fundamentals and there will be strategic complementarity: failing together results in a bailout. 2.5. Equilibrium Having described various ways to think about pay‐offs in the different games, we turn next to equilibrium. To do this, we employ the global games technology of Carlsson and van Damme (1993), developed in Morris and Shin (2003). The idea behind global games is to be able to analyse tractably games of coordination, where one player's action depends on his or her beliefs about other players' behaviour. In general, games of this type require agents not only to form beliefs about fundamentals but also to form ‘higher order’ beliefs about other agents’ beliefs (and therefore actions) and so on ad infinitum. Global games techniques make the analysis of problems like this tractable by assuming that individual players observe fundamentals with a small amount of noise. This allows the set of self‐fulfilling beliefs which prevail in equilibrium to be pinned down and, most usefully, allows us to analyse how the unique equilibrium of a coordination game varies with the parameters of the game. This is of particular use for policy applications, such as ours, where the coordination motives of private agents are important. See Morris and Shin (2000, 2003) for further discussion and applications. In each of the games described, suppose banks play threshold strategies given by h(ui)=safeifui<θ∗riskyifui≥θ∗ where θ* is a critical threshold level of fundamentals to be determined and ui is a given bank's private signal of fundamentals. In each game, the marginal gain from choosing the risky portfolio V(θ,l) satisfies ∂V(θ,l)/∂θ≥0 and ∂V(θ,l)/∂l≥0 ⁠, and there are dominance regions such that, for fundamentals θ∈[θ̲,θ¯] ⁠, there would be multiple equilibria were fundamentals observed without noise. Each game satisfies conditions set out in Morris and Shin (2003, Proposition 2.2) such that as the variance of the idiosyncratic signals received by banks becomes small, the threshold for risk taking θ* is determined by the solution to: ∫l=01V(θ∗,l)dl=0.(4) This threshold condition has the property that, when idiosyncratic noise is small, small changes in fundamentals either side of θ* will alter behaviour dramatically. In particular, if fundamentals transit from θ* − ɛ to θ* + ɛ for ɛ > 0, small banks would switch from holding safe portfolios operating at low leverage to risky portfolios operating at high leverage. To illustrate further, consider a simple numerical example for game 1 (systemic risk shifting). (See the Appendix for numerical examples for the other games.) 2.6. Numerical Example: Systemic Risk Shifting (Game 1) We choose simple functional forms for b(l) and R¯(θ,b) required for the systemic risk‐shifting game, namely b(l) = B − al, R¯(θ,b)=θ/b ⁠, where B, a are (weakly) positive parameters. With these functional forms, the parameter a ≥ 0 controls the strength of strategic complementarities. When a = 0, strategic complementarity disappears, and one bank's action is independent of another's. We use this feature in the next Section to discuss the role of policy in the presence of complementarities. When a > 0, the upper and lower dominance regions for the systemic risk‐shifting game are given by: θ¯=λ+B(R−λ),θ̲=λ+(B−a)(R−λ)<θ¯, respectively, such that for θ>θ¯ ⁠, it is always strictly dominant to choose the risky portfolio regardless of other banks' actions and, for θ < θ̲ ⁠, it is always strictly dominant to choose the safe portfolio regardless of other banks' actions. As a → 0, strategic complementarities disappear, and θ̲→θ¯≡θ^ ⁠, such that there is a single threshold for fundamentals, below (above) which the safe (risky) portfolio is the privately optimal choice. Using (4), the equilibrium condition for this game when a > 0 is given by the threshold θ* that lies in between the two dominance regions which solves: ∫l=01Vr(θ∗,l)dl=Vs⇔∫l=01λ(B−al)Rdλ−θ∗+(B−al)Rddl=λRdλ−(R−Rd). This condition defines the threshold as an implicit function of parameters. By writing the solution to this expression as θ∗=θ∗(λ,a)∈[θ̲,θ¯] we make clear the dependence of the threshold on the regulator's tool λ and the degree of strategic complementarities a. Equilibrium is illustrated schematically in Figure 9. As in the generic case above, a transition of fundamentals from θ* − ɛ to θ* + ɛ entails a switch in banks' portfolios from safe to risky. And because the risky portfolio entails a positive probability of default, 1 − b(l), the likelihood of large‐scale bank losses rises suddenly as fundamentals improve. Moreover, the probability of default 1 − b(l) is itself endogenous to the aggregate amount of risk taking in this game (l), where l = 1 for θ ≥ θ*, and l = 0 otherwise. Fig. 9. Open in new tabDownload slide Dominance Regions and Thresholds in the Risk‐shifting Game Fig. 9. Open in new tabDownload slide Dominance Regions and Thresholds in the Risk‐shifting Game Figure 10 contains a simple numerical example for this game, where, for the sake of illustration, we set the safe portfolio's return to 3%, the return on deposits to 1% and the parameters a, B such that the probability of a risky gamble paying off b is 50% when all banks choose the risky portfolio (a = 0.15, B = 0.65). Panel (a) illustrates the equilibrium threshold θ*, which is at the point where the integral of Vr crosses Vs. In panel (b), we plot the equilibrium threshold as a function of the policy maker's leverage requirement (solid line). As λ increases, leverage falls but by more for the risky portfolio, as: ∂xj∂λ=−xjλ−[E(Rj)−E(Rdj)]=−(xj)2E(Rdj)<0,j=r,s. As such: ∂xr∂λ−∂xs∂λ=(xs)2Rd−(xr)2bRd<0, by xr > xs and b < 1, which says that leverage for the risky portfolio falls by more than leverage for the safe portfolio as λ rises. The risky portfolio then becomes relatively less attractive at a given level of fundamentals, such that the threshold level of fundamentals above which banks choose the risky portfolio rises. For example, at λ = 0.20, the critical threshold for fundamentals is around 0.66. If b(0) = B = 0.65, the required return R¯(θ∗−ε,b) to choose the risky portfolio would be approximately 1.5%.30 By contrast, when λ is increased to a value of 0.3, this hurdle rate for risk taking would rise to be around 9.2%. This is because raising λ from 0.20 to 0.30 reduces bank leverage. For example, leverage under the risky portfolio falls from around 12 to around 5.5 as λ rises from 0.20 to 0.30. In other words, under this illustrative calibration, roughly halving leverage would raise the required threshold return for risk taking by around 7 percentage points. Below we examine what λ a regulator would choose in more detail, given a simple loss function. The leverage requirement affects banks' pay‐offs directly by affecting their profitability. But note also that a tighter leverage requirement affects each bank's expectation about what other banks will be likely to do when strategic complementarities are present. This means that the policy maker has traction through two channels: both the direct effect on pay‐offs, and an indirect effect through altering banks' expectations about each others' actions. This expectations channel gives the macro‐prudential policy maker added traction. To illustrate this further and to explore its implications for regulation, we turn next to a description of a simple policy problem. Fig. 10. Open in new tabDownload slide Equilibrium in Systemic Risk‐shifting Game Notes. (a) Shows the value of the integral of Vr over l ∈ [0, 1] and Vs as fundamentals θ vary. (b) Shows the equilibrium thresholds under micro‐prudential and macro‐prudential views of the world as leverage decreases (λ increases) respectively. Calibration details: b(l)=B−al ⁠, R¯(θ,b)=θ/b ⁠, B = 0.65, a = 0.15, R = 1.03, Rd = 1.01. The micro‐prudential view has a = 0. Fig. 10. Open in new tabDownload slide Equilibrium in Systemic Risk‐shifting Game Notes. (a) Shows the value of the integral of Vr over l ∈ [0, 1] and Vs as fundamentals θ vary. (b) Shows the equilibrium thresholds under micro‐prudential and macro‐prudential views of the world as leverage decreases (λ increases) respectively. Calibration details: b(l)=B−al ⁠, R¯(θ,b)=θ/b ⁠, B = 0.65, a = 0.15, R = 1.03, Rd = 1.01. The micro‐prudential view has a = 0. 3. Curbing the Credit Cycle: Micro and Macro‐prudential Policies In this Section, we examine the policy implications of our framework. We focus on what is distinct about macro‐prudential regulation – as opposed to micro‐prudential regulation, which has historically been concerned with the safety and soundness of individual financial institutions. At time 0, the regulator has a prior belief about the distribution of fundamentals and the threshold for risk taking, which we denote by θ′∈{θ^,θ∗} and discuss further below. The regulator has a simple loss function that takes the form:31 L=p(θ′)(1−b)+c(λ),(5) where p(θ′) ≡ Pr [θ > θ′], such that the first term is the regulator's assessment of the probability of bank distress (i.e. the product of the likelihood of excessive risk taking p(θ′) and the likelihood of a failed gamble) for belief θ′ and the second term is a generic convex cost to increasing the policy instrument, which we suppose takes the form c(λ) = (1/2)λ2. The first of these has a clear parallel with regulatory mandates to contain the likelihood of banking sector distress. The second captures the idea that the regulator cannot pursue reductions in leverage at any cost. These could represent a concern for maintaining credit supply while mitigating systemic bank risk.32 We will distinguish two regulatory regimes. First, we consider a micro‐prudential regulator mandated to mitigate bank risk at the level of the individual firm. In our framework we use the risk‐shifting game with a = 0 to capture the idea that micro‐prudential regulation is conducted ‘firm‐by‐firm’ and abstracts from the spillovers that may exist between institutions. This implies that a micro‐prudential regulator has belief θ′=θ^ ⁠.33 Second, we consider a macro‐prudential regulator, who explicitly takes a system‐wide view and internalises the implications of a > 0, and so has belief θ′ = θ*. This is clearly a somewhat stylised representation of how micro and macro‐prudential regulation could differ in practice. 3.1. Micro‐prudential Problem The micro‐prudential regulator minimises loss function (5) subject to θ′=θ^=λ+B(R−λ) derived above (subsection 9). The shape of the micro‐prudential loss function under our numerical example is illustrated in Figure 11 (dotted line). In Appendix A.5, we show that the formal minimisation problem results in the following policy rule implicitly defining the micro‐prudential setting of the policy instrument λ^ ⁠, arising from the first‐order condition of (5): λ^=φ^b(1−b),(6) Fig. 11. Open in new tabDownload slide Loss Functions for Micro and Macro‐prudential Regulators Notes. Figure shows loss functions for a = 0 (no strategic complementarities – ‘micro‐prudential’ view) and a = 0.15 (strategic complementarities – ‘macro‐prudential’ view). For other calibration details see the note to Figure 10. Fig. 11. Open in new tabDownload slide Loss Functions for Micro and Macro‐prudential Regulators Notes. Figure shows loss functions for a = 0 (no strategic complementarities – ‘micro‐prudential’ view) and a = 0.15 (strategic complementarities – ‘macro‐prudential’ view). For other calibration details see the note to Figure 10. where φ^b≡−(∂p/∂θ^)(∂θ^/∂λ)>0 ⁠. The left‐hand side of this expression is the marginal cost of tighter regulation and the right‐hand side is the marginal benefit, which is the reduced probability of bank failure. The coefficient φ^b determines the optimal reaction of the micro‐prudential regulator to changes in bank risk, 1 − b. When b rises, for example, so that bank risk falls, the micro‐prudential regulator reduces the strength of the micro‐prudential leverage restriction, λ^ ⁠, all else equal. The size of the coefficient is determined in part by the response of individual bank risk taking to changes in the instrument, ∂θ^/∂λ ⁠. 3.2. Macro‐prudential Problem The macro‐prudential regulator minimises (5) subject to the threshold for risk taking θ′ = θ* derived above (subsection 9) and accounting for the dependence of the probability of bank failure on the aggregate amount of risk taking. The macro‐prudential regulator attaches a probability to the risk‐taking portfolio being selected to be equal to the probability that fundamentals exceed the threshold Pr (θ ≥ θ*) ≡p(θ*), which is also the regulator's belief about the proportion of gambling banks. We capture this by writing the regulator's belief over the success probability of the risky portfolio as b(θ*) = B − ap(θ*). Figure 11 shows the macro‐prudential loss function under our numerical example (solid line). In Appendix A.6 we solve the minimisation problem to give the optimal macro‐prudential setting of the policy instrument λ* implicitly by: λ∗=φb∗[1−b(θ∗)]+φp∗p(θ∗),(7) where φb* ≡ − (⁠ ∂p/ ∂θ*)(⁠ ∂θ*/ ∂λ) and where φp* ≡ aϕb*. As in the micro‐prudential case (6), the left‐hand side is the marginal cost of tightening regulation and the right‐hand side is the marginal benefit. The coefficient ϕb* determines the response to bank risk 1 − b(θ*). The magnitude of this coefficient will in general differ from its micro‐prudential counterpart φ^b ⁠. For example, as illustrated in panel (b) of Figure 10, the sensitivity of the threshold to the policy instrument ∂θ*/ ∂λ is greater under the macro‐prudential view (solid line) than under the micro‐prudential view (dotted line) because of the additional strategic effect induced by complementarities.34 This tends to raise φb* in (7) relative to φ^b in (6). The macro‐prudential policy rule therefore responds by more for a given change in bank risk 1 − b(θ*). The strategic effect operating here reflects the role played by banks' expectations about what their competitors are also likely to do. Complementarities generate an expectations channel through which macro‐prudential policy could operate. Second, the macro‐prudential rule includes a response of the policy tool to p(θ*), the probability of system‐wide risk taking, with coefficient φp*. The intuition is simple. When this is higher, bank riskiness worsens endogenously. The macro‐prudential authority accounts for this endogenous component of risk when setting its policy tool. As φp* > 0, this constitutes an additional reason for the optimal macro‐prudential setting of the policy instrument to lie above the micro‐prudential setting in equilibrium, λ∗>λ^ ⁠. These results are apparent in Figure 11, which compares the two loss functions in our numerical example. How does this relate to the issue of practical policy design? The analysis suggests that both the level and the response of the policy instrument to economic fundamentals would differ under macro‐prudential, as opposed to purely micro‐prudential regulation. To be concrete, consider Basel III capital regulation.35 The Basel III package of capital reforms introduced a countercyclical capital buffer designed to be used to enhance resilience over the credit cycle. This has two effects. First, it raises the average level of capital over the cycle. Second, it induces a countercyclical response of policy to changes in economic conditions – in this case, to the so‐called credit‐to‐GDP gap. Our framework thus provides a rationale for the design of this tool. 4. Concluding Remarks: The Policy Landscape Where does the preceding discussion leave us? First, it indicates that, across countries and across a sweep of history, credit cycles measured by variation in the ratio of bank lending to GDP are both clearly identifiable and regular. Sustained growth in the ratio of bank lending to GDP has been strongly correlated with subsequent banking crises. Importantly, credit cycles are distinct from the business cycle in their frequency and amplitude. It is medium‐term fluctuations in credit that, over the last century, have most strikingly coincided with crises. Second, there are a number of mechanisms identified in the theoretical literature that might help explain such large fluctuations in credit. Each of the mechanisms we discuss features strategic complementarity in risk taking. Individual institutions face incentives that, collectively, may generate large expansions in credit aggregates. Because banks' lending decisions are then interdependent, regulatory approaches that focus on individual institutions in isolation may prove ineffectual in curbing the aggregate credit cycle. A macro‐prudential approach, which takes a system‐wide view, is needed to address such externalities. Part of the transmission of macro‐prudential policy to economic outcomes would work through the impact the policy has on individual market participants' expectations of what other market participants are doing – a form of ‘expectations channel’ for macro‐prudential policy. By influencing the costs of credit extension across the system, and therefore banks' expectations about others’ likely actions, macro‐prudential policy could gain greater traction over credit provision than micro‐prudential approaches could achieve alone. Third, our empirical analysis indicates that, while the synchronisation of credit cycles across countries has increased, significant cross‐country asymmetries remain. This suggests that, although a one‐size‐fits‐all policy makes little sense, care needs to be taken to ensure that national policy frameworks do not open up regulatory arbitrage opportunities for international banks. Basel III has made some initial promising steps in this regard (BIS, 2010). The framework includes an explicit reciprocity provision such that judgements on local credit conditions determine the amounts of capital international banks are required to have on their exposures in those countries. This reciprocity feature should help to reduce the arbitrage risks posed by the internationalisation of the credit cycle. Finally, especially at the outset, uncertainties about the role and efficacy of macro‐prudential policy will be considerable. Simplicity and humility will be needed. Simplicity to prevent confusion about the objectives and transmission channels for macro‐prudential policy. Humility because the state of macro‐prudential policy today has many similarities with the state of monetary policy just after the Second World War. Data are incomplete, theory patchy and policy experience negligible. Mistakes will be made. But the analysis in this study, empirical and theoretical, suggests that the biggest mistake would be not to try. Appendix A. Model Proofs and Numerical Examples A.1. Equilibrium Choice of Balance Sheet Size Under portfolio j, the bank's problem is to solve maxxjE(Rj)xj−E(Edj)dj,s.t.xj=dj+1,s.t.Vj≥λxj. The Lagrangian is L=(1+μj){[E(Rj)−E(Rdj)]xj+E(Rdj)}−μjλxj, where μj is the multiplier associated with the regulatory constraint. The first‐order condition for portfolio size is then (1+μj)[E(Rj)−E(Rdj)]≤μjλ, with equality if xj > 0, while the condition with respect to the multiplier yields: [E(Rj)−E(Rdj)]xj+E(Rdj)−λxj≥0, with equality if μj > 0. Note that firm value is increasing in portfolio size as long as E[Rj]−E[Rdj]>0 ⁠, but that the regulator's constraint increases more quickly in xj than does firm value under Assumption 1. Note also that at xj = 0, firm value is positive but the regulator's constraint is zero, such that as, xj, the regulator's constraint cuts firm value Vj at most once from below. Therefore, optimal portfolio size is given by the point at which the regulator's constraint binds, such that μj > 0 and the first‐order condition for μj binds with equality, implying xj=E(Rdj)λ−[E(Rj)−E(Rdj)]. As equity capital is normalised to unity, xj is also the bank's leverage. In general, it will exceed unity if and only if E(Rj)>λ. In each of the games that follow, we make use of variants of the assumptions set out in the text, summarised in Table A2. A.2. Game 1: Properties of Pay‐offs Under Assumptions 1–3, pay‐offs under risky and safe portfolios are Vr(θ,l)=λ×b(l)Rdλ−b(l)[R¯(θ,l)−Rd],Vs=λ×Rdλ−(R−Rd), respectively. By ∂R¯(θ,l)/∂θ>0 ⁠, it follows that ∂Vr(θ, l)/ ∂θ > 0. Next, using b(l)R¯(θ,l)=θ ⁠, ∂Vr(θ, l)/ ∂l > 0 follows if ∂Vr(θ,l)∂l=Vr(θ,l)1b∂b(l)∂l(1−xr)>0, which, by Assumption 2 (more risky lending depletes the pool of low default borrowers), is the case under Assumption 1 (which gives leverage in excess of unity, xr > 1). Next, we define the region of fundamentals for which there are multiple equilibria. The marginal gain to choosing the risky portfolio is V(θ,l)=Vr(θ,l)−Vs. When fundamentals are weaker than θ, it is dominant to choose the safe portfolio even when everyone else is choosing risky l = 1, where θ is defined by b(1)Rdλ−b(1)[R¯(θ̲,1)−Rd]=Rdλ−(R−Rd). When fundamentals exceed θ¯ ⁠, it is dominant to choose the risky portfolio even when no one else does so l = 0, where θ¯ is defined by b(0)Rdλ−b(0)[R¯(θ¯,0)−Rd]=Rdλ−(R−Rd). In the intermediate range θ∈[θ̲,θ¯] ⁠, there are multiple equilibria. In our numerical example, we have b(l) = B − a × l and R¯(θ,b)=θ/b(l) ⁠. Then, in this case, the region of multiple equilibria is bounded by θ¯=λ+B(R−λ),θ̲=λ+(B−a)(R−λ)<θ¯ (where the last inequality follows using the first inequality in Assumption 1). Equilibrium is then given by the solution to: ∫l=01Vr(θ∗,l)dl=Vs. A.3. Game 2: Properties of Pay‐offs The marginal return to choosing the risky portfolio is V(θ,l)=γ(Vr−Vs)+(1−γ)ρ(l). Using Vj = λxj, this reduces to V(θ,l)=γλ(xr−xs)+(1−γ)ρ(l). Hence, by ∂xr/ ∂θ > 0, we have ∂V(θ,l)/∂θ>0 and we also have ∂V(θ,l)/∂l>0 under Assumption 5 discussed in the text. Next, we define the region of multiple equilibria, which arises for θ∈[θ̲,θ¯] defined by xr(θ¯)=xs−1−γγλρ(0),xr(θ̲)=xs−1−γγλρ(1), which, by ρ(1) > ρ(0), implies θ̲<θ¯ ⁠. In the simple case where ρ(l)=l ⁠, equilibrium is given by: xr(θ∗)=xs−1−γγλ. If R¯(θ,b)=θ/b ⁠, we have simply that: θ∗=λ+bRd−bRdRdλ−(R−Rd)−1−γγλ.(A.1) This is increasing in λ for γ is sufficiently large, in which case we get a plot analogous to Figure 7(b). A.4. Game 3: Properties of Pay‐offs The two risky portfolios are xnr=Rdλ−[bR¯+(1−b)R̲−Rd],xr=Rdλb−(R¯−Rd). For the no bailout portfolio, Assumption 1 translates as λ>b(R¯−R̲)+(R̲−Rd). The bailout portfolio is larger than the no bailout portfolio if and only if λb<R¯−R̲, such that Assumption 1 and xnr<xr jointly require that b(R¯−R̲)+(R̲−Rd)<λ<b(R¯−R̲), which in turn requires that the net bad pay‐off R − Rd be sufficiently small/negative. Then as fundamentals improve, a bailout becomes more likely, such that the expected return to the risky portfolio rises (as the larger ‘bailout’ portfolio xr is held for a larger range of l). The marginal gain to risk taking in this game is as follows: V(θ,l)=λxnr−λxsif<a(θ)/(1−b)λxr−λxsif≥a(θ)/(1−b). Equilibrium solves: ∫l=0a(θ∗)/(1−b)(λxnr−λxs)dl+∫a(θ∗)/(1−b)l=1(λxr−λxs)dl=0, giving: a(θ∗)=(1−b)xr−xsxr−xnr.(A.2) A.5. Micro‐prudential Regulator's Problem Let p(θ^)≡Pr(θ>θ^) be the regulator's belief about a bank choosing the risky portfolio. Then, the regulator's objective is to solve: minλL(λ)=p(θ^)(1−b)+c(λ), s.tθ^=λ+B(R−λ). The first‐order condition is as follows: ∂p∂θ^∂θ^∂λ(1−b)+∂c∂λ=0. With c(λ)=(1/2)λ2 ⁠, this gives the ‘reaction function’ determining the micro‐prudential setting of the policy tool λ^ according to: λ^=φ^b−φ^bb, where φ^b≡−∂p∂θ^∂θ^∂λ>0, (where the inequality follows from ∂p/∂θ^=∂Pr(θ>θ^)/∂θ^=∂[1−Φ(θ^)]/∂θ^<0 ⁠, where Φ( · ) is the increasing cdf for fundamentals and ∂θ^/∂λ=1−B>0) ⁠. The second‐order condition is ∂2p∂θ^2∂θ^∂λ2(1−b)+∂2c∂λ^2, where ∂2p∂θ^2=−∂ϕ(θ^)∂θ^, in which φ(·) is the regulator's pdf over fundamentals. If, for example, θ^ lies above the mean value for fundamentals, and φ(·) is normal, then ∂2p/∂θ^2>0 ⁠, such that the second‐order condition is positive and −L(λ) is concave. A.6. Macro‐prudential Regulator's Problem Let p(θ*) ≡ Pr (θ > θ*) denote the macro‐prudential regulator's belief over banks choosing the risky portfolio. This is also the macro‐prudential regulator's expected proportion of risk‐taking banks, which in turn affects the probability of a given bank's risky portfolio paying off, that is, b(θ*) = B − ap(θ*). The objective is then to solve: minλL(λ)=p(θ∗)[1−b(θ∗)]+c(λ),s.t.θ∗=θ∗(λ,a). The first‐order condition is as follows: ∂p∂θ∗∂θ∗∂λ[1−b(θ∗)]+p(θ∗)a∂p∂θ∗∂θ∗∂λ+∂c∂λ=0. With c(λ) = (1/2)λ2, this gives the macro‐prudential setting of the policy tool λ* according to: λ∗=φb∗−φb∗b(θ∗)+φp∗p(θ∗), where φb∗≡−∂p∂θ∗∂θ∗∂λ,φp∗≡aφb∗. Using the definition of b, the first‐order condition can be written as: ∂p∂θ∗∂θ∗∂λ{1−[B−ap(θ∗)]}+ap(θ∗)∂p∂θ∗∂θ∗∂λ+∂c∂λ=∂p∂θ∗∂θ∗∂λ(1−B)+2ap(θ∗)∂p∂θ∗∂θ∗∂λ+∂c∂λ. The second‐order condition is then: ∂2p∂(θ∗)2∂θ∗∂λ2(1−B)+∂p∂θ∗∂2θ∗∂λ2(1−B)+2a∂p∂θ∗∂θ∗∂λ2+2ap(θ∗)∂2p∂(θ∗)2∂θ∗∂λ2+2ap(θ∗)∂p∂θ∗∂2θ∗∂λ2+∂2c∂λ2, which is positive in our numerical example. A.7. Numerical Example for Game 2 Above we showed that the threshold in this game is given by (A.1). Let the parameter values be the same as in the numerical example game 1, namely, R = 1.03, Rd = 1.01, b = 0.65. If the relative concern for firm value is γ = 0.775 and the regulator sets λ = 0.2, the threshold is θ* = 0.6987. As we let R(θ, b) = θ/b, this implies a threshold loan return of 0.6987/0.65, or around 7.5%. We can also solve for the optimal value of λ chosen by the regulator to minimise the loss function, just as for game 1. Panel (a) of Figure B1 in online Appendix B to this article contains a summary of the threshold as the policy instrument varies, for two different values of reputational concern γ. Panel (b) of the Figure shows the policy maker's loss function under the two values for γ. The optimal setting of λ varies with the concern for reputation. Figure B2 in online Appendix B to this article summarises the regulator's optimal choice of λ for different γ. As concern for reputation increases (γ falls), the effect is at first to encourage risk taking, which the regulator leans against by raising the level of the instrument λ. For sufficiently lower values of γ, however, risk taking becomes all but inevitable, at which point the regulator can do little to reduce risk taking and faces only costs of tight regulation – hence, the non‐monotonicity in Figure B3. A.8. Numerical Example for Game 3 As for the other games, suppose R = 1.03, Rd = 1.01 and B = 0.65. Under the risky portfolio, let R¯=1.15 (the risky upside return) and let R = 0 (the risky downside return: lose everything). Using (A.2), we learn that the critical proportion of gamblers t(θ*) = 0.1073. The critical threshold is inferred from this given a choice of functional form that maps θ* onto [0, 1] and satisfies ∂t(θ)/ ∂θ < 0, for example, t(θ) = [1 + exp (θ)]−1. In this example, with λ = 0.2, this would imply θ* = 2.12. As above, we can use the regulator's loss function to find an optimal value for the regulatory instrument λ. Panel (a) of Figure B3 in online Appendix B to this article shows the bank portfolio leverage under risky‐with‐bailout, risky‐without‐bailout and safe portfolios, (xr,xnr,xs) respectively. It is clear from this panel that bailouts raise the leverage associated with the risky portfolio (solid versus dashed line). Panel (b) shows the policy maker's loss function, minimised around λ* = 0.2, which entails leverage of around 5 under this calibration. Footnotes 1 " See IMF (2010). 2 " There are other potential non‐cyclical instruments of macro‐prudential policy, including instruments to tackle systemically important financial institutions or other structural vulnerabilities in the financial system. 3 " See also Morris and Shin (1998, 2000). 4 " Codes to replicate our findings are available online. 5 " The countries covered are as follows: Australia, Canada, Germany, Denmark, Spain, France, Japan, Switzerland, UK, Italy, Netherlands, Norway, Sweden and the US. 6 " Schularick and Taylor (2012) collect bank loans, ‘defined as the end‐of‐year amount of outstanding domestic currency lending by domestic banks to domestic households and non‐financial corporations (excluding lending within the financial system). Banks are defined broadly as monetary financial institutions and include savings banks, postal banks, credit unions, mortgage associations and building societies whenever the data are available. Total bank assets are then defined as the year‐end sum of all balance sheet assets of banks with national residency (excluding foreign currency assets)’. A full list of sources for Schularick and Taylor's data is available in the Appendix to their article. See in particular their Appendix B. 7 " We estimate the spectral density by smoothing the sample periodogram using a Parzen window with lag truncation parameter M. We experimented with various values of M. In general, this trades off the variance of the estimate of the density with bias. Higher values of M entail less bias but produce a greater variance and wider confidence intervals. The confidence intervals are plotted using the χ2 approximation discussed in Preistley (1999). 8 " We also experimented with the Hodrick–Prescott filter. With a high smoothing parameter, this approach produced broadly similar results to the band‐pass filter. See, for example, Harvey and Jaeger (1993) and Canova (1998) for discussions. 9 " In what follows, we use Christiano and Fitzgerald's (2003) optimal finite sample approximation to the band‐pass filter. 10 " For the UK, the variance of the medium‐term component of the credit cycle is around 1.8 times larger than the variance of the business cycle frequency component. 11 " Following Comin and Gertler (2006), we estimated the standard error of the standard deviation of the [8, 20] year cycle using GMM, where we used the Newey–West estimator of the covariance matrix. 12 " We also use Lopez‐Salido and Nelson (2010) definitions of crises dates for the US. 13 " Schularick and Taylor use lags of the real growth in bank loans as explanatory variables. We use the loan‐to‐GDP ratio instead, although the results are similar. 14 " The models include country effects, not reported. 15 " On average, across all countries, around 11% of years exhibit rapid credit growth without banking crises in the subsequent five years. 16 " There were also currency crises in 1947 and 1949. See Jorda et al. (2011b) for an examination of the role that external imbalances play in crisis episodes. 17 " The choice of trend is obviously critical for the results and this approach has clear drawbacks. Nevertheless, if credit booms and busts have roughly offsetting effects in the long run, then this provides a reasonably neutral way of estimating the trend. 18 " Carlsson and van Damme (1993) undertook an earlier analysis of global games. 19 " Though, see for example He and Krishnamurthy (2013) and Brunnermeier and Sannikov (forthcoming) for macroeconomic models with global, as opposed to local, dynamics. Gertler and Kiyotaki (2012) have developed a version of their model incorporating bank runs. 20 " Full dynamics could be incorporated straightforwardly by allowing the two‐period game to be repeated afresh each period of calendar time and allowing for some relationship between fundamentals across calendar time periods. 21 " This could differ from Rd because of limited liability. See below. 22 " It is a reinterpretation of the incentive compatibility constraint used to incorporate banks into macroeconomic models used by Gertler and Karadi (2010), Gertler and Kiyotaki (2011) and Gertler et al. (2012). It differs from regulatory ratio constraints – such as the Basel III capital ratio – which imposes a constraint on the book value of a bank's equity capital relative to its risk weighted assets. 23 " These are summarised in Table B2 of this article's online Appendix B. 24 " For instance, Dell'Ariccia et al. (2008) document how lending standards declined as the US credit boom inflated in the run‐up to the crisis. 25 " See, for example, Gibbons and Murphy (1990) and Murphy (1999) for evidence on the use of relative performance evaluation in financial firms. 26 " Under this structure, Pr(low ability|safe portfolio) =1/{1+[α/(1−α)][(1−l)/(1−σl)]} ⁠, which is increasing in l. 27 " See also Aikman et al. (2012). 28 " This is identical to Assumption 5 in game 2 but we re‐state it here for clarity. 29 " Returns to investment would also be high when fundamentals are strong, such that maintaining the flow of credit is more valuable. 30 " That is, 0.66/0.65, as R¯(θ∗,b)=θ∗/b in this example. 31 " It would be straightforward to allow the regulator to have preferences over the relative importance of the two objectives in (5), for example, L(λ) = δp(θ′)(1 − b) + (1 − δ)c(λ), for δ ∈ [0, 1]. 32 " The UK Financial Services and Markets Act 2000 required that the UK micro‐prudential regulator at the time, ‘[i]n discharging its general functions … must have regard to … the desirability of facilitating innovation in connection with regulated activities … [and] the international character of financial services and markets and the desirability of maintaining the competitive position of the United Kingdom’. See Part I.2 (3) of http://www.legislation.gov.uk/ukpga/2000/8/contents. 33 " We write ‘belief’ as this is the critical value for risk taking that arises when a = 0, or when there are no strategic complementarities. If a was in fact positive but the micro‐regulator perceived risk taking to occur for fundamentals above θ′=θ^=λ+B(R−λ) ⁠, the micro‐regulator uses a misspecified model. 34 " In addition, as θ∗<θ^ when there are complementarities, −∂p/∂θ∗=ϕ(θ∗)>ϕ(θ^)=−∂p/∂θ^ ⁠, where φ(·) is the policy maker's pdf of fundamentals as long as θ* lies above the mean, and, for example, φ(·) is the normal pdf. 35 " See ‘Basel III: a global regulatory framework for more resilient banks and banking systems’, available at http://www.bis.org/publ/bcbs189.pdf. " Correction note: This article was first published online on the 26th of March 2014. It was updated in June 2015 to correct the results displayed in Figure 3 and the description of them given in the first paragraph of subsection 1.2. 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We are grateful to the editor, Wouter den Haan, two anonymous referees and to Piergiorgio Alessandri, Richard Barwell, Salina Ladha, Roland Meeks, Victoria Saporta, Misa Tanaka, Alan Taylor, seminar participants at the Columbia University Center on Capitalism and Society, the Central Bank of Turkey and Maryland University for comments on earlier versions. Thanks to Vijay Balle and Clare Rogowski for excellent research assistance and to Moritz Schularick and Alan Taylor for making their data available to us. Disclaimer: opinions expressed in this work are those of the authors, and do not necessarily reflect those of the Bank of England, the MPC or the FPC. © 2013 Bank of England. The Economic Journal © 2013 Royal Economic Society

Rendahl,, Pontus

doi: 10.1111/ecoj.12115pmid: N/A

Abstract Solving dynamic models with inequality constraints poses a challenging problem for two major reasons: dynamic programming techniques are reliable but often slow, whereas Euler equation‐based methods are faster but have problematic or unknown convergence properties. This study attempts to bridge this gap. I show that a common iterative procedure on the first‐order conditions – usually referred to as time iteration – delivers a sequence of approximate policy functions that converges to the true solution under a wide range of circumstances. These circumstances extend to a large set of endogenous and exogenous state variables as well as a very broad spectrum of occasionally binding constraints. Dynamic models with inequality constraints are of considerable interest to a wide range of economists. The workhorse consumption–savings framework, for instance, is commonly augmented to include liquidity constraints that limit the extent to which agents can borrow; non‐cooperative dynamic games frequently observe enforcement constraints that bound the set of sub‐game perfect equilibria; and in the wake of the recent financial crisis, there has been a surge of interest in models incorporating collateral constraints that may propagate shocks by way of the ‘financial accelerator’.1 However, solving dynamic models with occasionally binding inequality constraints is no trivial matter. Dynamic programming techniques are reliable but often slow, whereas Euler equation‐based methods are known to be faster but have problematic or unknown convergence properties.2 This study aims to bridge this gap. I show that a common iterative procedure applied on the Euler equation – time iteration – delivers a sequence of approximate policy functions that converges to the true solution under a wide range of circumstances. The proposition extends to a large set of endogenous and exogenous state variables, and includes a broad spectrum of occasionally binding constraints.3 I exemplify the potential advantages of the method to alternative techniques by solving a simple real business cycle model with irreversible investments (McGrattan, 1996; Christiano and Fisher, 2000). The proposed method is generally faster than state‐of‐the‐art dynamic programming algorithms, and at least equally accurate. In the context of dynamic programming, dealing with occasionally binding constraints is generally straightforward. In one popular approach, the state and the choice space is confined to belong to a discrete grid, which is (almost trivially) delimited to rule out any violations of the constraint set (Imrohoroğlu, 1989; Hansen and Imrohoroğlu, 1992). The resulting procedure has the advantage of being simple, robust and essentially arbitrarily accurate but also comes at a very high cost in terms of computational power (Santos and Vigo‐Aguiar, 1998). To ease this computational burden, it has become increasingly common to treat the choice set as (if) convex (Johnson et al., 1993; Krusell and Smith, 1998). Function values in‐between grid points are then evaluated using some interpolation, or approximation, routine, and the maximisation step is carried out by constrained optimisation. Whereas these improvements have led to some significant gains in both speed and accuracy (Judd and Solnick, 1994), they are still not without complications of their own. An efficient implementation requires the approximation method to preserve certain desirable properties of the value function, such as differentiability and concavity, which adds to the computational complexity. And while most numerical optimisation routines are far more reliant on the slope rather than the level of the value function, dynamic programming infers the former from an approximation of the latter, which leaves the procedure vulnerable to the possibility of self‐propagating errors.4 A slope inferred from an approximation of a function can be orders worse than a direct approximation of the slope of the function, and errors may easily accumulate.5 Parts of these difficulties can be circumvented by operating directly on the Euler equation – or more generally – on the first‐order conditions.6 The optimisation step is then replaced by a collection of non‐linear equations, and the choice of approximation method is given a much larger degree of freedom. A monotone and continuous approximation of the slope of the value function, for instance, is isomorphic to a once continuously differentiable and concavity preserving approximation of the level of the value function, but comes at a much smaller computational cost. However, these approaches do not come without their own issues. Euler equation‐based methods often have problematic or unknown convergence properties and, without an educated initial guess for the optimal policy functions, convergence may fail.7 This study addresses some of these concerns. I show that any element in the sequence of approximate value functions defined by dynamic programming is differentiable when a general class of inequality constraints is considered, and I provide analytical expressions of their respective derivatives. Using these theoretical insights, an iterative procedure on the first‐order conditions, commonly known as time iteration, is derived. Given that this procedure is equivalent to value function iteration, it is, under mild initial conditions, a globally convergent method of finding the equilibrium functions for recursively defined, Pareto optimal problems. And due to the concavity of the problem, this turns out to be a very convenient and efficient technique from a computational perspective. The ideas developed in this study relate foremost to those of Coleman (1989, 1990, 1991), Deaton (1991) and Deaton and Laroque (1992). In influential work, Coleman (1989, 1990, 1991) argues that a certain iterative procedure applied on the Euler equation converges to the true solution of an infinite horizon problem. The proposed procedure, which Coleman refers to as ‘policy function iteration’, is identical to that explored in this study but the results are obtained using a different approach and, therefore, also carry different implications.8 In particular, Coleman shows that under the right conditions the Euler equation defines a ‘monotone map’, and Tarski's fixed‐point theorem applies.9 Convergence is therefore ensured even for non‐Pareto optimal economies, but is also restricted to a relatively simple one‐dimensional class of problems without occasionally binding constraints – see, in particular, Coleman (1991). This study, in contrast, abstracts from the analysis of sub‐optimal economies but instead allows for a much larger state and choice space, including multiple first‐order conditions and a broad collection of possibly binding inequality constraints. Deaton, on the other hand, does consider the possibility of occasionally binding constraints. In particular, Deaton (1991) studies a standard stochastic consumption–savings problem in the presence of liquidity constraints that preclude borrowing. By exploiting theoretical results developed in Deaton and Laroque (1992), Deaton shows that the same iterative procedure as advocated in Coleman (1989, 1990, 1991) defines a contraction mapping and convergence follows from the Banach fixed‐point theorem. This approach has received quite some attention in the literature and is particularly widely applied in consumption‐theoretic studies.10 I generalise Deaton's (1991) results in some dimensions but specify it in others. In particular, I expand the analysis to cover a much larger state and choice space, and consider a richer set of occasionally binding constraints. In contrast to Deaton (1991), however, this study abstracts from the difficulties that arise when the return function may be unbounded. 1. Analysis This Section presents the main propositions and discusses their implications. In the first part, I set up a general stochastic optimisation problem formulated as a Bellman equation. The problem is cast such that it includes the possibility of occasionally binding inequality constraints, subject to some standard constraint qualifications. I then proceed by showing that under some commonly imposed conditions on the primitives of the problem, any element in the sequence of successive approximations of the value function, {vn}, is differentiable with respect to the (vector of) endogenous state variables and analytic expressions for their respective derivatives are provided. Lastly, I show that as these derivatives solely depend on the policy and Lagrange multiplier functions associated with each vn, the first‐order conditions allow these functions to be updated by only using their values in the previous iteration. The method of successive approximations that maps vn−1 to vn can therefore be recast as a procedure that maps past policy and multiplier functions to the current policy and multiplier function, and with identical convergence properties. This latter procedure, it turns out, is identical to that of Coleman (1989, 1990, 1991), Deaton (1991) and Judd (1998 pp. 553–5, 601–2), and is referred to as time iteration. The last part of the Section discusses the practical implications of these results, and compares the resulting iterative procedure to standard value function iterations as well to other Euler equation‐based solution methods. 1.1. A General Problem This study is concerned with problems that can be formulated according to the following Bellman equation v(x,z)=maxy∈Γ(x,z)F(x,y,z)+β∫Zv(y,z′)Q(z,dz′),(1) where x ∈ X is the endogenous state, and z ∈ Z is the exogenous state with a law of motion determined by the stationary transition function Q. I employ the following standard assumptions. (i) X is a convex Borel set in Rℓ with Borel subsets X ⁠, and Z is a compact Borel set in Rk with Borel subsets Z ⁠. Denote the (measurable) product space of (X,X) and (Z,Z) as (S,S) ⁠. (ii) The transition function, Q(Z,Z) ⁠, has the Feller property.11 (iii) The feasibility correspondence Γ(x,z) : X × Z → 2X is non‐empty, compact valued and continuous. Moreover, the set A = {(y,x) ∈ X × X : y ∈ Γ(x,z)} is convex in x, for all z ∈ Z. (iv) The return function F(·,·,z):A→R is once continuously differentiable, jointly (strictly) concave in x and y, and bounded on A for all z ∈ Z. (v) The discount factor, β, is in the interval (0,1). It is important to note that the above definition of the feasibility correspondence includes the possibility of inequality constraints. Then, for any weakly concave and bounded v0, it follows that (Stokey et al. 1989, pp. 263–6) (i) The sequence of functions defined by vn+1(x,z)=maxy∈Γ(x,z)F(x,y,z)+β∫Zvn(y,z′)Q(z,dz′),(2) converges uniformly to the unique fixed point v. And gn+1(x,z)=argmaxy∈Γ(x,z)F(x,y,z)+β∫Zvn(y,z′)Q(z,dz′),(3) converges point‐wise to the unique fixed point g.12 (ii) v and vn are strictly concave. (iii) g and gn are continuous functions. To introduce the presence of inequality constraints more explicitly, I assume that all equality constraints have been substituted into the instantaneous return function, F. The remaining restriction on feasibility will then solely be summarised by the collection of functions mj(x,y,z), such that mj(x,y,z) ≤ 0, for j = 1,…,r. Assumption 1. The feasibility correspondence can be formulated as Γ(x,z)=y∈X:mj(x,y,z)≤0,j=1,…,r, and the functions mj(x,y,z), j = 1,…,r, are once continuously differentiable in x and y, and jointly (weakly) convex in x and y. Let Jx(x,y,z) denote the Jacobian matrix of the constraint vector m = [m1(·),m2(·),…,mr(·)]′ with respect to the first ℓ arguments (the xs). And let Jy(x,y,z) denote the Jacobian with respect to the ensuing ℓ arguments (the ys).13 Assumption 2. Linear independence constraint qualification (LICQ): The Jacobian of the p ≤ r binding constraints has full rank; i.e. rank{ Jy[x,gn(x,z),z]} = p, for each n∈N ⁠. Assumption 3. The following hold (i) Γ(x,z) ⊂ int(X), or (ii) X is compact and g(x,z) ∈ int(X). A few things ought to be noted here. Assumption 1 is actually quite weak. Although it is assumed that all equality constraints have been substituted into the instantaneous return function, none of the results hinges on this expositional simplification. Together with Assumption 1, Assumption 2 ensures that the duality gap is zero and that the Kuhn–Tucker (KT) conditions are sufficient. There are other, weaker, constraint qualifications that ensures the sufficiency of the KT conditions but LICQ is the weakest constraint qualification that ensures both the existence and uniqueness of the Lagrange multipliers (Wachsmuth, 2013). Together with Assumption 3, Assumption 2 also implies that a y^ exists such that mj(x,y^,z)<0 ⁠, for all x, z and j. This is normally known as Slater's condition. Part (i) of Assumption 3 implies part (ii) but the converse is not necessarily true. And lastly, the purpose of Assumption 3 is to ensure that it is the collection of constraints that restricts feasibility, and not the edge of the state space itself. Thus, while Assumptions 1 and 3 are largely expositional, Assumption 2 is more substantive and merits further discussion. First, and quite trivially, p, the number of binding constraints, must fall short of, or equal, the number of endogenous state variables, ℓ.14 One example of such a violation would be a standard neoclassical growth model with an irreversibility constraint on investments, kt+1 − (1 − δ)kt ≥ 0, and with a liquidity constraint on capital kt+1 ≥ ϕ.15 Then, for kt = ϕ/(1 − δ) both constraints simultaneously bind and LICQ is violated. While the KT conditions may still remain sufficient, the Lagrange multipliers are no longer unique as only their sum, and not their individual values, would distinctly influence the first‐order conditions. This argument can be made more general. If μ denotes a vector of Lagrange multipliers associated with some generic optimisation problem and Jy denotes the Jacobian of the associated constraint vector, then the vector v = Jyμ plays an integral part in the KT conditions; but the individual values of Jy and μ do not. However, if the rank of Jy falls short of the number of binding constraints, there are infinitely many vectors μ^ that satisfy the system of equations v=Jyμ^ ⁠, and the Lagrange multipliers are not unique. 1.2. Results Define the operator T on C1(S), the space of bounded, strictly concave and once continuously differentiable functions, as (Tf)(x,z)=maxy∈Γ(x,z)F(x,y,z)+β∫Zf(y,z′)Q(z,dz′).(4) That is, T is an operator that takes a function f ∈ C1(S) as an argument and maps it into a new function Tf. Although it is obvious that under the above conditions, T maps functions in C1(S) into functions in C(S) – the space of bounded, strictly concave and continuous functions – it will be shown that T : C1(S) → C1(S), which follows less straightforwardly.16 Under all the aforementioned assumptions it is possible to express the mapping in (4) as (Tf)(x,z)=minμ≥0maxy∈XL(x,y,z,μ)=maxy∈Xminμ≥0L(x,y,z,μ),(5) where L denotes the Lagrangian (or saddle function) L(x,y,z,μ)=F(x,y,z)+β∫Zf(y,z′)Q(z,dz′)−μ′m(x,y,z),(6) and μ is a (r × 1) column vector of Lagrange multipliers associated with the constraint vector m (see, for instance, Rockafellar, 1970). The ultimate goal of this Section is to show that time iteration yields a convergent sequence of policy functions. The following definition of time iteration will be used. Definition 1. Let Fx(x,y,z) denote the ℓ‐vector consisting of the partial derivatives of F with respect to its first ℓ arguments. And let Fy(x,y,z) denote the corresponding vector with respect to the ensuing ℓ arguments. Then, time iteration is the iterative procedure that finds the sequence [hn(x,z)]n=0∞ as y = hn+1(x,z) such that 0=Fy(x,y,z)−Jy(x,y,z)μn+1(x,z)+β∫Z{Fx[y,hn(y,z′),z′]−Jx[y,hn(y,z′),z′]μn(y,z′)}Q(z,dz′).(7) Although the notation may appear esoteric, time iteration can be thought of as using the Euler equation to find today's optimal policy, hn+1, given the policy of tomorrow, hn.17 Proposition 1. The n‐step value function, vn, is (once) continuously differentiable with respect to x ∈ int(X) and its vector of partial derivatives is given by vx,n(x,z)=Fx[x,gn(x,z),z]−Jx[x,gn(x,z),z]μn(x,z). Proof In Appendix A. As the space C1(S) is not complete in the sup norm, Proposition 1 does not imply that the limiting value function, v, is differentiable.18 Moreover, in the proposition above, strict concavity of the problem and full rank of Jy is assumed. This simplifies the proof given in Corollary 5, in Milgrom and Segal (2002, p. 597), which essentially is equivalent for x ∈ [0,1]. The final Proposition will show that the sequence of policy functions obtained by time iteration converges to the true policy function. Proposition 2. If the function y = hn+1(x,z) satisfies (7) with hn = gn, and the function gn+1(x,z) satisfies (3) with vn = Tvn−1, then hn+1(x,z) = gn+1(x,z). Proof In Appendix A. Thus, Proposition 2 reveals that if hn is the nth solution of time iteration and gn is the nth solution of dynamic programming, then hn and gn must coincide. And, as it is known that for all ε > 0 an Ns exists such that sups|g(s)−gn(s)|<ε for all n ≥ Ns, it follows from Proposition 2 that sups|g(s)−hn(s)|<ε for all n ≥ Ns – see, for instance, Theorem 3.8 in Stokey et al. (1989). As a consequence, the sequence {hn}n∈N converges to the unique function g.19 Finally, there is one additional remark to be made. As, gn converges to g, it follows that Fx[x,gn(x,z),z] converges to Fx[x,g(x,z),z]. Thus, if the constraint vector m(x,y,z) is independent of x, vx,n(x,z) is independent of μn and vx,n(x,z) → Fx[x,g(x,z),z].20 Hence, if convergence of gn is uniform, then v(x,z) is, under these additional conditions, indeed differentiable and its derivative is given by Fx[x,g(x,z),z].21 Thus, as a by‐product, this study provides a simple proof of the differentiability of the limiting value function, v, under state‐independent constraints, such as debt limits.22 1.3. Discussion The Propositions presented above establish an equivalence between value function and time iteration and, as such, neither method has any advantage over the other. It may therefore not be obvious how these ideas may help us in numerically computing dynamic models with occasionally binding constraints. Thus, to appreciate the usefulness of the techniques advocated in this study, it is important to develop an understanding of the challenges involved in numerical solution methods and how time iteration may help to alleviate some of these difficulties. To this end, I proceed in two steps. First, I discuss the challenges involved in numerical dynamic programming, and how Euler equation‐based methods can provide some relief. Second, I briefly discuss various ways of solving the Euler equation, and to which extent time iteration carries some advantages over alternative approaches. 1.3.1. Dynamic programming versus Euler equation‐based methods Many applications of dynamic programming rely on a discretised state and choice space and such a formulation makes any inequality constraint easy to implement. The resulting grid is simply delimited such that any violation of the constraint set is made impossible – see, for instance, Hansen and Imrohoroğlu (1992). However, to achieve any reasonable degree of accuracy, the discretisation must be made on a very fine grid and this causes the procedure to suffer immensely from the curse of dimensionality.23 Parts of these concerns can be avoided by relying on sophisticated approximation, or interpolation, methods. In particular, by defining the value function on a relatively coarse grid, function values in‐between grid points are then approximated and the choice space can be treated as convex (Krusell and Smith, 1998). In the context of the standard neoclassical growth model, Judd and Solnick (1994) showed that approximating the value function with a shape‐preserving spline on a grid consisting of 12 nodes performs as well as a discretisation technique using 1,200 nodes. However, these methods are still no panacea. First, efficient optimisation routines exploit information regarding the slope of the value function, and not its level. An incredibly accurate approximation of the level of the value function can still be orders off track when it comes to the derivative. Second, for an efficient implementation, the approximation procedure is usually confined to a computationally expensive class of methods. For instance, linear interpolations – which are known to be very efficient – create kinks in the value function and discontinuities in its derivative. These discontinuities may ultimately feed into the associated policy function and give rise to misleading results. Alternative approximation methods, such as orthonormal polynomials, are smoother but can on the other hand display pronounced inter‐nodal oscillations. These oscillations can easily translate into quite wild swings in the derivative of the value function (and even alter its sign) and may therefore pave the ground for erroneous or diverging solutions. Thus, the choice of approximation methods should therefore be confined to a relatively expensive class which preserves certain desirable properties of the value function, such as concavity and continuous differentiability. So in what way can Euler equation‐based methods circumvent these issues? Any approximation of the Euler equation – whether it is the policy function itself as in standard projection methods (Judd, 1992), or the entire conditional expectation as in the parameterised expectations algorithm (den Haan and Marcet, 1990) – is tantamount to an approximation of the derivative of the value function. Thus, an accurate approximation of the Euler equation will provide much more precise information of the slope of the value function than would an accurate approximation of its level. Furthermore, simple and efficient approximation methods, as those mentioned above, are much less capable of causing mischief when applied directly on the Euler equation. Linear interpolations, for instance, are not only continuous but also known to preserve monotonicity and positiveness, even in arbitrarily many dimensions (Judd, 1998). A continuous, positive and monotonically increasing approximation of the derivative of the value function is therefore as accurate as a once continuously differentiable and concavity preserving approximation of its level. But the former comes at a much smaller computational cost than the latter and there are, therefore, potentially large efficiency gains to be made. As a final remark in this Section, it appears appropriate to point out that what has been proven is a theoretical equivalence between time and value function iteration and not a practical numerical equivalence. Thus, once numerical approximations replace precise mathematical operations – i.e. the maximisation step and/or function evaluations – the contraction property may be lost. Stachurski (2008) shows how ‘non‐expansive’ numerical approximation methods preserve the contraction property under value function iteration. This result does not immediately carry over to time iteration. 1.3.2. Solution methods based on the Euler equation A substantial share of dynamic economic models can be summarised by a collection of first‐order conditions following xt+1=f(xt,xt+1,xt+2),(8) where xt ∈ X represents the state of the system. A Euler equation‐based solution technique attempts to find a policy function xt+1 = g(xt) such that g(xt)=f{xt,g(xt),g[g(xt)]},(9) for all xt ∈ X. There are several ways to accomplish this. In what follows I briefly discuss three classes of solution methods – one direct approach and two iterative procedures – under which, I believe, most practical implementations can be categorised and I juxtapose their respective advantages. Under the direct approach, the researcher decides on some approximation method for g(x) and finds the associated coefficients such that (9) holds according to some metric.24 This procedure is normally very fast but also quite fragile. In particular, if g(x) is approximated using N coefficients, the direct approach amounts to solving an N‐dimensional system of non‐linear equations. This is a non‐trivial task, in particular in higher dimensions, and convergence can easily fail. Iterative methods can to some extent avoid these issues. Fixed point iteration is initiated by some guess for the policy function, call this gn(x), which is then updated according to25 gn+1(xt)=f{xt,gn(xt),gn[gn(xt)]},(10) until gn+1 does not differ significantly from gn. In many applications, such as the simple case illustrated here, this procedure carries some immediate advantages over the direct approach as the updated guess, gn+1, can be found without resorting to any root‐finding operation at all. The cumbersome system of non‐linear equations is replaced by a simple, but repeated, evaluation of the Euler equation.26 Despite these salient properties there are no guarantees that the sequence of successive guesses obtained under fixed point iteration will eventually converge to the solution g, and oscillating or exploding sequences are frequent.27 Time iteration avoids these convergence issues. In particular, starting with some initial guess, gn(x), the sequence of successive approximations is then given by gn+1(xt)=f{xt,gn+1(xt),gn[gn+1(xt)]}.(11) Following Proposition 2, gn+1 → g, and time iteration is therefore a globally convergent solution method. However, these benefits do not come without costs. As the updated guess, gn+1, appears both in the left and right‐hand side of (11), some root‐finding operation is therefore unavoidable. But, in contrast to the direct approach, the root‐finding operation is confined to repeatedly solving a small‐scale system of non‐linear equations (of dimensionality at most equal to the number of first‐order conditions), instead of solving a much larger system only once.28 And in the one‐dimensional case, the method of endogenous grid points (Carroll, 2006) can avoid any numerical root‐finding operations altogether and each iteration of time iteration is equally fast as that of fixed point iteration but with superior convergence properties. 2. A Numerical Example This Section illustrates the ideas developed in this study by numerically solving a standard stochastic growth model with irreversible investment. The example is partly chosen on the basis that the underlying structure represents a fundamental building block of most modern macroeconomic models and partly as it includes a non‐trivial occasionally binding constraint. In addition, the model is parsimonious enough to be solved very accurately using standard dynamic programming techniques, which will therefore be used as a benchmark for comparisons. It should be emphasised already at this point that this Section is by no means intended as a large‐scale comparison of various solution algorithms.29 Instead, the purpose is to illustrate the applicability of the results derived in the preceding Section and how these ideas can be implemented in practice. As a by‐product of this exercise, however, some numerical results do emerge. 2.1. Model I follow McGrattan (1996) and Christiano and Fisher (2000) and consider a standard stochastic growth model with irreversible investment. As this economy is Pareto optimal, the model can be summarised by the Bellman equation v(k,z)=maxk′∈Γ(k)u[zf(k)+(1−δ)k−k′]+β∫z′∈Zv(k′,z′)Q(z,dz′),(12) with feasibility correspondence Γ(k)=[k′∈K:k′≥(1−δ)k].(13) The function u denotes the momentary utility function and f the production function. Both are assumed to satisfy standard regulatory and Inada conditions. The parameter δ represents the depreciation rate of capital. And the feasibility correspondence in (13) reflects the irreversible nature of investment.30 The state space K belongs to an interval, [k̲,k¯] ⁠, such that 0<k̲<k¯<∞ ⁠, and g(k,z) ∈ int(K). This conjecture will later be numerically verified, but holds quite naturally due to the Inada conditions imposed on u and f. In the numerical implementation, the state space is discretised into N uniformly spaced elements, K^={k1,k2,…,kN} ⁠, and function values in‐between grid points will be found by some approximation routine (see the more detailed description below). The stochastic element z follows a two‐state Markov process, z ∈ Z = {zb,zg}, with law of motion described by transition matrix P. As a consequence, any integral of the type ∫z′∈Zf(z′)Q(z,dz′) is from hereon written as ∑z′∈Zf(z′)Pz,z′ ⁠, where Pz,z′ denotes the probability of the economy transitioning from state z in period t to state z′ in period t + 1. 2.2. Solution Methods I solve the above model using two different solution methods: value function and time iteration. The two following subsections briefly discuss their precise numerical implementations. As any numerical implementation can only deal with the discretised state space K^×Z ⁠, the qualification ‘for (k,z)∈K^×Z’ is considered implied and is therefore omitted. 2.2.1. Value function iteration I initiate the value function iteration algorithm using the conjecture31 v0(k,z)=u[zf(k)](14) and iterate according to the familiar procedure in (2). Following Proposition 11, the optimisation step can be carried out by finding a g~n(k,z) which satisfies the (unrestricted) first‐order condition −u′[zf(k)+(1−δ)k−g~n(k,z)]+β∑z′∈Zvn−1′[g~n(k,z),z′]Pz,z′=0,(15) where vn−1′ refers to the derivative of vn−1 with respect to its first element. The optimal policy choice is then given by gn(k,z)=max[g~n(k,z),(1−δ)k] ⁠. And vn−1(k,z) is updated according to vn(k,z)=u[zf(k)+(1−δ)k−gn(k,z)]+β∑z′∈Zvn−1[gn(k,z),z′]Pz,z′.(16) Outside any k∈K^ ⁠, the value of vn(k,z) is given by a piece‐wise cubic Hermite interpolating polynomial (PCHIP). PCHIP sets the derivative at a ‘knot’ equal to the harmonic mean of the line secants between the knot itself and its two neighbouring knots.32 This procedure preserves monotonicity and prevents overshooting of the interpolant in‐between grid points. This latter property significantly reduces the risk of inter‐nodal oscillations. The unrestricted first‐order condition is solved using Newton's method with an analytically provided Jacobian. The procedure displays no issues of convergence for any parameterisation. The value function iteration algorithm is terminated once ε < 1(−6), where ε=max(k,z)∈K^×Z|ε(k,x)|,(17) and ε(k,z)=−u′[zf(k)+(1−δ)k−g~n(k,z)]+β∑z′∈Zvn′[g~n(k,z),z′]Pz,z′.(18) Lastly, following the discussion in Appendix B, I also consider a version of Howard's improvement algorithm (Howard, 1960). While the precise details of the procedure can be found in the Appendix, the algorithm proceeds similarly to that of value function iteration in finding gn above. But in contrast to merely updating vn−1, it ‘improves’ upon the value function through the iterative scheme vn,h+1(k,z)=u[zf(k)+(1−δ)k−gn(k,z)]+β∑z′∈Zvn,h[gn(k,z),z′]Pz,z′,(19) with vn,h[gn(k,z),z′] = vn−1[gn(k,z),z′], for h = 0. This improvement step is terminated once h = H, or the distance between vn,h+1 and vn,h is small – whichever comes first. The value function is then updated as vn = vn,H and the process repeats itself.33 Apart from significantly enhancing the convergence rate (Santos and Rust, 2003), Howard's improvement algorithm has an intuitive appeal. Given a candidate value and policy function, vn and gn, satisfying (16), the algorithm improves on vn by calculating the value function that emerges if the agent would operate on policy function gn, H number of times. The resulting value, here called vn,H, is then used in place of vn in the subsequent iteration of the standard algorithm and the process repeats itself. The enhanced convergence rate stems from the fact that the (computationally costly) maximisation step is only carried out infrequently, whereas the value function itself is ‘improved’ upon repeatedly. As a rule of thumb, I terminate the improvement step once H = 20 or the sup norm between vn,h+1 and vn,h is less than 1(−6). However, there are some variations across parameterisations in terms of which value of H is optimal and I alter H to attain a procedure as efficient as possible. The improvement algorithm is terminated using the same criterion as that of value function iteration. 2.2.2. Time iteration The time iteration algorithm is initiated using the conjecture v0′(k,z)=zf′(k)u′[zf(k)].(20) Similarly to value function iteration, I find g~n(k,z) as the solution to the (unrestricted) first‐order condition −u′[zf(k)+(1−δ)k−g~n(k,z)]+β∑z′∈Zvn−1′[g~n(k,z),z′]Pz,z′=0.(21) And the optimal policy choice is again given by gn(k,z)=max[g~n(k,z),(1−δ)k] ⁠. The Lagrange multiplier, μn(k,z), is trivially found as the ‘residual’ μn(k,z)=u′[zf(k)+(1−δ)k−gn(k,z)]−β∑z′∈Zvn−1′[gn(k,z),z′]Pz,z′,(22) and the derivative vn−1′ is then updated following Proposition 1 vn′(k,z)=[1+zf′(k)−δ]u′[zf(k)+(1−δ)k−gn(k,z)]−(1−δ)μn(k,z).(23) Outside of any k∈K^ ⁠, the value of vn′(k,z) is given by linear interpolation. In many aspects linear interpolation is a suitable choice of approximation to juxtapose the two solution methods: While PHCIP provides a C1 approximation, linear interpolation is C0. This will give some obvious speed advantages to time iteration, although without any a priori expected loss in accuracy. For the sake of further comparison, however, I will also report the results when vn′(k,z) is approximated using PCHIP. As with value function iteration, the unrestricted first‐order condition is solved using Newton's method with an analytically provided Jacobian. The procedure displays no issues of convergence for any parameterisation. Again, the time iteration algorithm is terminated once ε < 1(−6), where ε is defined in exactly the same way as in (17). I also employ the improvement algorithm discussed in Appendix B. The improvement algorithm proceeds very similarly to that of time iteration in finding gn above. In contrasts to merely updating vn−1′ ⁠, as is done in (23), I ‘improve’ upon the derivative through the iterative scheme vn,h+1′(k,z)=[1+zf′(k)−δ]u′[zf(k)+(1−δ)k−gn(k,z)]+gn′(k,z)×−u′[zf(k)+(1−δ)k−gn(k,z)]+β∑z′∈Zvn,h′[gn(k,z),z′]Pz,z′,(24) with vn,h′[gn(k,z),z′]=vn−1′[gn(k,z),z′] ⁠, for h = 0, and where gn′ denotes the derivative of the policy function gn with respect to its first argument (see Appendix B for a further discussion).34 This improvement step is terminated once h = H, or the distance between vn,h+1′ and vn,h′ is small – whichever occurs first. The derivative of the value function is then updated as vn′=vn,H′ and the process repeats itself.35 The disclaimer discussed in the previous Section in setting H applies here also. But as a rule of thumb, H = 20 performs generally well. Two final remarks ought to be noted. First, whenever H = 1, the second term in (24) is equal to gn′(k,z)×μn(k,z) ⁠. When the irreversibility constraint is binding, μn > 0 and gn′=(1−δ) ⁠, and (24) collapses into (23). Thus, with H = 1, the improvement algorithm is identical to that of time iteration. Second, the time iteration algorithm described above may appear different from that of Definition (7). From a theoretical perspective this difference is merely cosmetic: Substituting (23) into the definition of the multiplier reinstates the procedure in Definition (7). From a practical, or applied, perspective there may, however, be a difference. By approximating the derivative of the value function we avoid approximating both the policy and Lagrange multiplier function. This gives one functional approximation less to worry about. The procedure also has the pedagogical advantage of nesting the improvement algorithm described above. 2.3. Accuracy of Solutions Let q(k,z) denote an arbitrary and feasible policy, such that q(k,z) ∈ Γ(k,z), for all (k,z) ∈ K × Z.36 The fixed point associated with q is then defined as vq(k,z)=u[zf(k)+(1−δ)k−q(k,z)]+β∑z′∈Zvq[q(k,z),z′]Pz,z′.(25) In the case in which q indeed is optimal, q must equal to g, and vq must satisfy the Bellman equation in (12). However, for any two feasible policies q and q^ (which may or may not be optimal), vq(k,z)>vq^(k,z) implies that q(k,z) must be strictly preferred to q^(k,z) ⁠. As a consequence, we may assess the relative desirability – and therefore the relative accuracy – of any two feasible policies by simply comparing their associated fixed points. Evaluating accuracy in this way carries three important properties. First, the relative accuracy between any two arbitrary feasible policies q and q^ can be assessed without any reference to the ‘true’ or ‘optimal’ policy, g. Second, the relative accuracy of a policy is expressed in terms of the actual utility cost associated with its error and not by its distance to the ‘true’ policy function.37 Lastly, the cost of operating under policy q^ relative to policy q can be expressed in terms of consumption equivalents, which has a very natural interpretation. In particular, following Lucas (1987), the value associated with switching from policy q^ to policy q can be expressed in terms of the percent increase in consumption in each period for perpetuity according to η(k,z;q,q^)=lnvq(k,z)−lnvq^(k,z)1−γ×100,(26) where γ denotes the (constant) coefficient of risk aversion.38 To calculate a precise estimate of η, it is important to have a precise estimate of the fixed points vq and vq^ ⁠. To this end, I linearly interpolate a given policy function q:K^×Z→Γ onto a fine grid K¯ containing 1,000,000 nodes such that q:K¯×Z→Γ ⁠. Lastly, I use nearest‐neighbour interpolation to arrive at the policy function q:K¯×Z→Γ¯ ⁠, with Γ¯(k)=[k′∈K¯:k′≥(1−δ)k].(27) As this expanded policy function maps values of the very fine grid K¯×Z onto K¯ there is no need for any additional interpolation methods and a very accurate approximation to the fixed point in (25) is easily computed. I find an approximation by iterating 2,000 times on (25) using the initial guess v0q(k,z)=u[zf(k)]1−β,(k,z)∈K¯×Z.(28) It should be noted that while the above method provides a very precise procedure to assess the relative accuracy of two policies q and q^ ⁠, it does so at the cost of not providing any information about the absolute accuracy of any given policy. To attain the dual objective of assessing both relative as well as absolute accuracy of the suggested solution methods, I also solve the Bellman equation in (12) with Γ replaced by the fine grid Γ¯ and for (k,z)∈K¯×Z (see Appendix C for more details). Considering the large cardinality of K¯ ⁠, I interpret this benchmark solution to be exact.39 If I let g denote the policy function obtained from the ‘exact’ solution and q denote the policy function obtained from any other solution, I report accuracy following the three measures ηmax=max(k,z)∈K¯×Zη(k,z;g,q),ηmin=min(k,z)∈K¯×Zη(k,z;g,q),ηmean=∑(k,z)∈K¯×Zη(k,z;g,q)2,000,000, where, under the hypothesis that g indeed is optimal, η(k, z; g, q) ≥ 0 for all feasible policy function q, and all (k,z)∈K¯×Z ⁠. Lastly, notice that the linearity of η implies that η(k,z;q,q^)=η(k,z;g,q)−η(k,z;g,q^)(29) and that the mean consumption equivalent cost of policy q^ relative to policy q is therefore given by their differences in ηmean.40 2.4. Functional Forms and Parameterisations The momentary utility function belongs to the class of constant relative risk aversion, and the production function is of the standard Cobb–Douglas type u(c)=c1−γ1−γ,f(k)=kα,(30) with u(c) interpreted as ln(c) if γ is equal to 1. The transition matrix P is given by P=1+ρ21−ρ21−ρ21+ρ2,(31) where a typical element, Pi,j, is interpreted as the probability of state j occurring in the subsequent period given that the current state is i; Pr(z′=zj|z=zi) ⁠. The state space for the exogenous state is given as Z = [eσ,e−σ]. Table 1 Parameterisations Model . Parameters . . β . γ . α . δ . σ . ρ . k¯/kss . k¯/kss . (1) 1.03−14 1 0.3 0.02 0.23 0 1.9 0.3 (2) 1.03−14 10 0.3 0.02 0.23 0 3.8 0.005 (3) 1.03−14 1 0.05 0.02 0.0382 0 1.2 0.8 (4) 1.03−14 1 0.3 0.5 0.675 0 3.8 0.3 (5) 1.03−14 1 0.3 0.02 0.23 0.95 1.7 0.6 (6) 1.03−14 1 0.3 0.02 0.4 0 2.3 0.2 (7) 1.03−14 10 0.1 0.02 0.23 0.95 5.9 0.4 Model . Parameters . . β . γ . α . δ . σ . ρ . k¯/kss . k¯/kss . (1) 1.03−14 1 0.3 0.02 0.23 0 1.9 0.3 (2) 1.03−14 10 0.3 0.02 0.23 0 3.8 0.005 (3) 1.03−14 1 0.05 0.02 0.0382 0 1.2 0.8 (4) 1.03−14 1 0.3 0.5 0.675 0 3.8 0.3 (5) 1.03−14 1 0.3 0.02 0.23 0.95 1.7 0.6 (6) 1.03−14 1 0.3 0.02 0.4 0 2.3 0.2 (7) 1.03−14 10 0.1 0.02 0.23 0.95 5.9 0.4 Open in new tab Table 1 Parameterisations Model . Parameters . . β . γ . α . δ . σ . ρ . k¯/kss . k¯/kss . (1) 1.03−14 1 0.3 0.02 0.23 0 1.9 0.3 (2) 1.03−14 10 0.3 0.02 0.23 0 3.8 0.005 (3) 1.03−14 1 0.05 0.02 0.0382 0 1.2 0.8 (4) 1.03−14 1 0.3 0.5 0.675 0 3.8 0.3 (5) 1.03−14 1 0.3 0.02 0.23 0.95 1.7 0.6 (6) 1.03−14 1 0.3 0.02 0.4 0 2.3 0.2 (7) 1.03−14 10 0.1 0.02 0.23 0.95 5.9 0.4 Model . Parameters . . β . γ . α . δ . σ . ρ . k¯/kss . k¯/kss . (1) 1.03−14 1 0.3 0.02 0.23 0 1.9 0.3 (2) 1.03−14 10 0.3 0.02 0.23 0 3.8 0.005 (3) 1.03−14 1 0.05 0.02 0.0382 0 1.2 0.8 (4) 1.03−14 1 0.3 0.5 0.675 0 3.8 0.3 (5) 1.03−14 1 0.3 0.02 0.23 0.95 1.7 0.6 (6) 1.03−14 1 0.3 0.02 0.4 0 2.3 0.2 (7) 1.03−14 10 0.1 0.02 0.23 0.95 5.9 0.4 Open in new tab Table 1 summarises the various sets of parameterisations considered, together with the associated bounds on the endogenous state space, [k̲,k¯] ⁠. The parameterisations follow those of Christiano and Fisher (2000). I solve the models (1)–(7) using the two algorithms described above. The grid for capital contains N = 10, N = 100 and N = 1,000 equidistant nodes. 2.5. Numerical Results Tables 2 and 3 display the main results of the numerical exercise. The first three cells of each cluster provide the maximum, minimum and mean error respectively. The fourth cell provides the computation time in seconds, with and without the improvement algorithm (with the latter in parenthesis). The fifth cell provides the number of iterations executed until convergence, again with and without the improvement algorithm. If we start by comparing value function iteration with time iteration using linear interpolation, a few noticeable features emerge. Time iteration is as accurate as value function iteration for all parameterisations except for models (2) and (5), and is also markedly faster.41 It is an order of magnitude more accurate for model (4) at N = 10, but performs slightly worse across the board for model (5) – although with a sustained speed advantage.42 With respect to model (2), time iteration performs very poorly at small values of N, but remains roughly on par with value function iteration at higher values. However, even value function iteration performs intolerably badly for this particular model at the coarser grids, with the maximum error peaking at a striking 4.5%. There are two reasons underlying these inaccuracies. First, the ergodic set of capital for model (2) covers a very wide interval, ranging from 0.5% of the steady‐state level of capital, to 380%. A grid consisting of 10 nodes is therefore arguably far too coarse for any reasonable degree of accuracy. Second, a coefficient of risk aversion of 10 brings forth some significant non‐linearities in the model which appear difficult to approximate. These non‐linearities seem to cause more serious complications for the derivative of the value function – which is heavily exploited in time iteration – than they do for its level. At least at the very small values of N. Table 2 Comparison of Solution Methods, Models (1)–(4) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (1) Maximum error 5.2E‐3 3.7E‐5 3.7E‐5 6.2E‐3 3.7E‐5 3.7E‐5 4.1E‐4 3.7E‐5 3.7E‐5 Minimum error 1.3E‐3 1.7E‐6 1.6E‐6 1.7E‐3 2.9E‐6 1.6E‐6 8.4E‐5 1.7E‐6 1.6E‐6 Mean error 2.6E‐3 1.4E‐5 1.4E‐5 2.4E‐3 1.5E‐5 1.4E‐5 1.3E‐4 1.4E‐5 1.4E‐5 CPU time 1.35 1.32 1.80 0.95 0.92 1.22 1.03 1.12 1.55 (5.76) (5.89) (8.02) (3.37) (3.49) (5.17) (4.08) (4.03) (5.83) Iterations 14 17 17 9 9 9 8 9 9 (371) (361) (361) (287) (280) (280) (278) (280) (280) Model (2) Maximum error 4.5 2.8 8.2E‐2 36.3 2.85 3.0E‐3 21.3 9.6E‐1 3.6E‐3 Minimum error 2.0E‐2 2.8E‐6 9.9E‐7 7.5E‐1 1.4E‐3 1.0E‐6 9.0E‐4 1.0E‐6 9.9E‐7 Mean error 5.3E‐1 3.4E‐3 5.4E‐5 1.3 4.2E‐3 4.1E‐5 1.2E‐1 7.5E‐4 4.1E‐5 CPU time 3.56 2.77 2.43 2.19 1.71 2.71 2.09 2.13 3.8 (19.9) (28.9) (9.08) (8.05) (7.11) (11.1) (6.68) (7.18) (11.88) Iterations 28 20 19 36 20 19 24 19 19 (578) (668) (625) (773) (636) (625) (622) (625) (625) Model (3) Maximum error 2.7E‐7 7.3E‐8 7.4E‐8 2.5E‐6 7.3E‐8 7.5E‐8 2.2E‐7 7.3E‐8 7.5E‐8 Minimum error 2.3E‐7 2.6E‐8 2.7E‐8 2.2E‐2 2.6E‐8 2.7E‐8 1.8E‐7 2.6E‐8 2.7E‐8 Mean error 2.5E‐7 3.4E‐8 3.5E‐8 2.3E‐6 3.4E‐8 3.5E‐8 1.9E‐7 3.4E‐8 3.5E‐8 CPU time 0.87 0.89 2.76 0.53 0.56 2.47 0.62 0.70 2.82 (1.89) (2.04) (2.76) (1.71) (1.78) (2.47) (1.99) (2.02) (2.82) Iterations 9 10 135 10 10 129 10 13 129 (134) (135) (135)* (130) (129) (129)* (128) (129) (129)* Model (4) Maximum error 5.9E‐2 4.4E‐6 2.4E‐6 2.1E‐3 2.6E‐6 2.4E‐6 1.8E‐3 2.5E‐6 2.4E‐6 Minimum error 4.6E‐2 3.0E‐6 2.0E‐6 1.9E‐3 2.2E‐6 2.0E‐6 1.6E‐3 2.2E‐6 2.0E‐6 Mean error 4.7E‐2 3.1E‐6 2.1E‐6 1.9E‐3 2.3E‐6 2.1E‐6 1.7E‐3 2.3E‐6 2.1E‐6 CPU time 0.22 0.29 0.59 0.20 0.41 0.55 0.21 0.42 0.62 (0.50) (0.48) (0.59) (0.39) (0.39) (0.55) (0.41) (0.42) (0.62) Iterations 7 7 27 8 11 28 7 28 28 (29) (27) (27)* (28) (28) (28)* (28) (28)* (28)* Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (1) Maximum error 5.2E‐3 3.7E‐5 3.7E‐5 6.2E‐3 3.7E‐5 3.7E‐5 4.1E‐4 3.7E‐5 3.7E‐5 Minimum error 1.3E‐3 1.7E‐6 1.6E‐6 1.7E‐3 2.9E‐6 1.6E‐6 8.4E‐5 1.7E‐6 1.6E‐6 Mean error 2.6E‐3 1.4E‐5 1.4E‐5 2.4E‐3 1.5E‐5 1.4E‐5 1.3E‐4 1.4E‐5 1.4E‐5 CPU time 1.35 1.32 1.80 0.95 0.92 1.22 1.03 1.12 1.55 (5.76) (5.89) (8.02) (3.37) (3.49) (5.17) (4.08) (4.03) (5.83) Iterations 14 17 17 9 9 9 8 9 9 (371) (361) (361) (287) (280) (280) (278) (280) (280) Model (2) Maximum error 4.5 2.8 8.2E‐2 36.3 2.85 3.0E‐3 21.3 9.6E‐1 3.6E‐3 Minimum error 2.0E‐2 2.8E‐6 9.9E‐7 7.5E‐1 1.4E‐3 1.0E‐6 9.0E‐4 1.0E‐6 9.9E‐7 Mean error 5.3E‐1 3.4E‐3 5.4E‐5 1.3 4.2E‐3 4.1E‐5 1.2E‐1 7.5E‐4 4.1E‐5 CPU time 3.56 2.77 2.43 2.19 1.71 2.71 2.09 2.13 3.8 (19.9) (28.9) (9.08) (8.05) (7.11) (11.1) (6.68) (7.18) (11.88) Iterations 28 20 19 36 20 19 24 19 19 (578) (668) (625) (773) (636) (625) (622) (625) (625) Model (3) Maximum error 2.7E‐7 7.3E‐8 7.4E‐8 2.5E‐6 7.3E‐8 7.5E‐8 2.2E‐7 7.3E‐8 7.5E‐8 Minimum error 2.3E‐7 2.6E‐8 2.7E‐8 2.2E‐2 2.6E‐8 2.7E‐8 1.8E‐7 2.6E‐8 2.7E‐8 Mean error 2.5E‐7 3.4E‐8 3.5E‐8 2.3E‐6 3.4E‐8 3.5E‐8 1.9E‐7 3.4E‐8 3.5E‐8 CPU time 0.87 0.89 2.76 0.53 0.56 2.47 0.62 0.70 2.82 (1.89) (2.04) (2.76) (1.71) (1.78) (2.47) (1.99) (2.02) (2.82) Iterations 9 10 135 10 10 129 10 13 129 (134) (135) (135)* (130) (129) (129)* (128) (129) (129)* Model (4) Maximum error 5.9E‐2 4.4E‐6 2.4E‐6 2.1E‐3 2.6E‐6 2.4E‐6 1.8E‐3 2.5E‐6 2.4E‐6 Minimum error 4.6E‐2 3.0E‐6 2.0E‐6 1.9E‐3 2.2E‐6 2.0E‐6 1.6E‐3 2.2E‐6 2.0E‐6 Mean error 4.7E‐2 3.1E‐6 2.1E‐6 1.9E‐3 2.3E‐6 2.1E‐6 1.7E‐3 2.3E‐6 2.1E‐6 CPU time 0.22 0.29 0.59 0.20 0.41 0.55 0.21 0.42 0.62 (0.50) (0.48) (0.59) (0.39) (0.39) (0.55) (0.41) (0.42) (0.62) Iterations 7 7 27 8 11 28 7 28 28 (29) (27) (27)* (28) (28) (28)* (28) (28)* (28)* Notes *For models (3) and (4), the improvement algorithm did not converge for N = 1,000 for either algorithm. Similar difficulties were encountered for model (4) using time iteration (PCHIP) with N = 100. Open in new tab Table 2 Comparison of Solution Methods, Models (1)–(4) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (1) Maximum error 5.2E‐3 3.7E‐5 3.7E‐5 6.2E‐3 3.7E‐5 3.7E‐5 4.1E‐4 3.7E‐5 3.7E‐5 Minimum error 1.3E‐3 1.7E‐6 1.6E‐6 1.7E‐3 2.9E‐6 1.6E‐6 8.4E‐5 1.7E‐6 1.6E‐6 Mean error 2.6E‐3 1.4E‐5 1.4E‐5 2.4E‐3 1.5E‐5 1.4E‐5 1.3E‐4 1.4E‐5 1.4E‐5 CPU time 1.35 1.32 1.80 0.95 0.92 1.22 1.03 1.12 1.55 (5.76) (5.89) (8.02) (3.37) (3.49) (5.17) (4.08) (4.03) (5.83) Iterations 14 17 17 9 9 9 8 9 9 (371) (361) (361) (287) (280) (280) (278) (280) (280) Model (2) Maximum error 4.5 2.8 8.2E‐2 36.3 2.85 3.0E‐3 21.3 9.6E‐1 3.6E‐3 Minimum error 2.0E‐2 2.8E‐6 9.9E‐7 7.5E‐1 1.4E‐3 1.0E‐6 9.0E‐4 1.0E‐6 9.9E‐7 Mean error 5.3E‐1 3.4E‐3 5.4E‐5 1.3 4.2E‐3 4.1E‐5 1.2E‐1 7.5E‐4 4.1E‐5 CPU time 3.56 2.77 2.43 2.19 1.71 2.71 2.09 2.13 3.8 (19.9) (28.9) (9.08) (8.05) (7.11) (11.1) (6.68) (7.18) (11.88) Iterations 28 20 19 36 20 19 24 19 19 (578) (668) (625) (773) (636) (625) (622) (625) (625) Model (3) Maximum error 2.7E‐7 7.3E‐8 7.4E‐8 2.5E‐6 7.3E‐8 7.5E‐8 2.2E‐7 7.3E‐8 7.5E‐8 Minimum error 2.3E‐7 2.6E‐8 2.7E‐8 2.2E‐2 2.6E‐8 2.7E‐8 1.8E‐7 2.6E‐8 2.7E‐8 Mean error 2.5E‐7 3.4E‐8 3.5E‐8 2.3E‐6 3.4E‐8 3.5E‐8 1.9E‐7 3.4E‐8 3.5E‐8 CPU time 0.87 0.89 2.76 0.53 0.56 2.47 0.62 0.70 2.82 (1.89) (2.04) (2.76) (1.71) (1.78) (2.47) (1.99) (2.02) (2.82) Iterations 9 10 135 10 10 129 10 13 129 (134) (135) (135)* (130) (129) (129)* (128) (129) (129)* Model (4) Maximum error 5.9E‐2 4.4E‐6 2.4E‐6 2.1E‐3 2.6E‐6 2.4E‐6 1.8E‐3 2.5E‐6 2.4E‐6 Minimum error 4.6E‐2 3.0E‐6 2.0E‐6 1.9E‐3 2.2E‐6 2.0E‐6 1.6E‐3 2.2E‐6 2.0E‐6 Mean error 4.7E‐2 3.1E‐6 2.1E‐6 1.9E‐3 2.3E‐6 2.1E‐6 1.7E‐3 2.3E‐6 2.1E‐6 CPU time 0.22 0.29 0.59 0.20 0.41 0.55 0.21 0.42 0.62 (0.50) (0.48) (0.59) (0.39) (0.39) (0.55) (0.41) (0.42) (0.62) Iterations 7 7 27 8 11 28 7 28 28 (29) (27) (27)* (28) (28) (28)* (28) (28)* (28)* Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (1) Maximum error 5.2E‐3 3.7E‐5 3.7E‐5 6.2E‐3 3.7E‐5 3.7E‐5 4.1E‐4 3.7E‐5 3.7E‐5 Minimum error 1.3E‐3 1.7E‐6 1.6E‐6 1.7E‐3 2.9E‐6 1.6E‐6 8.4E‐5 1.7E‐6 1.6E‐6 Mean error 2.6E‐3 1.4E‐5 1.4E‐5 2.4E‐3 1.5E‐5 1.4E‐5 1.3E‐4 1.4E‐5 1.4E‐5 CPU time 1.35 1.32 1.80 0.95 0.92 1.22 1.03 1.12 1.55 (5.76) (5.89) (8.02) (3.37) (3.49) (5.17) (4.08) (4.03) (5.83) Iterations 14 17 17 9 9 9 8 9 9 (371) (361) (361) (287) (280) (280) (278) (280) (280) Model (2) Maximum error 4.5 2.8 8.2E‐2 36.3 2.85 3.0E‐3 21.3 9.6E‐1 3.6E‐3 Minimum error 2.0E‐2 2.8E‐6 9.9E‐7 7.5E‐1 1.4E‐3 1.0E‐6 9.0E‐4 1.0E‐6 9.9E‐7 Mean error 5.3E‐1 3.4E‐3 5.4E‐5 1.3 4.2E‐3 4.1E‐5 1.2E‐1 7.5E‐4 4.1E‐5 CPU time 3.56 2.77 2.43 2.19 1.71 2.71 2.09 2.13 3.8 (19.9) (28.9) (9.08) (8.05) (7.11) (11.1) (6.68) (7.18) (11.88) Iterations 28 20 19 36 20 19 24 19 19 (578) (668) (625) (773) (636) (625) (622) (625) (625) Model (3) Maximum error 2.7E‐7 7.3E‐8 7.4E‐8 2.5E‐6 7.3E‐8 7.5E‐8 2.2E‐7 7.3E‐8 7.5E‐8 Minimum error 2.3E‐7 2.6E‐8 2.7E‐8 2.2E‐2 2.6E‐8 2.7E‐8 1.8E‐7 2.6E‐8 2.7E‐8 Mean error 2.5E‐7 3.4E‐8 3.5E‐8 2.3E‐6 3.4E‐8 3.5E‐8 1.9E‐7 3.4E‐8 3.5E‐8 CPU time 0.87 0.89 2.76 0.53 0.56 2.47 0.62 0.70 2.82 (1.89) (2.04) (2.76) (1.71) (1.78) (2.47) (1.99) (2.02) (2.82) Iterations 9 10 135 10 10 129 10 13 129 (134) (135) (135)* (130) (129) (129)* (128) (129) (129)* Model (4) Maximum error 5.9E‐2 4.4E‐6 2.4E‐6 2.1E‐3 2.6E‐6 2.4E‐6 1.8E‐3 2.5E‐6 2.4E‐6 Minimum error 4.6E‐2 3.0E‐6 2.0E‐6 1.9E‐3 2.2E‐6 2.0E‐6 1.6E‐3 2.2E‐6 2.0E‐6 Mean error 4.7E‐2 3.1E‐6 2.1E‐6 1.9E‐3 2.3E‐6 2.1E‐6 1.7E‐3 2.3E‐6 2.1E‐6 CPU time 0.22 0.29 0.59 0.20 0.41 0.55 0.21 0.42 0.62 (0.50) (0.48) (0.59) (0.39) (0.39) (0.55) (0.41) (0.42) (0.62) Iterations 7 7 27 8 11 28 7 28 28 (29) (27) (27)* (28) (28) (28)* (28) (28)* (28)* Notes *For models (3) and (4), the improvement algorithm did not converge for N = 1,000 for either algorithm. Similar difficulties were encountered for model (4) using time iteration (PCHIP) with N = 100. Open in new tab Table 3 Comparison of Solution Methods, Models (5)–(7) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 8.4E‐7 9.7E‐7 3.7E‐4 1.0E‐6 9.0E‐7 1.7E‐6 8.5E‐7 1.1E‐6 Minimum error 6.5E‐5 8.7E‐8 8.3E‐8 2.5E‐4 2.0E‐7 1.1E‐7 6.9E‐7 8.4E‐8 1.9E‐7 Mean error 1.1E‐4 1.6E‐7 1.6E‐7 3.2E‐4 3.0E‐7 1.9E‐7 1.0E‐6 1.6E‐7 2.9E‐7 CPU time 1.15 1.16 6.34 0.76 0.89 2.9 0.93 1.07 4.96 (4.87) (5.08) (6.34) (3.20) (3.26) (4.59) (3.49) (3.69) (4.96) Iterations 13 13 312 10 11 135 10 11 251 (314) (312) (312)* (256) (251) (251) (251) (252) (251)* Model (6) Maximum error 2.2E‐3 1.4E‐4 1.4E‐4 8.2E‐3 1.4E‐4 1.4E‐4 6.6E‐4 1.4E‐4 1.4E‐4 Minimum error 1.2E‐4 9.4E‐5 9.4E‐5 1.9E‐3 9.6E‐5 9.4E‐5 1.2E‐4 9.4E‐5 9.4E‐5 Mean error 2.8E‐4 1.2E‐4 1.2E‐4 2.9E‐3 1.2E‐4 1.2E‐4 1.7E‐4 1.2E‐4 1.2E‐4 CPU time 1.30 1.35 1.75 1.05 0.96 1.25 1.14 1.18 1.62 (5.37) (5.76) (7.69) (3.52) (3.56) (5.09) (3.83) (4.10) (5.9) Iterations 10 10 10 10 9 9 9 9 9 (368) (368) (368) (308) (294) (294) (293) (294) (294) Model (7) Maximum error 2.5E‐1 5.7E‐4 3.0E‐5 1.1E‐1 3.5E‐4 3.0E‐5 2.7E‐3 3.0E‐5 3.0E‐5 Minimum error 5.4E‐2 1.6E‐5 2.3E‐6 4.7E‐2 1.1E‐4 2.3E‐6 1.2E‐3 2.4E‐6 2.3E‐6 Mean error 1.1E‐1 2.2E‐5 1.0E‐5 9.7E‐2 2.5E‐4 1.0E‐5 1.8E‐3 1.0E‐5 1.0E‐5 CPU time 1.45 1.42 1.84 1.31 1.31 2.07 1.39 1.58 2.52 (3.93) (4.03) (5.79) (4.56) (4.6) (7.33) (4.36) (5.07) (7.73) Iterations 20 19 19 24 22 22 18 19 19 (399) (387) (386) (422) (386) (385) (379) (385) (385) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 8.4E‐7 9.7E‐7 3.7E‐4 1.0E‐6 9.0E‐7 1.7E‐6 8.5E‐7 1.1E‐6 Minimum error 6.5E‐5 8.7E‐8 8.3E‐8 2.5E‐4 2.0E‐7 1.1E‐7 6.9E‐7 8.4E‐8 1.9E‐7 Mean error 1.1E‐4 1.6E‐7 1.6E‐7 3.2E‐4 3.0E‐7 1.9E‐7 1.0E‐6 1.6E‐7 2.9E‐7 CPU time 1.15 1.16 6.34 0.76 0.89 2.9 0.93 1.07 4.96 (4.87) (5.08) (6.34) (3.20) (3.26) (4.59) (3.49) (3.69) (4.96) Iterations 13 13 312 10 11 135 10 11 251 (314) (312) (312)* (256) (251) (251) (251) (252) (251)* Model (6) Maximum error 2.2E‐3 1.4E‐4 1.4E‐4 8.2E‐3 1.4E‐4 1.4E‐4 6.6E‐4 1.4E‐4 1.4E‐4 Minimum error 1.2E‐4 9.4E‐5 9.4E‐5 1.9E‐3 9.6E‐5 9.4E‐5 1.2E‐4 9.4E‐5 9.4E‐5 Mean error 2.8E‐4 1.2E‐4 1.2E‐4 2.9E‐3 1.2E‐4 1.2E‐4 1.7E‐4 1.2E‐4 1.2E‐4 CPU time 1.30 1.35 1.75 1.05 0.96 1.25 1.14 1.18 1.62 (5.37) (5.76) (7.69) (3.52) (3.56) (5.09) (3.83) (4.10) (5.9) Iterations 10 10 10 10 9 9 9 9 9 (368) (368) (368) (308) (294) (294) (293) (294) (294) Model (7) Maximum error 2.5E‐1 5.7E‐4 3.0E‐5 1.1E‐1 3.5E‐4 3.0E‐5 2.7E‐3 3.0E‐5 3.0E‐5 Minimum error 5.4E‐2 1.6E‐5 2.3E‐6 4.7E‐2 1.1E‐4 2.3E‐6 1.2E‐3 2.4E‐6 2.3E‐6 Mean error 1.1E‐1 2.2E‐5 1.0E‐5 9.7E‐2 2.5E‐4 1.0E‐5 1.8E‐3 1.0E‐5 1.0E‐5 CPU time 1.45 1.42 1.84 1.31 1.31 2.07 1.39 1.58 2.52 (3.93) (4.03) (5.79) (4.56) (4.6) (7.33) (4.36) (5.07) (7.73) Iterations 20 19 19 24 22 22 18 19 19 (399) (387) (386) (422) (386) (385) (379) (385) (385) Notes *For model (5), the improvement algorithm did not converge for N = 1,000 for either algorithm. Similar difficulties were encountered for model (4) using time iteration (PCHIP) with N = 100. Open in new tab Table 3 Comparison of Solution Methods, Models (5)–(7) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 8.4E‐7 9.7E‐7 3.7E‐4 1.0E‐6 9.0E‐7 1.7E‐6 8.5E‐7 1.1E‐6 Minimum error 6.5E‐5 8.7E‐8 8.3E‐8 2.5E‐4 2.0E‐7 1.1E‐7 6.9E‐7 8.4E‐8 1.9E‐7 Mean error 1.1E‐4 1.6E‐7 1.6E‐7 3.2E‐4 3.0E‐7 1.9E‐7 1.0E‐6 1.6E‐7 2.9E‐7 CPU time 1.15 1.16 6.34 0.76 0.89 2.9 0.93 1.07 4.96 (4.87) (5.08) (6.34) (3.20) (3.26) (4.59) (3.49) (3.69) (4.96) Iterations 13 13 312 10 11 135 10 11 251 (314) (312) (312)* (256) (251) (251) (251) (252) (251)* Model (6) Maximum error 2.2E‐3 1.4E‐4 1.4E‐4 8.2E‐3 1.4E‐4 1.4E‐4 6.6E‐4 1.4E‐4 1.4E‐4 Minimum error 1.2E‐4 9.4E‐5 9.4E‐5 1.9E‐3 9.6E‐5 9.4E‐5 1.2E‐4 9.4E‐5 9.4E‐5 Mean error 2.8E‐4 1.2E‐4 1.2E‐4 2.9E‐3 1.2E‐4 1.2E‐4 1.7E‐4 1.2E‐4 1.2E‐4 CPU time 1.30 1.35 1.75 1.05 0.96 1.25 1.14 1.18 1.62 (5.37) (5.76) (7.69) (3.52) (3.56) (5.09) (3.83) (4.10) (5.9) Iterations 10 10 10 10 9 9 9 9 9 (368) (368) (368) (308) (294) (294) (293) (294) (294) Model (7) Maximum error 2.5E‐1 5.7E‐4 3.0E‐5 1.1E‐1 3.5E‐4 3.0E‐5 2.7E‐3 3.0E‐5 3.0E‐5 Minimum error 5.4E‐2 1.6E‐5 2.3E‐6 4.7E‐2 1.1E‐4 2.3E‐6 1.2E‐3 2.4E‐6 2.3E‐6 Mean error 1.1E‐1 2.2E‐5 1.0E‐5 9.7E‐2 2.5E‐4 1.0E‐5 1.8E‐3 1.0E‐5 1.0E‐5 CPU time 1.45 1.42 1.84 1.31 1.31 2.07 1.39 1.58 2.52 (3.93) (4.03) (5.79) (4.56) (4.6) (7.33) (4.36) (5.07) (7.73) Iterations 20 19 19 24 22 22 18 19 19 (399) (387) (386) (422) (386) (385) (379) (385) (385) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 8.4E‐7 9.7E‐7 3.7E‐4 1.0E‐6 9.0E‐7 1.7E‐6 8.5E‐7 1.1E‐6 Minimum error 6.5E‐5 8.7E‐8 8.3E‐8 2.5E‐4 2.0E‐7 1.1E‐7 6.9E‐7 8.4E‐8 1.9E‐7 Mean error 1.1E‐4 1.6E‐7 1.6E‐7 3.2E‐4 3.0E‐7 1.9E‐7 1.0E‐6 1.6E‐7 2.9E‐7 CPU time 1.15 1.16 6.34 0.76 0.89 2.9 0.93 1.07 4.96 (4.87) (5.08) (6.34) (3.20) (3.26) (4.59) (3.49) (3.69) (4.96) Iterations 13 13 312 10 11 135 10 11 251 (314) (312) (312)* (256) (251) (251) (251) (252) (251)* Model (6) Maximum error 2.2E‐3 1.4E‐4 1.4E‐4 8.2E‐3 1.4E‐4 1.4E‐4 6.6E‐4 1.4E‐4 1.4E‐4 Minimum error 1.2E‐4 9.4E‐5 9.4E‐5 1.9E‐3 9.6E‐5 9.4E‐5 1.2E‐4 9.4E‐5 9.4E‐5 Mean error 2.8E‐4 1.2E‐4 1.2E‐4 2.9E‐3 1.2E‐4 1.2E‐4 1.7E‐4 1.2E‐4 1.2E‐4 CPU time 1.30 1.35 1.75 1.05 0.96 1.25 1.14 1.18 1.62 (5.37) (5.76) (7.69) (3.52) (3.56) (5.09) (3.83) (4.10) (5.9) Iterations 10 10 10 10 9 9 9 9 9 (368) (368) (368) (308) (294) (294) (293) (294) (294) Model (7) Maximum error 2.5E‐1 5.7E‐4 3.0E‐5 1.1E‐1 3.5E‐4 3.0E‐5 2.7E‐3 3.0E‐5 3.0E‐5 Minimum error 5.4E‐2 1.6E‐5 2.3E‐6 4.7E‐2 1.1E‐4 2.3E‐6 1.2E‐3 2.4E‐6 2.3E‐6 Mean error 1.1E‐1 2.2E‐5 1.0E‐5 9.7E‐2 2.5E‐4 1.0E‐5 1.8E‐3 1.0E‐5 1.0E‐5 CPU time 1.45 1.42 1.84 1.31 1.31 2.07 1.39 1.58 2.52 (3.93) (4.03) (5.79) (4.56) (4.6) (7.33) (4.36) (5.07) (7.73) Iterations 20 19 19 24 22 22 18 19 19 (399) (387) (386) (422) (386) (385) (379) (385) (385) Notes *For model (5), the improvement algorithm did not converge for N = 1,000 for either algorithm. Similar difficulties were encountered for model (4) using time iteration (PCHIP) with N = 100. Open in new tab When piece‐wise cubic Hermite interpolation replaces linear interpolation in time iteration, some of these results change. For N = 10, time iteration consistently outperforms value function iteration – roughly by order one or two – for all parameterisations apart from model (2). It performs no worse than value function iteration at the finer grids, apart from, again, model (2) where it consistently, although moderately, actually performs better. For model (7) time iteration is roughly an order more accurate than value function iteration when N is equal to 100. These gains in accuracy associated with Hermite interpolation are non‐trivial. But they do also come at the expense of computational speed. Piece‐wise cubic Hermite interpolation is slower than linear interpolation and computation time, for time iteration increases across all models and for all grids (with N = 10 for model (2) being the only exception). However, even under this more demanding interpolation scheme, time iteration still remains faster than value function iteration for all models apart from (4) and (7), and for almost all grid sizes (N = 1,000 for models (2) and (3) is the exception). In general, time iteration is faster than value function iteration in 15 of the 21 possible cases, and equally or more accurate in 20. The speed advantage of time iteration relative to value function iteration under this interpolation scheme stems from the superior performance of the non‐linear equation solver: PCHIP renders a continuous (albeit kinked) Jacobian for time iteration but imposes some challenging discontinuities for the Jacobian associated with value function iteration.43 It is therefore unclear whether these speed gains would remain in models that rely on search‐based, rather than gradient‐based solution algorithms. Lastly, an interesting, although disconcerting, observation is that accuracy barely changes as N increases from 100 to 1,000. With the notable exception of model (2), this pattern emerges across all parameterisations and all solution methods.44 I can see two possible explanations of this phenomenon. First, there might be something inherent in the numerical implementations which prevents them from ever approaching the exact solution, even as the mesh size approaches zero. This would be a serious concern. Alternatively, and second, the ‘exact’ solution may be further from the truth than initially thought – or at least further from the truth than the alternative solution methods. To see how the latter explanation could generate the above observation, recall that the policy functions obtained have been interpolated and discretised onto the very same grid that defines the ‘exact’ solution. Thus, any measure of accuracy – i.e. the cost of operating on any policy other than the ‘exact policy’ – will, by construction, always be positive, and the ‘exact policy’ will always outperform any other alternative. But if the ‘exact’ solution is not exact enough, any other policy that is actually closer to the truth will appear inferior, even when it is not. To discriminate between these two hypotheses I interpolate and discretise all policy functions (including the ‘exact’) on an even finer grid consisting of 10,000,000 nodes in capital. I subsequently calculate the associated fixed points as described in subsection 2.3, and re‐calculate the associated accuracy measures. The results for the parameterisations where the problem is the most pronounced are provided in Table 4. Perhaps unsurprisingly the relative performance of the solutions is largely unchanged. Time iteration performs roughly as well as value function iteration for N = 100 and N = 1,000 irrespective of interpolation method. Linear interpolation loses somewhat in relative accuracy to value function iteration at N = 10, whereas Hermite interpolation instead makes some headway. But more importantly, the absolute performances of the solutions have changed quite dramatically. Both value function iteration and time iteration deliver a more accurate solution using a grid consisting of 100 nodes than discretised value function iteration on a grid using 1,000,000 nodes. I have repeated this exercise for all parameterisations in Appendix D with very similar results: Value function iteration and time iteration using linear interpolation outperform the discretisation algorithm in four of seven cases when N = 100, and in six of seven cases when PCHIP is used. While these numerical results are by no means a centrepiece of the analysis in this study, they are quite surprising and suggest that the grid size must be much larger than what has been previously used in the literature when using discretisation methods as a benchmark for accuracy.45 Table 4 Comparison of Different Solution Methods, Models (5) and (6) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 −1.8E‐7 −1.9E‐7 3.7E‐4 −2.1E‐8 −1.9E‐7 1.3E‐6 −1.9E‐7 −1.9E‐7 Minimum error 6.4E‐5 −3.8E‐6 −3.8E‐6 2.5E‐4 −3.7E‐7 −3.8E‐6 −3.1E‐6 −3.8E‐6 −3.8E‐6 Mean error 1.1E‐4 −5.0E‐7 −5.0E‐7 3.2E‐4 −3.6E‐7 −5.0E‐7 3.7E‐7 −5.0E‐7 −5.0E‐7 Model (6) Maximum error 2.1E‐3 −6.1E‐5 −6.2E‐4 8.0E‐3 −6.0E‐5 −6.2E‐5 5.0E‐4 −6.2E‐5 −6.2E‐5 Minimum error −9.5E‐5 −9.9E‐5 −9.9E‐5 1.7E‐3 −9.9E‐5 −9.9E‐5 −9.5E‐5 −9.9E‐5 −9.9E‐5 Mean error 8.0E‐5 −7.9E‐5 −7.9E‐5 2.7E‐3 −7.8E‐5 −7.9E‐5 −3.0E‐5 −7.9E‐5 −7.9E‐5 Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 −1.8E‐7 −1.9E‐7 3.7E‐4 −2.1E‐8 −1.9E‐7 1.3E‐6 −1.9E‐7 −1.9E‐7 Minimum error 6.4E‐5 −3.8E‐6 −3.8E‐6 2.5E‐4 −3.7E‐7 −3.8E‐6 −3.1E‐6 −3.8E‐6 −3.8E‐6 Mean error 1.1E‐4 −5.0E‐7 −5.0E‐7 3.2E‐4 −3.6E‐7 −5.0E‐7 3.7E‐7 −5.0E‐7 −5.0E‐7 Model (6) Maximum error 2.1E‐3 −6.1E‐5 −6.2E‐4 8.0E‐3 −6.0E‐5 −6.2E‐5 5.0E‐4 −6.2E‐5 −6.2E‐5 Minimum error −9.5E‐5 −9.9E‐5 −9.9E‐5 1.7E‐3 −9.9E‐5 −9.9E‐5 −9.5E‐5 −9.9E‐5 −9.9E‐5 Mean error 8.0E‐5 −7.9E‐5 −7.9E‐5 2.7E‐3 −7.8E‐5 −7.9E‐5 −3.0E‐5 −7.9E‐5 −7.9E‐5 Notes This Table does not include the computational time nor the number of iterations as these are identical to those in Table 3. The disclaimer that applies for Tables 2 and 3 applies here also. Open in new tab Table 4 Comparison of Different Solution Methods, Models (5) and (6) Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 −1.8E‐7 −1.9E‐7 3.7E‐4 −2.1E‐8 −1.9E‐7 1.3E‐6 −1.9E‐7 −1.9E‐7 Minimum error 6.4E‐5 −3.8E‐6 −3.8E‐6 2.5E‐4 −3.7E‐7 −3.8E‐6 −3.1E‐6 −3.8E‐6 −3.8E‐6 Mean error 1.1E‐4 −5.0E‐7 −5.0E‐7 3.2E‐4 −3.6E‐7 −5.0E‐7 3.7E‐7 −5.0E‐7 −5.0E‐7 Model (6) Maximum error 2.1E‐3 −6.1E‐5 −6.2E‐4 8.0E‐3 −6.0E‐5 −6.2E‐5 5.0E‐4 −6.2E‐5 −6.2E‐5 Minimum error −9.5E‐5 −9.9E‐5 −9.9E‐5 1.7E‐3 −9.9E‐5 −9.9E‐5 −9.5E‐5 −9.9E‐5 −9.9E‐5 Mean error 8.0E‐5 −7.9E‐5 −7.9E‐5 2.7E‐3 −7.8E‐5 −7.9E‐5 −3.0E‐5 −7.9E‐5 −7.9E‐5 Grid size, N . Value function iteration . Time iteration (linear) . Time iteration (PCHIP) . 10 . 100 . 1,000 . 10 . 100 . 1,000 . 10 . 100 . 1,000 . Model (5) Maximum error 6.9E‐4 −1.8E‐7 −1.9E‐7 3.7E‐4 −2.1E‐8 −1.9E‐7 1.3E‐6 −1.9E‐7 −1.9E‐7 Minimum error 6.4E‐5 −3.8E‐6 −3.8E‐6 2.5E‐4 −3.7E‐7 −3.8E‐6 −3.1E‐6 −3.8E‐6 −3.8E‐6 Mean error 1.1E‐4 −5.0E‐7 −5.0E‐7 3.2E‐4 −3.6E‐7 −5.0E‐7 3.7E‐7 −5.0E‐7 −5.0E‐7 Model (6) Maximum error 2.1E‐3 −6.1E‐5 −6.2E‐4 8.0E‐3 −6.0E‐5 −6.2E‐5 5.0E‐4 −6.2E‐5 −6.2E‐5 Minimum error −9.5E‐5 −9.9E‐5 −9.9E‐5 1.7E‐3 −9.9E‐5 −9.9E‐5 −9.5E‐5 −9.9E‐5 −9.9E‐5 Mean error 8.0E‐5 −7.9E‐5 −7.9E‐5 2.7E‐3 −7.8E‐5 −7.9E‐5 −3.0E‐5 −7.9E‐5 −7.9E‐5 Notes This Table does not include the computational time nor the number of iterations as these are identical to those in Table 3. The disclaimer that applies for Tables 2 and 3 applies here also. Open in new tab 3. Concluding Remarks Dynamic models with inequality constraints pose a challenging problem for two major reasons: dynamic programming techniques are reliable but often slow, whereas Euler equation‐based methods are faster but have problematic or unknown convergence properties. This study has, at least partly, addressed these concerns. I have shown that a common iterative procedure applied on the first‐order conditions – known in the literature as time iteration – delivers a sequence of approximate policy functions that converges to the true solution under a wide range of circumstances. These circumstances extend to a finite, but large set of endogenous and exogenous state variables, and include a broad spectrum of occasionally binding constraints. I exemplified the potential advantages of the method to alternative techniques by solving a simple real business cycle model with irreversible investments. In general, time iteration turns out to be as accurate as state‐of‐the‐art value function iteration algorithms, but almost indiscriminately faster. Appendix A. Proofs Lemma A1. The minimiser, μ(x,z), of (5) is a continuous function with respect to x and z. Proof By the definition of a saddle function, the fact that μ ≥ 0 and mj(x,y^,z)<0 ⁠, for all x, z and j, it follows that (Tf)(x,z)≥L(x,y^,z,μ)≥F(x,y^,z)+β∫Zf(y^,z′)Q(z,dz′)−μ′m(x,y^,z).(A.1) Which implies that μ is bounded. Denote the upper bound as μ¯ ⁠. Given a certain μ∈[0,μ¯]r ⁠, define g~(x,z,μ) as g~(x,z,μ)=argmaxy∈XL(x,y,z,μ).(A.2) By Berge's theorem of the maximum, L[x,g~(x,z,μ),z,μ] is a continuous function. Hence, the set of minimisers μ(x,z) that solve the dual problem min0≤μ≤μ¯L[x,g~(x,z,μ),z,μ],(A.3) is an upper hemicontinuous correspondence in x and z. By Assumptions 2 and 3, μ(x,z) is single valued and consequently a continuous function in x and z (Wachsmuth, 2013). A.1. Proof of Proposition 1 Proof It is sufficient to show that T : C1(S) → C1(S). Define the saddle function L[x,g(x,z),z,μ(x,z)]=(Tf)(x,z)=F[x,g(x,z),z]+β∫Zf[g(x,z),z′]Q(z,dz′)−μ(x,z)′m[x,g(x,z),z].(A.4) Pick an x ∈ int(X) and an x^∈X such that xj=x^j for all j ≠ i and x^i=xi+ε ⁠, where xi denotes the ith element of the vector x. For notational convenience, denote the policy and multiplier functions from (5) as g, μ and g^ ⁠, μ^ for (x,z) and (x^,z) respectively. The definition of a saddle function implies L(x^,g,z,μ^)≤L(x^,g^,z,μ^)≤L(x^,g^,z,μ),(A.5) and L(x,g^,z,μ)≤L(x,g,z,μ)≤L(x,g,z,μ^).(A.6) Combine these two expressions and divide by x^i−xi L(x^,g,z,μ^)−L(x,g,z,μ^)x^i−xi≤(Tf)(x^,z)−(Tf)(x,z)x^i−xi≤L(x^,g^,z,μ)−L(x,g^,z,μ)x^i−xi.(A.7) By Lemma A1 and the results on page 6, the functions g and μ are continuous. Consequently the limits of g^ and μ^ exist and are limx^→xg^=g,limx^→xμ^=μ ⁠. Hence, limx^→xL(x^,g,z,μ^)−L(x,g,z,μ^)x^i−xi=limx^→xL(x^,g^,z,μ)−L(x,g^,z,μ)x^i−xi=∂L(x,g,z,μ)∂xi.(A.8) By the Pinching (squeeze) theorem limx^→x(Tf)(x^,z)−(Tf)(x,z)x^i−xi=∂L(x,g,z,μ)∂xi=∂(Tf)(x,z)∂xi.(A.9) Thus, (Tf)x(x,z)=Lx(x,g,z,μ)=Fx(x,g,z)−Jx(x,g,z)μ(x,z).(A.10) If v0 is a weakly concave and differentiable function, the desired result is achieved. A.2. Proof of Proposition 2 Proof A sufficient condition for a maximum is a saddle point of the Lagrangian L(x,y,z,μ)=F(x,y,z)+β∫Zvn(y,z′)Q(z,dz′)−μn+1′m(x,y,z).(A.11) By Proposition 1, the value function vn(y, z′) is differentiable and by Assumption 3, given minimisers μn+1, sufficient conditions for a saddle point are thus 0=Fy(x,y,z)+β∫Zvx,n(y,z′)Q(z,dz′)−Jy(x,y,z)μn+1.(A.12) By Proposition 1, this can be re‐written as 0=Fy(x,y,z)−Jy(x,y,z)μn+1(x,z)+β∫Z{Fx[y,hn(y,z′),z′]−Jx[y,hn(y,z′),z′]μn(y,z′)}Q(z,dz′).(A.13) Due to strict concavity, the solution is unique and hn+1(x,z) = gn+1(x,z). Appendix B. An Improvement Algorithm Value function iterations are considered prohibitively costly in many practical applications as convergence occur at a linear rate. Howard's improvement algorithm (Howard, 1960) can at least partly circumvent this issue, as convergence can be shown to be quadratic (Puterman and Brumelle, 1979; Santos and Rust, 2003). This part of the Appendix shows, albeit somewhat heuristically, that some of the ideas of Howard (1960) can be extended to the current setting of time iteration. Let gn denote the policy function associated with Tv~n−1 ⁠. Here, v~n−1 denotes some candidate value function and T is the operator defined as in (4). Howard's improvement algorithm then suggests to find improvements to Tv~n−1 ⁠, which I will call v~nh+1 ⁠, through the iterative procedure v~nh+1(x,z)=F[x,gn(x,z),z]+β∫Zv~nh[gn(x,z),z′]Q(z,dz′),(B.1) with v~nh=v~n−1 ⁠, for h = 0. This procedure is then repeated until h = H, or, say, until the distance between v~nh+1 and v~nh is sufficiently small.46 The updated v~n is given as v~n=v~nH ⁠, and the new policy function gn+1 solves Tv~n ⁠: Tv~n=maxy∈Γ(x,z)F(x,y,z)+β∫Zv~nH(y,z′)Q(z,dz′).(B.2) The procedure repeats itself until {v~n} converges. As previously mentioned, v~n converges to v at a quadratic rate. Under standard value function iteration any element in the sequence {vn} is known to be concave and differentiable (Proposition 1). However, the same cannot be said about any element in {v~n} ⁠.47 From a theoretical perspective this may cause problems as concavity and differentiability are properties often exploited to solve the optimisation problem in (B.2). In practice, however, it is quite common to ignore these issues and proceed under the hypothesis that each element v~nH is known to be concave and differentiable, even if this may not be the case. The optimisation problem in (B.2) is then solved by finding the solution to the first‐order conditions 0=Fy(x,y,z)−Jy(x,y,x)μn+1(x,z)+β∫Zv~x,nH(y,z′)Q(z,dz′),(B.3) where v~x,nH denotes – following the notation used throughout this study – the ℓ‐vector of derivatives of v~nH ⁠. A sufficient, albeit strong, condition to ensure that each v~nh+1 indeed is differentiable – and this is where the heuristic part comes in – is that gn, the policy function, is differentiable. Let Gx,n denote the Jacobian of the policy function gn. The ℓ‐vector of derivatives of v~nh+1 is then recursively given by v~x,nh+1(x,z)=Fx[x,gn(x,z),z]+Gx,n(x,z)Fy[x,gn(x,z),z]+β∫Zv~x,nh[gn(x,z),z′]Q(z,dz′).(B.4) Interestingly, (B.3) and (B.4) suggest an obvious improvement algorithm to augment time iteration. First, given a candidate v~x,n−1H ⁠, we can find gn as the solution to (B.3). This step would be identical to the updating step in Howard's improvement algorithm – Equation (B.2) – under the familiar assumptions of concavity and differentiability. Second – defining v~x,n=v~x,n−1H and v~x,nh=v~x,n ⁠, for h = 0 – we would use (B.4) to repeatedly find improvements of the derivative of v~n ⁠, and iterate until h = H, or until the distance between v~x,nh+1 and v~x,nh is sufficiently small. This step differs from the analogous step in Howard's algorithm in that we never bother to either find, nor improve upon, the value function itself, but only its derivative. While perhaps obvious, it is worth emphasising that under the same assumptions as those commonly imposed in many practical applications – i.e. differentiability of the policy function gn, and concavity of v~nh – the sequence of policy functions {gn} that emerges under the derived improvement algorithm will converge to the limiting function g. And convergence will occur at exactly the same rate as that of Howard's improvement algorithm. Appendix C. A Benchmark Solution For the benchmark solution I discretise the state space K into a set K¯ containing 1,000,000 equidistant nodes K¯={k1,k2,…,kN} ⁠. For each (k,z)∈K¯×Z and each n = 0,1, … , I solve Tvn−1(k,z)=maxk′∈Γ¯(k)u[zf(k)+(1−δ)k−k′]+β∑z′∈Zvn−1(k′,z′)Pz,z′,(C.1) with feasibility correspondence Γ¯(k)=[k′∈K¯:k′≥(1−δ)k].(C.2) The solution is then improved upon until h = 400 using vnh(k,z)=u[zf(k)+(1−δ)k−gn(k,z)]+β∑z′∈Zvnh−1(k′,z′)Pz,z′,(C.3) with vnh−1=vn−1 for h = 1, and where gn(k,z) is the maximiser of (C.1). The updated value function is then set to vn=vn400 ⁠, and the procedure repeats itself until ‖vn+1 − vn‖ < 1(−9). The initial guess for the value function is taken from the solution of the algorithm in subsection 2.2.1 using 10,000 nodes. The maximisation step is carried out by a rather brute force search algorithm which guarantees global optimality. As this procedure is known to converge to the true solution as the mesh size of the discretisation approaches zero (Santos and Vigo‐Aguiar, 1998), I consider the solution as virtually exact. Footnotes 1 " See, for instance, Deaton (1991), Aiyagari (1994) and Ludvigson and Michaelides (2001) for examples of liquidity constraints; Kehoe and Perri (2002), Cooley et al. (2004) and Jermann and Quadrini (2012) for examples of enforcement constraints; and Gertler and Kiyotaki (2010), Gertler and Karadi (2011) and de Groot (2011) for examples of collateral constraints. Studies that incorporate collateral constraints frequently assume that these always bind to simplify computation. The ‘financial accelerator’ is commonly attributed to Kiyotaki and Moore (1997) and Bernanke et al. (1999). 2 " To be clear, dynamic programming techniques refer to the method of successive approximations of the value function, with its various numerical implementations. 3 " Occasionally binding or inequality constraints are used interchangeably throughout this study. 4 " It should be noted here that the choice of approximation and optimisation method is not independent. For instance, if one would use a gradient‐based optimisation routine, which is normally very fast, it would also be advisable to employ an (at least) once continuously differentiable and concave approximation method, such as a shape‐preserving spline – see, for instance, Judd and Solnick (1994) and Judd (1998). Alternatively, a search‐based optimisation routine, which instead is quite slow, would leave greater freedom in terms of suitable approximation methods. 5 " See, for instance, Judd (1998, pp. 213, 438) for a discussion of these issues. 6 " See Bizer and Judd (1989), Coleman (1990), den Haan and Marcet (1990) and Baxter (1991) for early applications of this approach. Davig (2004), Davig and Leeper (2007), Kumhof and Ranciere (2011) and Malin et al. (2011) provide some recent examples. See McGrattan (1996), Judd (1998) and Christiano and Fisher (2000) for a detailed discussion. 7 " Christiano and Fisher (2000), for instance, use the solution to a log‐linearised version of their problem as an initial guess for the policy function. Despite this, some of the algorithms they explore fail to converge. 8 " Time iteration, or policy function iteration, can be described as the solution to a finite horizon problem using the first‐order conditions while letting the horizon approach infinity. As in Judd (1998), I prefer the term ‘time iteration’ to ‘policy function iteration’ to avoid confusion with Howard's improvement algorithm (Howard, 1960), which is sometimes referenced under the latter name (Santos and Rust, 2003). 9 " For these reasons, Coleman's method is sometimes referred to as the ‘monotone map method’ (Davig, 2004; Davig and Leeper, 2007; Davig et al., 2012). 10 " See, for instance, Ludvigson (1999), Ludvigson and Michaelides (2001), Szeidl (2002), Haliassos and Michaelides (2003), Carroll (2006) and Carroll (2009). 11 " Alternatively one may assume that Z is countable and Z contains all subsets of Z. 12 " Following standard notation, g is the argmax of (1). It is important to note that if X is compact, then convergence of gn is uniform. 13 " The Jacobian is here defined as an (ℓ × r)‐matrix, where ℓ is the number of endogenous state variables and r is the number of inequality constraints. 14 " The rank of the Jacobian of the constraint vector must, by construction, always fall short of, or equal, the number of endogenous state variables. 15 " See Section 2 for a clarification of the notation used here. The parameter ϕ is an arbitrary positive constant. 16 " See, for instance, Stokey et al. (1989) for the former claim. The latter claim follows from the proof of Proposition 1 in Appendix A. 17 " In the theoretical part of this paper I use h to denote the policy function obtained through time iteration, and g as the policy function obtained through dynamic programming. The goal is to show that hn = gn. 18 " See Rincon‐Zapatero and Santos (2009) for a result related to the limiting value function. 19 " If X is compact, Ns is independent of s and convergence is uniform. 20 " This particular case covers state‐independent inequality constraints, such as borrowing or short‐selling constraints. 21 " This follows as for any sequence of functions such that fn converges point‐wise to f, and fn′ exists, is continuous and converges uniformly to the function g, then f is differentiable with derivative g (Schroder, 2008). 22 " In fact, this result holds under weaker assumptions than previously stated; in particular, the uniqueness of the Lagrange multipliers is no longer necessary and LICQ could be replaced by a less strict constraint qualification, such as Slater's condition. 23 " Some numbers may illustrate this point quite clearly. Suppose that a grid consisting of a thousand nodes in each endogenous state variable is known to produce an accurate solution to some dynamic problem. Then, a reasonably small‐scaled problem with three‐state variables suggests a meshed grid of 1,0003, or one billion, nodes. 24 " For example, (9) should hold exactly on a finite grid of points in X according to the collocation method. In the Galerkin method, (9) holds as a weighted average on the entire state space, X. 25 " See, for Instance, Judd (1998, pp. 599–601). 26 " Fixed point iteration nests the ideas of the parameterised expectations algorithm by den Haan and Marcet (1990). In particular, as gn+1 only shows up in the left‐hand side, one could instead approximate the entire right‐hand side as f^n(xt)=f{xt,gn(xt),gn[gn(xt)]} ⁠, which is then updated as f^n+1(xt)=f{xt,f^n(xt),f^n[f^n(xt)]} ⁠. 27 " A remedy to this issue is to consider a dampening parameter η ∈ (0,1). In particular, let g^n+1(x) denote the left‐hand side of (10), then the ‘dampened’ update guess is given by gn+1(x)=ηg^n+1(x)+(1−η)gn(x) ⁠. However, in some cases η must be set at a low value close to zero which may slow the algorithm down considerably. 28 " Malin et al. (2011, p. 237), state this difference as ‘Thus we trade off solving non‐linear systems of much smaller size by the necessity to having to solve the systems many times (as opposed to only once)’. 29 " See Christiano and Fisher (2000) or Aruoba et al. (2006) for comprehensive work in this direction. 30 " An alternative formulation consistent with the notation in the previous Section would be Γ(k) = [k′ ∈ K:m(k,k′,z) ≤ 0], with m(k,k′,z) = (1 − δ)k − k′. 31 " In the numerical implementation, this conjecture proved to be simpler for the non‐linear equation solver than the more commonly used v0(k,z) = u[zf(k)]/(1 − β). 32 " A knot is the same as a node; i.e. an element of K^ ⁠. Derivatives at end points are set by the one‐sided line secant. For more information about this approximation method, see Kahaner et al. (1989) or Moler (2004). 33 " For simplicity I let vn,H denote the improved value function at the iteration at which the procedure was terminated even if less than H iterations were executed. 34 " An approximation for the derivative of gn(·,z) on K^ is computed using forward difference for the first node, backward difference for the last node and the central difference for all other nodes. 35 " As previously, I let vn,H′ denote the improved derivative at the iteration at which the procedure was terminated even if less than H iterations were executed. 36 " I denote an arbitrary and feasible policy as q to underline its distinction from the optimal policy g. 37 " I use the words ‘true’ and ‘optimal’ in quotation marks here as virtually all comparative studies on this topic use some stand‐in for the true function, which is never recovered (Christiano and Fisher, 2000). 38 " If utility is logarithmic, i.e. if γ → 1, the relevant metric is η=(1−β)(vq−vq^)×100 ⁠. 39 " Christiano and Fisher (2000) consider the same solution exact when K¯ contains 40,000 nodes. 40 " As an example, suppose that ηmean(q^)=5 ⁠, and ηmean(q) = 3, then ηmean(q,q^)=−2 ⁠. That is, the superiority of policy q relative to q^ is worth a 2% rise in consumption for perpetuity. Notice however that the same logic does not apply for ηmax or ηmin. 41 " With the only exceptions of N = 100 for model (4) and N = 1,000 for model (7) for which value function iteration is slightly faster. 42 " Speed should not be neglected here. For instance, time iteration with 100 nodes is both faster and more accurate for model (5) than value function iteration with 10 nodes. 43 " I have also tried using a cubic spline as an alternative to PCHIP. The C2 property of the spline alleviates the issues raised here but causes some other due to inter‐nodal oscillations; indeed, for some parameterisations the derivative of the value function took on negative values on parts of the state space, which led the solver to a complex (absorbing) terrain. In general, PCHIP performed quite a lot better than a cubic spline across all models. 44 " Accuracy actually declines for model (5) when using time iteration with piece‐wise Hermite interpolation. I have also tried increasing N to 10,000 with unaltered results. 45 " Christiano and Fisher (2000), for instance, treat the discretised value function iteration algorithm using 40,000 nodes as ‘exact’ for precisely the same parameterisations as those explored here. McGrattan (1996) uses 214 = 16,384 nodes for a very similar exercise. 46 " For notational convenience I will let v~nH denote the outcome of this procedure, irrespective of the stopping criterion. 47 " Unless, of course, H = 1. References Aiyagari , R.S. ( 1994 ). ‘ Uninsured idiosyncratic risk and aggregate saving ’, Quarterly Journal of Economics , vol. 109 ( 3 ), pp. 659 – 84 . 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( 2008 ). ‘ Continuous state dynamic programming via nonexpansive approximation ’, Computational Economics , vol. 31 ( 2 ), pp. 141 – 60 . Google Scholar Crossref Search ADS WorldCat Stokey , N.L. , Lucas , J.R.E. and Prescott , E.C. ( 1989 ). Recursive Methods in Economic Dynamics , Cambridge MA : Harvard University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Szeidl , A. ( 2002 ). ‘On ergodic distributions and buffer stock saving models’ , manuscript, Harvard University. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Wachsmuth , G. ( 2013 ). ‘ On LICQ and the uniqueness of lagrange multipliers ’, Operations Research Letters , vol. 41 ( 1 ), pp. 78 – 80 . Google Scholar Crossref Search ADS WorldCat Author notes " This study is a substantially revised version of a chapter in my Ph.D dissertation written at the European University Institute, previously circulated under the title ‘Inequality Constraints in Recursive Economies’. I thank Wouter den Haan, Dirk Krueger, Albert Marcet, Karel Mertens, Morten Ravn, Sanne Zwart and two anonymous referees for helpful comments and suggestions. The usual disclaimer applies. © 2013 Royal Economic Society

Ambrus,, Attila;Rozen,, Kareen

doi: 10.1111/ecoj.12103pmid: N/A

Abstract This article shows that when multiple selves' preferences are aggregated into a decision, even if the researcher has a fully specified theory of how preferences get aggregated, there are typically no testable implications without restricting the number of selves. This points to the importance of collecting reliable information on the number of selves in interpersonal and intrapersonal decision‐making contexts. We establish the result through a linear relationship between the number of selves and the set of choice functions an aggregator is guaranteed to rationalise. The latter relates to a choice function's number of independence of irrelevant alternatives violations, a new measure of irrationality. Since the seminal work of May (1954), a growing number of papers have proposed models of multi‐self decision making, with the primary motivation of accommodating context‐dependent behaviour (thereby relaxing the axiom of independence of irrelevant alternatives (IIA), which requires that if an alternative is chosen from a set, it is also chosen from any subset in which it is contained).1,2 Formal models of multi‐self decision making include among others – Kalai et al. (2002), Fudenberg and Levine (2006), Manzini and Mariotti (2007) and Green and Hojman (2009) in economics; Tversky (1969), Shafir et al. (1993) and Tversky and Simonson (1993) in psychology; and Kivetz et al. (2004) in marketing. A parallel literature, including for example, Apps and Rees (1988), Chiappori (1988), Browning and Chiappori (1998) and Cherchye et al. (2007), studies interpersonal aggregation of preferences in a household or a larger community. Many papers in the literature assume a particular method of aggregating preferences but do not put a priori restrictions on the number of selves involved in the decision. Each of these practices is potentially justified. In interpersonal contexts, the decision‐making procedure can be observable, or suggested by theoretical considerations. In intrapersonal contexts, theoretical considerations, experimental data or neuroscience research can suggest a certain method of preference aggregation for different selves, or motivations, of the individual. At the same time, the researcher might not know the number of selves relevant for the decision; such data limitations include the possibility of unobserved selves, or at the extreme, having no available data on the number of selves. In this article, we examine whether specifying a certain method of preference aggregation generates testable predictions on choice behaviour without putting an a priori restriction on the number of selves. For this, we need to know what choice behaviour can be explained by a given number of selves. For some aggregators, this is easy to determine. For example, if the decision maker's (DM) method of aggregating the utilities of her selves is simple utilitarianism, then the set of choice functions is exactly the set of rational choice functions – regardless of the number of selves. But what if, in analogy to models of relative utilitarianism (Karni, 1998), each self's utility is normalised by her range of utilities over the choice set? Or if the aggregator is the ‘normalised contextual concavity model’ proposed in Kivetz et al. (2004)? In order to investigate this question, we lay out in Section 1 a framework that incorporates various models of multi‐self decision making which have been proposed in the literature. In particular, we model the DM or group as a collection of selves (of possibly different types) and an aggregation rule f (decision‐making method) which combines the selves' utility functions in a possibly context‐dependent way. That is, given a choice set A, and selves S, an aggregator f specifies an aggregate utility for every alternative in A. Each aggregator in the framework captures a particular theory of multi‐self decision making. We examine a broad class of aggregators characterised by five simple properties from social choice theory and show that many models of multi‐self decision making proposed in the existing literature can be formally translated into an aggregator satisfying our axioms. Since our results apply for a broad class of multi‐self models, we provide a meta‐analysis of various models proposed in the literature, and offer a way to characterise the explanatory power of such models. An important feature of the set of aggregators that we focus on is that aggregation can depend on cardinal information in the selves' utilities. This is partly motivated by the fact that many existing models of multi‐self decision making make use of cardinal information embedded in different selves' utility functions. Furthermore, the use of cardinal information (intensity of preferences) is natural to assume in intrapersonal decision making and in certain interpersonal decision‐making situations as well, such as household decisions. A second feature of the aggregators we consider, related to cardinality, is the possibility of compromise among selves. As opposed to the models provided in Kalai et al. (2002) and Cherepanov et al. (2013), but in accordance with models proposed in Tversky (1969), Tversky and Kahneman (1991), Kivetz et al. (2004), Fudenberg and Levine (2006), Green and Hojman (2009) and others, all of the selves in our framework are ‘active’ for every possible choice set. However, the weights allocated to different selves by the aggregator can depend on the choice set. This means that the model can capture behaviour as in Fudenberg and Levine (2006), where a long‐run self must exert more costly self‐control when more appealing options are available to a short‐run self; or Shafir et al. (1993), where the primary rationales for purchasing may depend on the set of available products. Our primary goal was to investigate the behaviours a model of multi‐self decision making can rationalise (explain). Formally, the DM's behaviour is described by a choice function c that specifies the alternative she selects from each subset of the grand set of alternatives X. For a given model of aggregation f, the DM's choice behaviour is rationalised by a finite collection of selves S if she selects the unique maximiser of the selves' aggregate utility from every choice set. The DM's choice behaviour need not satisfy IIA. In Section 2 we define a measure of a choice function's irrationality: the number of IIA violations it exhibits. Our main results, in Sections 3 and 4, establish that for a large class of multi‐self models, including various models proposed in the existing literature, if there is no restriction on the number of selves then the model can rationalise any choice function. For example, in the important class of scale‐invariant aggregators (for which the unit of utility measurement does not change the ordinal rankings in the aggregation), whenever two simple types of irrational behaviour can be rationalised on a triple of alternatives, the aggregator can rationalise any behaviour over any set of alternatives. Furthermore, we show that in a formal sense, aggregators satisfying the above property are generic; therefore, one generically cannot have testable predictions without restricting the number of selves. The lesson to draw from this is that to offer a refutable theory of multi‐self decision making, it is not enough to impose a concrete method of aggregating different selves' utilities; it is also important to fix the number of selves a priori (e.g. as in a ‘dual‐self’ model) in order to restrict the set of rationalisable behaviours. This need not be the case outside the class of models studied here: Manzini and Mariotti (2007) and de Clippel and Eliaz (2012), for example, have shown that their models can explain only certain types of irrational behaviours, even when using arbitrarily many rationales. We establish our theorems by finding a simple linear relationship between the number of selves in the model, and the number of IIA violations the choice function can have while still guaranteeing that it can be rationalised. Our results can be seen as drawing a connection between the complexity of a rationalisation and the extent to which the choice behaviour in question deviates from rationality, as measured by the number of IIA violations.3 Our results differ from Kalai et al. (2002), who examine the complexity of a rationalisation as a function of the number of alternatives available. They say a collection of preference orderings rationalises a choice function if the choice from each set is optimal for some preference, and show it suffices to posit as many selves as there are alternatives to explain any behaviour. To rationalise a choice function, they assign each utility function the sets over which it acts as dictator, which amounts to modifying the method of aggregation. In contrast, this article studies the set of behaviours rationalisable by a fixed aggregator. Our results also differ from those in the household choice literature, such as Chiappori (1988), Browning and Chiappori (1998), Chiappori and Ekeland (2006) and Cherchye et al. (2007, 2011). These works focus on rationalising demand in a market environment. The results they obtain are non‐parametric in the sense that they do not rely on the particular functional specification chosen for the preferences or for the intra‐household allocation process. Our finite (though abstract) choice setting is closer in spirit to that of Cherchye et al., who study a global revealed preference framework assuming a finite set of demand and price observations. Green and Hojman (2009) also study a class of aggregation methods. Because they model a DM as a probability distribution over all possible ordinal preference rankings, their framework is difficult to compare to models with a discrete number of cardinal selves but is related to models in the voting literature (Saari, 1999). Extending results from that literature, they show that if choice is determined by a voting rule satisfying a monotonicity property, then their model can explain any choice behaviour.4 The rest of their paper focuses on welfare analysis. 1. Framework We observe a collective choice behaviour on a finite set of alternatives X. Denote by P(X) the set of non‐empty subsets of X. The collective choice function c : P(X)→X identifies the alternative c(A) ∈ A chosen from each A ∈ P(X). A rationalisation of the collective choice function consists of a collection of selves and a model of aggregation that combines the utilities of different selves in a possibly menu‐dependent way into an aggregate utility function. In an interpersonal context, selves represent different individuals. In an intrapersonal context, selves represent the DM's conflicting motivations or priorities. The aggregator corresponds to a method of ‘sorting out’ priorities of different selves to come to a decision. In order for our framework to encompass as many of the multi‐self models proposed in the existing literature as possible, we permit selves to have ‘types’ and consider potentially asymmetric aggregators that treat selves differently according to their type. Formally, a model of aggregation ( f,T ) specifies a set T of the possible types a self may take, and a function f that gives the aggregate utility for every alternative a in every choice set A, for any (finite) grand set of alternatives X and any collection of selves defined over X and T. A single self s is given by a pair (u,t), where u:X→R is a utility function and t ∈ T is the self's type. Hence, each self is an element of RX×T ⁠. A collection of selves S is an unordered list of selves.5 Formally, for a given grand set of alternatives X and set of possible types T, a collection of selves S is an element of S(X,T)=∪n=1∞Sn(X,T) ⁠, where Sn(X,T) is the set of all unordered lists of selves over X that contain n elements. We denote the number of selves in a particular collection S by |S|, or simply n when no confusion would arise.6 The aggregator f specifies an aggregate utility for every alternative a in every choice set A, given any (finite) grand set of alternatives X, set of types T, and collection of selves S. Formally, the domain over which f is defined is {a,A,S,X,T|X∈X,S∈S(X,T),A∈P(X),a∈A} ⁠, where X is the set of conceivable finite grand sets of alternatives. Since the choice set A is one of the arguments of the function, f aggregates the utilities of the selves in a possibly context‐dependent way.7 An aggregation rule may be seen as a particular theory of how selves are activated by choice sets: the aggregator determines the weight each self receives on the choice set as a function of its utility levels over the alternatives. Formally, the grand set of alternatives X is an argument of the aggregator, not only because the evaluation of an alternative a ∈ A might depend on alternatives outside the choice set A but also because this enables a ‘comparative static’: we study how the number of selves needed to rationalise a choice rule depends on the size of X. For simplicity, we will suppress notational dependence of f on X and T, writing simply f(a,A,S), whenever doing so would not cause confusion. Given a model, we say that a collection of selves rationalises a choice function if from every choice set, the alternative that maximises the aggregated utility is precisely the one selected by the choice function.8 Note that this definition requires a unique maximiser of aggregate utility. Definition 1. A model ( f,T) rationalises a choice function c(·) on X if there exists a finite collection of selves S∈S(X,T) such that for every A ∈ P(X), c(A) = argmaxa ∈ A f(a,A,S). 1.1. The Class of Models Studied We study a class F of models of multi‐self aggregation satisfying the following properties, most of which are familiar from the theory of social choice. These properties are satisfied by several previously proposed multi‐self models. In the resulting class of models, aggregation of utilities is cardinal and the framing effect of a choice set operates only through the utility levels of the different selves. Before introducing these properties, it will be useful to define the following notation. For any collections of selves S=〈s1,…,s|S|〉 and S′=〈s1′,…,s|S′|′〉 in S(X,T) ⁠, we denote by 〈S,S ′ 〉 the combined collection (s1,…,s|S|,s1′,…,s|S′|′)∈S(X,T) ⁠. P1. (Neutrality) For any permutation π: X → X, f(a, A, S) = f(π(a), π(A), 〈(u ∘π−1,t) 〉(u,t) ∈ S). P2. (Consistency) For any s =(u,t), u(a) ≥ u(b) if and only if f(a,A,s) ≥ f(b,A,s). P3. (Reinforcement) If both f(a, A, S) ≥ f(b, A, S) and f(a,A,S ′) ≥ f(b,A,S ′) then f(a, A, 〈S,S ′ 〉 ⁠) ≥ f(b,A, 〈S,S ′ 〉 ⁠), with strict inequality if one of the above is strict. P4. (Continuity to near‐indifferent additions) If f(a, A, S) > f(b, A, S), then for any k ∈ Z there exists δ > 0 such that f(a, A, 〈S,S ′ 〉 ⁠) > f(b, A, 〈S, S ′ 〉 ⁠) for any S′∈Sk(X) with the property that maxa,b∈A,A⊆X,s′∈S′|f(a,A,s′)−f(b,A,s′)|<δ ⁠. P5. (Profile equivalence) If u(a) = u(a′) for all (u, t) ∈ S then f(b, A ∪{a}, S) = f(b, A ∪ {a′ },S) for all b ∈ A. While these properties are not without loss of generality, they are satisfied by many multi‐self models that have been proposed in the literature. Neutrality implies that the names of alternatives do not affect their ranking (only utilities affect rankings). Consistency requires respecting the preference of a lone self. Reinforcement requires that if two separate collections of selves S and S ′ each prefer the alternative a to the alternative b, then the combined collection of selves, obtained by merging collections of selves S ′ and S, also prefers a to b. Consistency and reinforcement together imply Pareto‐optimality. Continuity to near‐indifferent additions introduces a cardinal feature into the method of aggregation. It does not require that f (or the ordering of the alternatives implied by f) be continuous in the utilities of selves, for a fixed number of selves; it only requires that if a collection of selves leads to a strict aggregate preference for a over b, then that preference is not overturned when adding selves for which the aggregate utility difference between alternatives is sufficiently small. That is, preference intensity matters. In view of consistency and reinforcement, assuming P4 is weaker than assuming full continuity.9 Finally, profile equivalence says that aggregation is only affected by the set of available utility levels of the alternatives in a given choice set. In particular, choice is not affected by which of two alternatives is adjoined to a set, as long as those two alternatives yield exactly the same utility to all of the selves. This means that adding ‘duplicate’ elements to a set, which replicate the exact utility levels of some element already in the set, does not affect the rankings of alternatives. However, increasing the size of a set can still affect the DM when the new elements change the set of possible utility levels. For ease of exposition, in the main text, we also restrict attention to aggregators for which the aggregate utility of an alternative in a choice set A is independent of alternatives outside of A (see online Appendix E for an extension of our results without imposing this assumption). P6. (Independence of unavailable alternatives) Let X,X ′ be two grand sets of alternatives and consider any A ⊆X∩ X ′. Take any collection of types (t1,…,tn) and any two collections (u1,…,un) and (u1′,…,un′) of utility functions over X and X ′, respectively. If ui(x)=ui′(x) for each x ∈ A and each i, then the aggregator satisfies f(·,A,〈(ui,ti)〉i,X,T)=f(·,A,〈(ui′,ti)〉i,X′,T). 1.2. Examples of Aggregators The following are examples of context‐dependent aggregators satisfying P1–P6 that are equivalent or closely related to models proposed in the existing literature. In the first four examples, the aggregator treats all types symmetrically, so we may take the type set T to be a singleton. Example 1. (Utilitarianism) The aggregate utility of an alternative a in a choice set A is given by ∑(u,t)∈ Su(a). Note that the utility of an alternative is independent of the choice set within which it is evaluated. Example 2. (Generalisation of Tversky (1969)) The aggregate utility of an alternative a in a choice set A is ∑(u,t)∈ SΦ(maxb∈Au(b)−minb∈Au(b))u(a), where the contribution of a self to the aggregate utility depends via Φ on the range of u over choice set A. For binary choice sets, this reduces to the additive difference model of Tversky (1969), which was proposed to explain intransitive pairwise choice through the aggregation of criterion‐by‐criterion comparisons of alternatives.10 If Φ in that model is increasing, utility functions with a greater intensity of preference over the set A receive greater weight in the aggregate utility. The case Φ(x) = x is Köszegi and Szeidl's (2013) focus‐weighted model. If Φ is decreasing, the model may be seen as a context‐dependent version of the models of relative utilitarianism in Karni (1998), Dhillon and Mertens (1999) and Segal (2000), where a DM's weight in society is normalised by her utility range over the grand set. Example 3. (Nash bargaining solution with an endogenous disagreement point) The aggregate utility of an alternative a in a choice set A is Π(u,t)∈S(κ+u(a)−mina′∈Au(a′)) ⁠, where κ is any positive constant to ensure each term is strictly positive. This example, which specifies the worst outcome as the disagreement point, is similar to Kaneko and Nakamura (1979), although they assume the utility of the worst outcome is the same in all choice sets. A more general theory of context‐dependent disagreement points in the bargaining solution is offered by Conley et al. (1997). Example 4. (Loss aversion of Tversky and Kahneman (1991), with endogenous reference point) The aggregate utility of an alternative a in a choice set A is given by ∑(u,t)∈Sm(u(a))+∑u∈Uℓ(u(a)−r({u(a′)}a′∈A)) ⁠, where r(·) determines the reference point against which u(a) is evaluated; m(·) captures the impact of absolute valuations on aggregate utility; and the loss aversion function ℓ(·) satisfies the properties proposed by Tversky and Kahneman (1991): steeper disutility from losses than utility from gains, and weakly diminishing sensitivity. The above model has been applied in various forms. In Orhun (2009), each u can be interpreted as the valuation of alternatives under some attribute. Orhun (2009) finds the optimal product line for a model corresponding to the case where m is linear, ℓ is the standard kinked‐linear loss aversion function (i.e. ℓ(x) = x for x > 0, ℓ(x) = λx for x < 0 and some λ > 1) and r is a weighted average of valuations. Kivetz et al. (2004) consider goods (e.g. laptops) which have defined attribute levels (e.g. processor speed) and posit utility levels (‘partworths’) for a given attribute. Their contextual concavity model specifies r(·) ≡ min(·), m(·) ≡ 0 and ℓ(·) ≡ (·)ρ for some concavity parameter ρ. They also introduce a type‐dependent version of their model, where the concavity parameter ρ depends on the type of attribute to which the self corresponds. Fudenberg and Levine (2006) propose a dual‐self impulse control model with a long‐run self exerting costly self‐control over a short‐run self. The reduced‐form model they derive has an analogous representation in our framework, with two selves: the long‐run self, with utility given by ulr (the expected present value of the utility stream induced by the choice in the present), and the short‐run self, with utility function usr (the present period consumption utility).11 Using our terminology, there are two types of selves, long run (lr) and short run (sr), and their reduced form representation assigns to alternative a the aggregate utility ulr(a) − C(a), where term C(a) depends on the attainable utility levels for the short‐run self and is labelled as the cost of self‐control. For example, using Fudenberg and Levine's (2006) parameterisation, C(a)=γ[maxa′∈Ausr(a′)−usr(a)]ψ ⁠. More generally, there may be multiple long‐run considerations and multiple short‐run temptations. Example 5. (Costly self‐control aggregators) The set of possible types is T = {lr,sr} and the aggregate utility of an alternative a in a choice set A is f(a,A,S)=∑(u,lr)∈Su(a)−∑(u,sr)∈Sγ(maxa′∈Au(a′)−u(a))ψ. Of course, given the utilitarian aggregation of the long‐run selves, they could equivalently be represented using a single utility function. 2. Counting IIA Violations The examples of decision rules presented in the previous Section violate the IIA because they are context dependent. IIA requires that if a ∈ A ⊂ B and c(B) = a then c(A) = a. This says that if an alternative is chosen from a set, then it should be chosen from any subset in which it is contained. It is well known that a choice function can be rationalised as the maximisation of a single preference relation if and only if it has no violations of IIA. In the next Section, we connect the set of choice functions that an aggregator can rationalise with n selves to the number of IIA violations that a choice function exhibits. To do this, we formally define an accounting procedure for the number of IIA violations. The number of IIA violations can be determined straightforwardly for choice functions over three‐element sets; for example, if the choice over pairs is transitive but the second‐best element according to the pairs is selected from the triple, there is one violation to the property of IIA. For a larger set of alternatives, there are different plausible ways to define the number of violations. For example, suppose that c({a,b,c,d,e,f})=d,c({a,b,c,d,e})=b,.3pcc({a,b,c,d})=b,.7pcc({b,c,d})=c. In light of c({a,b,c,d,e,f}) = d, IIA dictates that the last three choices should be d (but they are not). In light of c({a,b,c,d,e}) = b, IIA dictates that the choice from {b,c,d} should be b (but it is not) and the IIA implication for {b,c,d} is again violated in light of c({a,b,c,d}) = b. Hence, one way of counting would indicate five IIA violations with respect to the above four choice sets. However, according to our counting procedure, there are two IIA violations in this example: only the choices from {a,b,c,d,e} and {b,c,d} are associated with violations. The reason is that while c({a,b,c,d}) = b does contradict c({a,b,c,d,e,f}) = d, the intermediate choice c({a,b,c,d,e}) = b itself implies by IIA that c({a,b,c,d}) = b. The idea is that one ‘resets’ the point from which the IIA implication must hold: if c(B) is chosen from B but is not chosen from A ⊂ B, then for all subsets of A in which c(A) is contained, one expects c(A) to continue being chosen. With this idea in mind, our accounting procedure counts the number of such resets, associating an IIA violation with a choice set when it is the largest set whose choice violates the IIA implication coming from a superset. Definition 2. (IIA violation) The set A causes an IIA violation under the choice function c(·) if (1) there exists B such that A ⊂ B and c(B) ∈ A∖{c(A)}, and (2) for every A′ such that A ⊂ A′ ⊂ B, c(A′) ∉ A. Then, the total number of IIA violations is defined in the natural way. Definition 3. (Number of IIA violations) The total number of IIA violations of a choice function c(·) is given by IIA(c) = #{A ∈ P(X)|A causes an IIA violation}. The sketch of proof for our main result in Section 4 illustrates the connection between this definition and our rationalisation procedure. There are other plausible measures for the number of IIA violations implied by a choice function. One alternative measure would be the minimal number of sets at which the choice function would have to be changed to make it rational. This measure can in general be either larger or smaller than our measure of the number of IIA violations.12 3. Scale‐invariant Models We begin by introducing our results for a special class F* of type‐independent aggregators satisfying P1–P6 and taking the form f(a,A,S)=∑(u,t)∈Sg(a,{u(a′)}a′∈A) ⁠, where the function g satisfies g(a,{αu(a′)}a′∈A)=ϕ(α)g(a,{u(a′)}a′∈A) for all α∈R and some invertible and odd ϕ:R→R ⁠. This says that the unit in which the preference intensity of different selves is measured does not affect rankings. This class includes utilitarianism as well as various menu‐dependent variations. As previously noted, utilitarianism explains only rational choice behaviour. This Section shows that being able to explain only a limited set of behaviours is a non‐generic feature of aggregators in this class. Consider the following model of reference‐dependent aggregation in F* ⁠. Example 6. (Simple reference dependence) The aggregate utility of an alternative a in a choice set A is ∑(u,t)∈ S(u(a) − mean u(A))ρ, where ρ is an odd integer and mean u(A) is a geometric or arithmetic mean over the set {u(a′)}a′∈A ⁠. This is a reference‐dependent variation of the CRRA form, where the origin is shifted. The reference dependence in Example 6 permits that model to rationalise a much wider array of behaviour than can utilitarianism. To understand why, let us first examine choice behaviour over only three alternatives. There are three possible kinds of irrational choice functions defined over a three‐element set. One possibility is transitive choice, where the second‐best element (from the transitive ranking) is chosen from the triple; another is transitive choice, where the worst element is chosen from the triple; and the third is intransitive choice. Using the model in Example 6, it is easy to construct rationalisations for all three of these behaviours. The first part of the following theorem shows that if a model of aggregation in F* can rationalise the last two irrational behaviours over a triple of alternatives, then it can rationalise any choice function defined over any space of alternatives. The second part of the theorem shows that a generic aggregator in F* (including Example 6) can rationalise any choice behaviour with a uniform bound on the number of utility functions needed. To describe the metric for which genericity is defined, note that by scale invariance there is a natural bijection (simply by scaling the utility functions inputted) between (1) models in F* applied to pairs and triples of elements, and (2) the set of pairs of operators Ω={O1,O2|O1:Δ2→R2,O2:Δ3→R3} ⁠, where Δ2,Δ3 are the two‐ and three‐dimensional simplexes respectively. The distance between two such pairs (O1,O2) and (O1′,O2′) is defined as maxi=1,2supx∈Ri|Oi(x)−Oi′(x)| ⁠. Theorem 1. Let X be a finite grand set of alternatives. Then, (i) Take any model in F* and any x,y,z ∈ X. If the model can rationalise both (1) intransitive choice over x,y,z and (2) transitive choice over x,y,z where the worst pairwise element is best in the triple {x,y,z}, then the model can rationalise any choice function c defined over X. (ii) The set of models in F* that can rationalise any choice function c using at most 1 + 5IIA(c) utility functions is open and dense. The proof of this theorem appears in the Appendix, and is discussed in the next Section. Theorem 1 formalises the sense in which only being able to explain rational choice behaviour is fragile. Once certain types of irrational behaviours can be explained over three alternatives, an additive and scale‐invariant model can rationalise any choice behaviour with sufficiently many selves. Moreover, the ability to explain any behaviour is generic in this class, with at most five ‘good reasons’ needed for every ‘mistake’ made. Note that the result gives a lower bound on the set of behaviours a generic aggregator in F* can rationalise, thereby providing a linear connection between the complexity of the observed behaviour (as measured by the number of IIA violations) and the degree of freedom in the model (as measured by the number of utility functions). Given n utility functions, a generic aggregator in F* can rationalise any choice function c, defined on any finite grand set of alternatives X, that has at most (n − 1)/5 IIA violations. Thus, in spite of having a structured form, essentially any aggregator in F* can rationalise any choice function with sufficiently many utility functions. In other words, a model of decision making satisfying the above properties must put a priori restrictions on the number of utility functions in order to be falsifiable. Given a model of aggregation and any triple of alternatives, it is very easy to check whether the model can rationalise the two irrational behaviours described in part (i) of Theorem 1 However, the proof of 1 also reveals a simple sufficient condition for checking whether a model f is of the generic type in part (ii). It suffices to find a single self‐defined over a triple {x,y,z} for which f ‘stretches’ the utility differences over pairs f(x,{x,z},s)−f(z,{x,z},s)≠f(x,{x,y},s)−f(y,{x,y},s)+f(y,{y,z},s)−f(y,{y,z},s), and for which f's evaluation of alternatives in the triple is not fixed by the pairwise rankings, f(x,{x,y,z},s)−f(y,{x,y,z},s)+f(x,{x,y,z},s)−f(z,{x,y,z},s)≠f(x,{x,y},s)−f(y,{x,y},s)+f(x,{x,z},s)−f(z,{x,z},s). For example, defining the above self using the utility function u(y) = 4 > u(z) = 2 >u(x) = 1 shows that the model in Example 6 using an arithmetic mean is in the generic class. In contrast, utilitarianism and generalisations of the form f(a,A,s) = u(a) + h(A), where the choice set cannot change intensity of preference within a set, fail the sufficient condition (and, in fact, explain only rational choice). The proof shows that the sufficient condition is satisfied generically. Nonetheless, it is not necessary – even aggregators that fail to satisfy the condition may be able to rationalise all choice behaviours. As seen from our upcoming results, the model of Example 2 using linear ϕ can rationalise any behaviour with five utility functions per IIA violation but fails the sufficient condition. 4. A Rationalisation Theorem and Procedure We begin with an illustrative example before presenting our main result. Recall the model in Example 2, where the aggregate utility of an alternative a ∈ A is f(a,A,S)=∑(u,t)∈SΦmaxb∈Au(b)−minb∈Au(b)u(a), for some monotonic function Φ. Let us suppose Φ is increasing, and examine how this aggregator behaves on an arbitrary three‐element set of alternatives {a,b,c}. In particular, define a collection S of five selves having the following five utility functions defined on {a,b,c} (in each column, the alternative on the left receives the utility number to its right): u1 . u2 . u3 . u4 . u5 . b 2 b 2 c 2 a,c 2 a 2 c 1 a 1 b 1 b 0 b,c 0 a 0 c 0 a 0 u1 . u2 . u3 . u4 . u5 . b 2 b 2 c 2 a,c 2 a 2 c 1 a 1 b 1 b 0 b,c 0 a 0 c 0 a 0 u1 . u2 . u3 . u4 . u5 . b 2 b 2 c 2 a,c 2 a 2 c 1 a 1 b 1 b 0 b,c 0 a 0 c 0 a 0 u1 . u2 . u3 . u4 . u5 . b 2 b 2 c 2 a,c 2 a 2 c 1 a 1 b 1 b 0 b,c 0 a 0 c 0 a 0 It is easy to verify that a is chosen from {a,b}. Indeed, f(a,{a,b},S) = 4Φ(2) + Φ(1) and f(b,{a,b},U) = 2Φ(2) + 3Φ(1). Hence, f(a,{a,b},S) > f(b,{a,b},S) since Φ(2) > Φ(1). In any other choice set, all alternatives have the same aggregate utility: f(a,{a,c},S)=f(c,{a,c},S)=2Φ(0)+Φ(1)+2Φ(2),f(b,{b,c},S)=f(c,{b,c},S)=3Φ(1)+2Φ(2),f(a,{a,b,c},S)=f(b,{a,b,c},S)=f(c,{a,b,c},S)=5Φ(2). That is, under the collection of selves S, alternative a receives strictly higher aggregate utility than b in the choice set {a,b} and there is complete indifference in all other choice problems. We will call such a collection S defined on a three‐alternative set {a,b,c} a triple basis for this aggregator f. Triple bases can serve as building blocks for rationalisations of choice functions on arbitrary spaces of alternatives. To illustrate, take X = {x1,x2,…,xn} and define the choice function c(·) as selecting the alternative with the smallest index in every choice set, with the exception that c({xi,xj}) = xj for one pair i < j. This choice function has one IIA violation, corresponding to the set {xi,xj}. Using the triple basis above, we construct a collection of five selves S{xi,xj} having the following utility functions over X: u1 . u2 . u3 . u4 . u5 . xi 2 xi 2 X ∖{xi,xj} 2 X ∖{xi} 2 xj 2 X ∖{xi,xj} 1 xj 1 xi 1 xi 0 X ∖{xj} 0 xj 0 X ∖{xi,xj} 0 xj 0 u1 . u2 . u3 . u4 . u5 . xi 2 xi 2 X ∖{xi,xj} 2 X ∖{xi} 2 xj 2 X ∖{xi,xj} 1 xj 1 xi 1 xi 0 X ∖{xj} 0 xj 0 X ∖{xi,xj} 0 xj 0 u1 . u2 . u3 . u4 . u5 . xi 2 xi 2 X ∖{xi,xj} 2 X ∖{xi} 2 xj 2 X ∖{xi,xj} 1 xj 1 xi 1 xi 0 X ∖{xj} 0 xj 0 X ∖{xi,xj} 0 xj 0 u1 . u2 . u3 . u4 . u5 . xi 2 xi 2 X ∖{xi,xj} 2 X ∖{xi} 2 xj 2 X ∖{xi,xj} 1 xj 1 xi 1 xi 0 X ∖{xj} 0 xj 0 X ∖{xi,xj} 0 xj 0 This is constructed by letting the choice from {xi,xj}, which is xj, play the role of a in the triple basis; letting the unchosen alternatives in {xi,xj}, which is only xi, play the role of b; and letting the alternatives outside {xi,xj}, which consists of X ∖{xi,xj}, play the role of c. Using S{xi,xj} ⁠, how does f evaluate the alternatives in each choice set A ⊆ X? Since any x ∈ X ∖{xi,xj} has the same utility as c in the calculations above, it is easy to see f(·,A,S{xi,xj}) is constant for any set A ≠ {xi,xj}. Since xj plays the role of a and xi plays the role of b, the previous calculations imply that f(xj,{xi,xj},S{xi,xj})>f(xi,{xi,xj},S{xi,xj}) ⁠. Thus, the utility functions in S{xi,xj} rationalise the choice from {xi,xj} and have no impact on other choice sets. Since the collection S{xi,xj} has implications only for the IIA violation {xi,xj}, one needs an additional self to rationalise the remaining ‘rational’ choices. We construct a final self s* whose utility function u* has sufficiently small range to not overturn any strict preferences induced from S{xi,xj} ⁠, and which has the ranking u*(x1) > u*(x2) >⋯ > u*(xn) derived from standard revealed preference. By construction, the selves 〈s*,S{xi,xj}〉 rationalise c(·). 4.1. Rationalisability Result Observe that the triple basis S given above would still be a triple basis for the generalised additive difference model if we were to scale all the utilities by a common constant. Loosely speaking, this means that for any δ, the collection of selves S rationalises being indifferent among all alternatives in subsets of {a,b,c} except for having a δ‐amount of strict preference within one pair. This is a property we term triple solvability and is formally defined below for any model of aggregation. Definition 4. Given a triple {a,b,c} and model (f,T), the collection of selves S∈S({a,b,c,},T) is a triple basis if f(a,{a,b},S) > f(b,{a,b},S) and f(·,A,S) is constant for all other A ⊆{a,b,c}. The model ( f,T ) is triple solvable with k utility functions if for every δ > 0, there is a triple basis S∈Sk({a,b,c},T) with maxa,b∈A,A⊆{a,b,c},s∈S|f(a,A,s)−f(b,A,s)|<δ ⁠. Given a model, it is easy to check for the existence of a triple basis. Indeed, triple bases can be found for the models featured earlier.13 For scale‐invariant aggregators, which satisfy the property that measuring utilities in a different unit does not change the ordering implied by the aggregator, checking the property is particularly simple, since it then suffices to construct one triple basis which can be scaled as needed. More generally, it is easy to see from our construction that it suffices for there to be triple bases using only |X| − 2 δs, where each is smaller than the amount of strict preference under the previous δs. It turns out that triple solvability holds broadly among the class of models featured here, and in fact models in the class F* generically satisfy this property. The fact that these examples illustrate various models of multi‐self decision making proposed in the literature suggests that this property, which can be checked simply by looking at choice behaviour on three‐element sets, holds broadly. As our next result shows, this behavioural property has strong implications for the explanatory power of a model. Theorem 2. Suppose the model (f,T)∈F is triple solvable with kf selves. Then, for any choice function c, defined on any finite grand set of alternatives X, no more than 1 + kfIIA(c) selves are needed to rationalise c. We sketch below the proof of Theorem 2, describing our general rationalisation method and its connection to our definition of an IIA violation. Note that an alternative statement of the result is as follows: using n selves, the model (f,T) can rationalise any choice function c, defined on any finite grand set of alternatives X, which has at most (n − 1)/kf IIA violation. Hence, the result also gives a lower bound on the set of rationalisable behaviour for a fixed number of selves, providing a linear connection between the number of IIA violations and the degree of freedom in the model (as measured by the number of selves). In online Appendix C, we apply this result to understand a generalised Strotzian model: a DM chooses a menu knowing her choice from the menu is a compromise among multiple motivations. Behaviour that is interpreted differently by the literature on choice over menus can arise from ‘anticipated’ IIA violations. Note that for each model ( f,T), the proportionality constant kf is independent of the size of the alternative space X and can be calculated using any triple of alternatives (it is simply the number of selves in a triple basis). This means that the number of selves that are sufficient to rationalise a choice function on the alternative space X does not increase if the choice function is extended to a larger alternative space X^ in a manner such that no additional IIA violations are created. This formalises the sense in which the size of the rationalisation depends directly on the complexity of the behaviour and not the size of the alternative space; the size of the alternative space matters only in the sense that it bounds the number of IIA violations that are possible. 4.1.1. Sketch of proof: a universal rationalisation method Suppose f is triple solvable with kf utility functions. Given an arbitrary X and any choice function c defined on X, the procedure works as follows. We examine all possible choice sets in X from smallest to largest, first going through all choice sets of size two, then all choice sets of size three, etc. We ignore any choice set that does not cause an IIA violation. For each choice set A causing an IIA violation, the construction creates a collection of selves SA whose utility functions, defined on X, correspond to those of a triple basis: c(A) plays the role of the preferred element a in {a,b}, A∖{c(A)} plays the role of the unchosen element b in this pair, and X ∖A plays the role of the third element c. The properties P1 and P5 (neutrality and profile equivalence) imply that the selves SA behave similarly to the triple basis: (i) under the model, the selves SA imply an aggregate strict preference for c(A) in every subset of A in which it is contained; and (ii) the selves SA are completely indifferent between all options for all other choice sets; that is, sets not containing c(A) or sets containing some element of X ∖A. Remember that a set A causes an IIA violation if there is a superset B such that c(B) ∈ A∖{c(A)}, and there is no set in between A and B (in terms of containment) which has an IIA implication for A. When we rationalise the choice from A using the selves SA, there is a ‘trickle‐down’ effect: following IIA, those selves continue to prefer c(A) in subsets of A. Another IIA violation may occur for some subset of A′ ⊂ A, where c(A) is available but not chosen. In the recursive procedure, the selves SA′ corresponding to the IIA violation in the smaller set A′ are constructed first. Upon reaching the set A, the triple basis used to generate SA must be indifferent enough over the alternatives so that the trickle‐down effect of SA does not overturn the strict preference of SA′. This is possible by P4 (continuity to near‐indifferent additions). Since the selves SA only induce strict aggregate preference in subsets of A which contain c(A), P3 (reinforcement) implies they do not affect the aggregate preference over alternatives in other sets. In particular, the selves SA do not interfere with selves constructed for any other IIA violations. Similarly, the selves SA′ do not interfere with the choice from larger sets, such as A or X. Of course, since the selves constructed for IIA violations have no implications for larger sets (such as X), there must be some self which accounts for those remaining choices as well. The construction is completed by creating an extra self s* which, using P4, is indifferent enough never to overturn any strict preferences from selves associated with IIA violations. Using P2 (consistency), this self is constructed via standard revealed preference, by allocating the highest utility to c(X), the next highest utility to X ∖{c(X)} etc. To summarise, since there are IIA(c) violations in the observed choice behaviour, and each triple basis has kf selves, this procedure thus generates 1 + kfIIA(c) selves. This collection of selves rationalises the observed choice behaviour c(·) under the model. As shown more formally in the Appendix, the selves generated for a set which causes an IIA violation ensure that the choice from that set is indeed picked under the model. Furthermore, their only other effect is to ensure that particular alternative continues to be picked in subsets in which it is contained – unless there is a subset which causes another IIA violation. In that case, selves are constructed which overturn any aggregate preferences coming from larger sets. All ‘rational’ choices which are not otherwise covered by the selves corresponding to IIA violations are implied by the final, rational self s*.14 It is easy to see that the proposed rationalisation procedure can be modified to generate rationalisations of choice correspondences, by extending our definition of IIA violations for choice functions to count both violations of Sen's α and Sen's β (axioms that, when taken together, are equivalent to rational choice behaviour for correspondences). Theorem 1, for scale‐invariant and type‐independent aggregators, is proved in five steps. The first is knowing that if f is triple solvable with k selves, we can rationalise any choice function c with 1 + kIIA(c) selves. This is simply 2. The next step is showing that if a certain matrix – constructed by permuting possible aggregate utility differences given various rankings of three alternatives a,b and c – has a non‐zero determinant, then the aggregator f is triple solvable. Next, we prove part (i) by showing that if the two types of irrational behaviours can be explained, then the above matrix has non‐zero determinant. To prove part (ii), we first show that if the sufficient condition described after Theorem 1 is satisfied, then the above matrix has a non‐zero determinant and the aggregator is triple solvable with five utility functions. Finally, we prove the sufficient condition is generically satisfied. For intuition on why the bound is five, notice that checking whether a collection constitutes a triple basis requires checking five aggregate utility differences: the aggregate utility difference between any two pairs of alternatives within the set {a,b,c} and the aggregate utility difference between the alternatives within each of the three pairs {a,b}, {b,c} and {a,c}. It turns out that a generic model in F* ‘stretches’ utility differences in a non‐linear, menu‐dependent fashion, and that under scale invariance, having five selves provides enough degrees of independence to ensure that a triple basis can be constructed. 5. Discussion This article studies a framework that encompasses many multi‐self models proposed in the literature. Our results have implications in both interpersonal and intrapersonal decision making, calling attention to the importance of collecting reliable information on the number of selves (motivations, in the interpersonal context) participating in the decision. We identify a class of models for which it is important to impose a priori restrictions on the number of selves in order to ensure falsifiability; outside this class of models, one can find examples where such restrictions are not needed. To our knowledge, the models treated here have not been characterised from a choice‐theoretic perspective. Indeed, the fact that all selves contribute to aggregate utility on every choice set can make it difficult to construct a rationalisation of a choice function, or ascertain whether a rationalisation exists. The proof of our result provides a universal procedure for constructing such a rationalisation. Appendix A Proof of Theorem 2. For an arbitrary choice function c, we will construct a collection of 1 + kIIA(c) selves which will be shown to rationalise c. This implies the claim in the theorem. In particular, we construct k selves for each set with which an IIA violation is associated, and an extra self for X. Let I1={A11,…,Ai11} be the subsets of X such that there is an IIA violation associated with the set, but there is no proper subset of the set with which an IIA violation is associated. For j ≥ 2, let Ij={A11,…,Aij+11} be the subsets of X such that there is an IIA violation associated with the set, but there is no proper subset of the set outside ⋃l=1j−1Il with which an IIA violation is associated. Let j* be the largest j such that Ij ≠ ∅. We will now iteratively construct a collection of k selves for each set associated with an IIA violation, starting with sets in I1. Consider any collection of k selves S¯1=〈s¯11…,s¯k1〉 that is a triple basis over {a,b,c} (existence follows from triple solvability). For every A ⊂ I1, construct now the following collection SA=〈s1A,…skA〉 where each self siA has type t¯i and utility function uiA defined over X by uiA(x)=u¯i1(a)ifx=c(A)u¯i1(b)ifx∈A,x≠c(A)u¯i1(c)ifx∉A. Suppose now that SA is defined for every A∈⋃k=1jIk for some j ≥ 1. Let Sk be the collection of selves Sk=〈SA1k,…,SAikk〉 ⁠, for k = 1,…,j. Let S^j=〈S1,…,Sj〉 ⁠. By P4, there exists δ > 0 such that for any δ‐indifferent collection of k selves S ′, f(a,A,S^j)>f(b,A,S^j) implies f(a,A,〈S^j,S′〉)>f(b,A,〈U^j,U′〉) ⁠. Then by P3 and P6, we know f(a,A,〈S^j,S~1,…,S~m〉)>f(b,A,〈S^j,S~1,…,S~m〉)impliesf(a,A,〈S^j,S~1,…,S~m,S′〉)>f(b,A,〉S^j,S~1,…,S~m,S′〉) for any (exactly) indifferent collections of selves S~1,…,S~m. Let now Ij+1={A11,…,Aij+11} be the subsets of X such that there is an IIA violation associated with the set but there is no proper subset of the set outside Ij with which an IIA violation is associated. By triple solvability with k selves, there is a δ‐indifferent collection of k selves S¯j+1=〈s¯1j+1,…,s¯kj+1〉 that is a triple basis over {a,b,c}. For every A ∈ Ij+1, construct the collection of selves SA=〈s1A,…,skA〉 where each self siA has type t¯ij+1 and the utility function uiA over X defined by 〈 for every i = 1,…,k. Let Sj+1 be the collection 〈Sj,SA11,…,SAij+11〉 ⁠. The above procedure generates a collection of kIIA(c) selves in j * steps. Then by P3 and P4, there is δj*>0 such that for any δj* ‐‐ indifferent self s*, f(a,A,Sj*)>f(b,A,Sj*) implies f(a,A,〈Sj*,s*〉)>f(b,A,〈Uj*,s*〉) ⁠. Finally, construct one more self s* in the following way. Let a1 = c(X) and ak = c(X ∖{a1,a2,…ak−1}) for 2 ≤ k ≤ n. Fix some t* ∈ T and let s* = (t*,u*), where we construct u*:X→R such that u*(a1) > u*(a2) > ⋯ > u*(an) and u* is δj* ‐‐ indifferent. We show that the collection Sc≡〈Sj*,s*〉 rationalises c under aggregator f. Observation A1. For any set A which is an IIA violation, by P1, P5 and construction of SA, f(a,B,SA) = f(b,B,SA)∀ B and a,b ∈ B such that B∖A ≠ ∅ or c(A) ∉ B, and f(c(A),B,SA) >f(b,B,SA) = f(b′,B,SA)∀b,b′ ∈ B∖{c(A)} and B such that B∖A = ∅ and c(A)∈ B. We will now show that the choice induced by f from any choice set is equal to the choice implied by c. First, note that this holds for X, since by Observation A1, f(a,X,SA) = f(b,X,SA) for every a,b ∈ X and every A with which there is an IIA violation associated. Moreover, f(c(X),X,s*)> f(a,X,s*)∀a ∈ X ∖{c(X)} by P2. Then repeated application of P3 implies f(c(X),X,Sc) > f(a,X,Sc)∀a ∈ X ∖{c(X)}. Next, consider any A⊊X which causes an IIA violation. Suppose A ∈ Ij. Observation A1 implies that for any B∈(⋃l=1jIl)\A ⁠, f(a,A,SB) = f(a′,A,SB)∀a,a ′ ∈ A, and f(c(A),A,SA) > f(a,A,SA)∀a ∈ A. Then repeated implication of P3 implies f(c(A),A,Sj) > f(a,A,Sj) ∀ a ∈ A. By construction then f(c(A),A,Sc) > f(a,A,Sc)∀a ∈ A. Finally, consider a set A that does not cause an IIA violation. There are several cases. Case 1. For all a ∈ A, there is no B ⊃ A such that a = c(B). Then by construction u*(c(A)) > u*(a)∀a ∈ A∖{c(A)}. Moreover, by Observation A1, f(a,A,SB) = f(a ′,A,SB)∀a,a ′ ∈ A and B with which an IIA violation is associated. Repeated use of P3, together with P2, implies f(c(A),A,Sc) > f(a,A,Sc)∀a ∈ A. Case 2. There is a unique a ∈ A such that for some B ⊃ A, c(B) = a. First, note that a = c(A) is necessary, otherwise A would have caused a violation. There are two subcases: Case 2a. For every B such that B ⊃A and c(B) = a, B did not cause an IIA violation. This means that for all B ⊃A, c(B) ∉ A∖{c(A)}. So just like in Case 1, u*(c(A)) > u*(a) for all a ∈ A∖{c(A)}, and f(a,A,SB) = f(a′,A,SB)∀a,a′ ∈ A and B with which an IIA violation is associated. Hence, f(c(A),A,Sc) > f(a,A,Sc) for all a ∈ A. Case 2b. There is B ⊃ A with c(B) = a such that B caused an IIA violation. Consider any smallest such B, and suppose B ∈ Ij. By Observation A1, for any A∈⋃l=1jIl either f(c(A),A,SB) > f(a,A,SB) for all a ∈ A, or f(a,A,SB) = f(a ′,A,SB) for all a,a ′ ∈ A. But then repeated application of P3 implies that f(c(A),A,Sj) > f(a,A,Sj) for all a ∈ A. By construction, f(c(A),A,Sc) > f(a,A,Sc) for all a ∈ A. Case 3. There are at least two elements in A that have each been chosen in some superset. First, note that one of those elements must be c(A), otherwise A would have caused an IIA violation. Let {bi}i be the set of elements other than c(A) such that bi ∈ A and bi = c(Bi) for some Bi ⊃ A. Drop any bi s such that Bi ⊃ Bm for some m and call the remaining set {bj}. Because A did not cause an IIA violation by assumption, it must be that for each bj there is Aj′ such that A⊂Aj′⊂Bj and c(Aj′)∈A ⁠. Because Bj does not contain any Bk, we know c(Aj′)=c(A) ⁠. For each j there may be multiple such Aj′s; consider only the maximal Aj′ with respect to the minimal Bj. Now by maximality, for any A′′ such that Aj′⊂A″⊂Bj ⁠, c(A′′) ∉ A. If there is A′′ such that c(A″)∈Aj′ ⁠, then c(A′′) ≠ c(A), by maximality of Aj′ ⁠. If for every A′′ it is the case that c(A″)∉Aj′ ⁠, then once again Aj′ caused an IIA violation with respect to B. Either way, we added selves to ensure choice c(A) for every Aj′ ⁠. Thus, a should be the choice from A unless selves were added for some B′ between some Aj′ and A for which c(B ′) ∈ A∖{a}. This is impossible by minimality of the Bj s. Proof of Theorem 1. Theorem 1 follows from Theorem 2 and the following three Lemmas. Let X = {a,b,c} and take any f∈F* ⁠. For compactness, we use the notation x1 = f(a,{a,b,c},S) −f(b,{a,b,c},S), x2 = f(b,{a,b,c},S) − f(c,{a,b,c},S), x3 = f(a,{a,c},S) − f(c,{a,c},S), x4 = f(b,{b,c},S)− f(c,{b,c},S), and x5 = f(a,{a,b},S) − f(b,{a,b},S). Lemma A1. If x3 ≠ x4 +x5, and if any one 2x1 + x2 − x3 − x5 = 0, x1 + 2x2 − x3 − x4 = 0, or x1 − x2 + x4 − x5 = 0 fails, then the aggregator is triple solvable (with kf at most 2 + 3|S|). Proof Consider the columns in Table A1. Table A1 Possible Matrix Columns for the Proof of Lemma A1 1:S . 2:(bc)(a) . 3:(ab)(c) . 4:(abc) . 5:(acb) . 6:a∼b ≻ c . 7:a ≻ b∼c . x1 x1 + x2 −x1 x2 −x1 − x2 0 x1 x2 −x2 x1 + x2 −x1 − x2 x1 x1 0 x3 x5 x4 −x5 −x4 x1 x1 x4 −x4 x3 −x3 x5 x1 0 x5 x3 −x5 x4 −x3 0 x1 1:S . 2:(bc)(a) . 3:(ab)(c) . 4:(abc) . 5:(acb) . 6:a∼b ≻ c . 7:a ≻ b∼c . x1 x1 + x2 −x1 x2 −x1 − x2 0 x1 x2 −x2 x1 + x2 −x1 − x2 x1 x1 0 x3 x5 x4 −x5 −x4 x1 x1 x4 −x4 x3 −x3 x5 x1 0 x5 x3 −x5 x4 −x3 0 x1 Table A1 Possible Matrix Columns for the Proof of Lemma A1 1:S . 2:(bc)(a) . 3:(ab)(c) . 4:(abc) . 5:(acb) . 6:a∼b ≻ c . 7:a ≻ b∼c . x1 x1 + x2 −x1 x2 −x1 − x2 0 x1 x2 −x2 x1 + x2 −x1 − x2 x1 x1 0 x3 x5 x4 −x5 −x4 x1 x1 x4 −x4 x3 −x3 x5 x1 0 x5 x3 −x5 x4 −x3 0 x1 1:S . 2:(bc)(a) . 3:(ab)(c) . 4:(abc) . 5:(acb) . 6:a∼b ≻ c . 7:a ≻ b∼c . x1 x1 + x2 −x1 x2 −x1 − x2 0 x1 x2 −x2 x1 + x2 −x1 − x2 x1 x1 0 x3 x5 x4 −x5 −x4 x1 x1 x4 −x4 x3 −x3 x5 x1 0 x5 x3 −x5 x4 −x3 0 x1 Column 1 lists aggregate utility values under S. By neutrality, if we can generate column 1, we can also generate the second column using the permutation (bc)(a) over alternatives, the third column using the permutation (ab)(c) over alternatives, etc. We generate columns 6 and 7 using profile equivalence to evaluate f ∘u and f ∘u ′ for the utility functions u and u ′ given by those headers. Determinants of three possible 5 × 5 matrices, each composed of five of the columns above, are Det(1|3|5|6|7)=x12(x1+2x2−x3−x4)(2x1+x2−x3−x5)(x3−x4−x5),Det(1|2|5|6|7)=x12(2x1+x2−x3−x5)(x3−x4−x5)(x1−x2+x4−x5),Det(2|3|4|6|7)=−x12(x1+2x2−x3−x4)(x3−x4−x5)(x1−x2+x4−x5). To prove the result, it suffices to show that there exists S such that defining x1,x2,…,x5 as above, one of the determinants above must be non‐zero. If one of those determinants is non‐zero, then we have find a vector (c1,c2,c3,c4,c5) such that the non‐singular matrix times (c1,c2,c3,c4,c5) is equal to (0,0,0,0,β) for some β ≠ 0. Using scaling, each ci can be pulled in so that the utilities of selves corresponding to the ith column are multiplied by ci. The resulting collection is a triple basis (and therefore we can get triple solvability through scaling that triple basis). The proof is completed in light of the linear dependence of the equations 2x1 + x2 − x3 − x5 = 0, x1 + 2x2 − x3 − x4 = 0 and x1 − x2 + x4 − x5 = 0: if any one of these fails, there must be a second which fails too. Lemma A2. Suppose there exist selves S defined over {a,b,c} such that x3 ≠ x4 + x5 and which rationalise under f the choice behaviour where the worst element in the transitive pairwise ranking is best in the triple. Then either 2x1 + x2 ≠ x3 + x5 or x1 + 2x2 ≠ x3 + x4.15 Proof By neutrality and symmetry of the condition x3 − x4 − x5 ≠ 0, there are two types of choice behaviour we must examine to prove the result: Case 1. a ≻Pb ≻Pc on the pairs, and c ≻Tb ⪰Ta on the triple. That is, x3,x4,x5 > 0, with x1 ≤ 0 and x2 < 0. But then 2x1 + x2 ≠ x3 + x5, since LHS < 0 and RHS > 0. Case 2. a ≻Pb ≻Pc on the pairs, and c ≻Ta ⪰Tb on the triple. That is, x3,x4,x5 > 0, with x1 ≥ 0, x2 < 0. If we can find S such that f rationalises this behaviour using the selves S, then observe that x1 + 2x2 is negative. Hence, x1 + 2x2 ≠ x3 + x4 because RHS > 0. Say that f∈F* is non‐degenerate if for some utility function u on X = {a,b,c}, we have x3 ≠ x4 + x5 and 2x1 + x2 ≠ x3 + x5 using the collection S consisting of one self with utility u. We formally establish that for any fixed scaling function ϕ(α) the property that an additive, neutral and scale‐invariant aggregator f∈F* is non‐degenerate holds generically. In order to define a topology on F* ⁠, we transform the latter set of aggregators to a convenient representation. Note that for a fixed scaling function, specifying the aggregated utilities of n alternatives for selves in the n‐dimensional simplex determines the aggregated utilities of n alternatives for all possible selves over n alternatives, since any self is a scalar multiple of exactly one self from the simplex. Hence, with respect to a grand set of alternatives with three elements, there is a natural bijection β between additive and scale‐invariant aggregators, and the set of pairs of operators Ω=(O1,O2|O1:Δ2→R2;O2:Δ3→R3), where O1 determines how a self's utilities get aggregated in pairs, and O2 determines how a self's utilities get aggregated in the triple. Define metric d on Ω such that the distance between (O1,O2) and (O1′,O2′) is defined as maxi=1,2supx∈Ri|Oi(x)−Oi′(x)| ⁠. Lemma A3. Given the topology induced by d, the pairs of operators in Ω that are associated with non‐degenerate aggregators in F* is open and dense relative to Ω. Proof For any utility function v and f∈F* ⁠, let s be a self with utility v and define 〈 1. Openness. Suppose f is non‐degenerate and let u:{a,b,c}→R satisfy the non‐degeneracy requirement. Note that u cannot be fully indifferent. Let εi=Γil(f,u)−Γir(f,u) for i ∈ {1,2} and let ε=max(|ε1|,|ε2|) ⁠. Next, for every i,j ∈ {a,b,c} such that i ≠ j, let αij be such that αij(u(i),u(j)) ∈ Δ2. Note that the terms αij are uniquely defined. Similarly, let αabc be such that αabc(u(a),u(b),u(c)) ∈ Δ3. Let α=max(|αab|,|αac|,|αbc|,|αabc|) ⁠. Since s is not an indifferent self, α > 0. Then for δ<ε/8α it holds that Γil(f′,u)≠Γir(f′,u) for i ∈ {1,2} for every f ′ such that |β(f)−β(f′)|<δ ⁠, since each term given f′ in the above inequalities can differ from the corresponding term given f by at most ε/8 ⁠. 2. Denseness. Let δ > 0. Consider a self s with utility u ∈ Δ3 over {a,b,c} such that u(a) >u(b) > u(c). For every i,j ∈ {a,b,c} such that i ≠ j, let αij satisfy αij(u(i),u(j)) ∈ Δ2. Let α=max(|αab|,|αac|,|αbc|) ⁠. If non‐degeneracy holds, there is trivially a point in the δ‐neighbourhood of β(f) corresponding to a non‐degenerate aggregator. Otherwise, let ε∈(0,δ/α) be such that ε≠|Γil(f,u)−Γir(f,u)| for i ∈ {1,2}. Take any f′∈F* for which (i) for triples, f ′ is equivalent to f; and (ii) for a pair {x,y}, given any utility function v over {x,y} for which v(x) ≥ v(y), f ′(x,{x,y},v) = f(x,{x,y},v) and f ′(y,{x,y},v) = f(y,{x,y},v) if v(x) − v(y) < u(a) − u(c), but f′(x,{x,y},v) = f(x,{x,y},v) + ɛ and f ′(y,{x,y},v) = f(y,{x,y},v) if v(x) − v(y) ≥ u(a) − u(c). Thus, with respect to selves for which the utility difference between elements of the pair is at least u(a) − u(c), aggregate utility is ɛ > 0 higher than what f yields for the preferred alternative (but is the same for other alternative) – otherwise f′ is equivalent to f. By construction, |β(f′)−β(f)|<δ ⁠. Also, Γ1l(f′,v)=Γ1l(f,v)+ε ⁠, Γ1r(f′,v)=Γ1r(f,v), Γ2l(f′,v)=Γ2l(f,v), and Γ2r(f′,v)=Γ2r(f,v)+ε. Then ε≠|Γil(f,v)−Γir(f,v)| for i ∈ {1,2} implies that Γil(f′,v)≠Γir(f′,v) for i ∈ {1,2}. Hence, f′ is non‐degenerate. Footnotes 1 " The IIA condition is also known as Sen's α (Sen, 1971) and for single‐valued choice, is equivalent to being able to describe choices as the maximisation of a strict, complete and transitive preference. 2 " Another approach allows for context dependence by considering extended choice situations where behaviour can depend on unspecified ancillary conditions or frames (Salant and Rubinstein, 2008; Bernheim and Rangel, 2009). While information effects can explain some context dependence (Sen, 1993), they cannot explain many systematic violations of IIA (Tversky and Simonson, 1993). 3 " Measuring the complexity of a rationalisation by the number of selves is akin to measuring the complexity of an automata by the number of states; for example, see Salant (2011) in the context of decision making. 4 " This paper's result on rationalisation is independent of their monotonicity theorem. 5 " In combinatorics, this object is also referred to as a multiset. 6 " Though aggregation in our framework is cardinal, the model has the ‘ordinal’ feature that there can be many ‘equivalent’ representations of an aggregator in this context. In particular, if f rationalises the choice function c using the selves S, then so does any increasing transformation of f. Similarly, given any representation S and f, one can obtain an equivalent representation by applying a monotone transformation of utilities in S, if a corresponding transformation is applied to the aggregation function f as well. 7 " We can permit aggregators with restricted domains: let R^X be a convex subset of RX and let Sn=(×i=1nR^X)×T ⁠. 8 " We note that an aggregator f encodes additional information, such as the ranking of unchosen alternatives in each set, that might be observable using a larger data set than that provided by a choice function. However, using only simple revealed preference on the choice from a menu, only the best choice from each set (i.e. the choice function) is elicited in light of the potential menu‐dependence of choices. 9 " P2 means that a fully indifferent self leads to aggregate indifference and iterating this and using P3 means that adding a finite number of indifferent selves does not affect an existing strict preference; if we were to assume full continuity on top of this, P4 would be implied. 10 " Tversky (1969) accounts for intransitive pairwise choice behaviour by positing utilities v1,v2,…,vn and an odd ϕ:R→R such that x ≻y if and only if Σi=1nϕ(vi(xi)−vi(yi))>0 ⁠. Observe that a is preferred to b in the pair {a,b} if and only if Σu∈UΦ(|u(a)−u(b)|)(u(a)−u(b)) ⁠, where each summand is an odd function of u(a)−u(b). 11 " The long‐run self's utility is equal to the short‐run self's utility plus the expected continuation value induced by the choice. If the latter can take any value, then ulr is not restricted by the short‐run utility usr. If continuation values cannot be arbitrary (e.g. they have to be non‐negative) then usr restricts the possible values of ulr, hence U has a restricted domain. In Fudenberg and Levine (2006) the utility functions also depend on a state variable y. Here, we suppress this variable, instead make the choice set explicit. 12 " Indeed, suppose that pairwise choices exhibit the transitive ranking a preferred to b preferred to c. Under our measure, there is one violation of IIA if c({a,b,c}) = b, which is defeated once in the pair {b,c}, and two violations of IIA if c({a,b,c}) = c, which is defeated twice. The alternative measure counts one violation either way. To see that the alternative measure can also be larger, consider the choice function over {a,b,c,d,e} which chooses the alphabetically lowest alternative in all sets, except that b is chosen in three‐element sets in which it is contained as well as from the pair {a,b}. The alternative measure counts four violations (e.g. one could switch choices on the sets {a,b,c}, {a,b,d}, {a,b,e} and {a,b} to a), while ours counts three (not considering {a,b} a violation). 13 " Solvability of the simple reference dependence model will follow from the sufficient condition it satisfies. For the case of the contextual concavity model of Kivetz et al. (2004), the following is a triple basis for any ρ ≠ 1: u1(a) = 4, u1(b) = 3, u1(c) = 1, u2(a) = 3, u2(b) = 1, u2(c) = 2, u3(a) = 3, u3(b) = 4, u3(c) = u4(a) = 1, u4(b) = u4(c) = 3, u5(a) = 2, u5(b) = 1, u5(c) = 3, u6(a) = 1, u6(b) = 2, u6(c) = 4. For the case of loss aversion with kinked linear ℓ and parameter 2, the following is a triple basis (there is some rounding error): u1(a) = −2.112, u1(b) = −1.275, u1(c) = 7.225, u2(a) = 0, u2(b) = 1.445, u2(c) = 1, u3(a) = 6, u3(b) = 7.225, u3(c) = 4, u4(a) = −4.766, u4(b) = −2.938, u4(c) =0, u5(a) = 5, u5(b) = −5.981, u5(c) = 2.814. For bargaining with endogenous disagreement point, the following is a triple basis (there is some rounding error): u1(a) = 2.847, u1(b) = 1, u1(c) = 7.634, u2(a) = 0, u2(b) = 4.288, u2(c) = 1, u3(a) = 6, u3(b) = −0.129, u3(c) = 4, u4(a) = −4.651, u4(b) = −0.949, u4(c) = 0, u5(a) = 5, u5(b) = −1.619, u5(c) = −15.8. 14 " While our goal was to show that any behaviour can be rationalised with sufficiently many selves, it is easy see that the above construction also goes through using a refinement of our definition of IIA violations, showing even more behaviour is rationalisable with a given number of selves. 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Google Scholar Crossref Search ADS WorldCat Author notes " We are grateful to Geoffroy de Clippel, Eddie Dekel, Drew Fudenberg, John Geanakoplos, Dino Gerardi, Tzachi Gilboa, Jerry Green, Daniel Hojman, Gil Kalai, Bart Lipman, Philippe Mongin, Wolfgang Pesendorfer, Ben Polak, Ariel Rubinstein, Philipp Sadowski, Larry Samuelson, Rani Spiegler, Tomasz Strzalecki and especially the editor and referees for valuable comments and suggestions. We also thank seminar audiences at Brown, Harvard, MIT, Montreal, NYU, UCL, Yale and the North American Summer Meeting of the Econometric Society. © 2013 Royal Economic Society

Burke,, Marshall;Gong,, Erick;Jones,, Kelly

doi: 10.1111/ecoj.12149pmid: N/A

Abstract We examine how variation in local economic conditions has shaped the AIDS epidemic in Africa. Using data from over 200,000 individuals across 19 countries, we match biomarker data on individuals' serostatus to information on local rainfall shocks, a large source of income variation for rural households. We estimate infection rates in HIV‐endemic rural areas increase by 11% for every recent drought, an effect that is statistically and economically significant. Income shocks explain up to 20% of variation in HIV prevalence across African countries, suggesting existing approaches to HIV prevention could be bolstered by helping households manage income risk better. The relationship between income and health has long been of interest to economists and a lengthy literature documents strong linkages between economic conditions and many important health outcomes (Currie, 2009). There has been much less progress, however, in understanding the economic foundations of the HIV/AIDS epidemic, one of the most important global health challenges. Such an understanding might yield particular dividends in sub‐Saharan Africa (SSA), where over a million people continue to become newly infected with the disease each year (UNAIDS, 2010). In this article, we explore the role of negative income shocks in shaping the evolution of the HIV/AIDS epidemic in Africa. Such shocks represent a well‐documented challenge to poor households around the world. Lacking access to formal savings and insurance, income shortfalls often force poor households to make difficult trade‐offs between short‐run consumption and longer‐run earnings and human capital accumulation (Rosenzweig and Wolpin, 1993; Ferreira and Schady, 2009; Maccini and Yang, 2009). Recent indirect evidence suggests that variation in income could also affect important disease outcomes, either by altering individual sexual behaviour (Robinson and Yeh, 2011b; Baird et al., 2012; Kohler and Thornton, 2012), or by affecting other phenomena such as migration or marriage timing that play a documented role in disease transmission (Lurie et al., 2003; Clark, 2004; Oster, 2012). Were income variation to play a role in HIV outcomes through any of these mechanisms, it would suggest that addressing income risk could play an important role in comprehensive HIV prevention strategies.1 Using one of the most widespread sources of income variation in the developing world – rainfall‐related shocks to agriculture – we directly assess the effect of negative income shocks on HIV outcomes across the African continent. We use the exogenous timing of rainfall events to develop an annual measure of shocks that is orthogonal to time‐invariant determinants of disease outcomes. Our definition of a shock is annual rainfall below the 15th percentile of the historical distribution of rainfall for a local area. Using data on roughly 2,000 individuals across 19 African countries, we compare the HIV status of individuals randomly exposed to a higher number of recent shocks (past 10 years) to the status of nearby individuals exposed to fewer recent shocks. We find that exposure to recent negative rainfall shocks substantially increases HIV infection rates in rural areas with high baseline HIV prevalence. Exposure to a single additional shock leads to a significant 11% increase in overall HIV infection. These results are robust to a variety of ways of constructing the shock measure, to a variety of controls, and to a set of placebo tests. Consistent with expectations, we find little effect of shocks in urban areas (where incomes should be less sensitive to rainfall) and in low‐prevalence regions (where there exists less HIV to be transmitted). We show that these individual‐level results are mirrored in the broader cross‐country patterns of HIV prevalence observed in SSA. Using country‐level data from UNAIDS, we show that exposure to shocks at the country level is also associated with significantly higher levels of HIV infection, and that our shock measure explains 14–21% of the cross‐country variation in HIV prevalence across SSA. This provides somewhat independent evidence on the role of shocks in shaping HIV outcomes, and implies that meteorological bad luck earlier on in the AIDS epidemic could have played a substantial role in shaping how the epidemic progressed over the following decades. While these reduced form results provide direct causal evidence that negative shocks substantially increase equilibrium HIV infection rates, they provide limited insight into the many channels through which shocks might shape HIV risk. For instance, adults may respond to shocks by temporarily migrating in search of work (Skoufias, 2003), or school‐aged girls may respond by marrying at an earlier age to increase economic security (Jensen and Thornton, 2003), both behaviour that is associated with an increased risk of HIV (Lurie et al., 2003; Clark, 2004). Alternatively, women may increase their sexual activity in response to economic hardship in order to obtain transfers (both monetary and in kind) from their male partners (Swidler and Watkins, 2007; Dinkelman et al., 2008; Robinson and Yeh, 2011b; LoPiccalo et al., forthcoming). This ‘transactional sex’ has been documented among women who are not commercial sex workers in numerous African countries and is believed to be a key driver in the AIDS epidemic (UNAIDS, 2010), a fact that has motivated numerous recent attempts to address the link between income and sexual behaviour through cash transfers (Baird et al., 2011; Handa et al., 2012; Kohler and Thornton, 2012; de Walque et al., 2012). While we are unable to definitively isolate the mechanism by which shocks increase HIV, we show that our data are largely inconsistent with either a migration or an early‐sexual‐debut explanation. In particular, we show that shocks do not induce earlier marriage or increased time away from one's village. Furthermore, we show that the effects of shocks on HIV are larger for men working outside of agriculture (whose purchasing power would have declined the least), evidence that is broadly consistent with an outward shift in the supply of transactional sex. This work contributes to the literature within and outside of economics that seeks to understand why the AIDS epidemic has disproportionately affected SSA. Our results provide strong evidence that a primary source of income variation for rural Africans – rainfall‐related variation in agricultural productivity – could be an important contributing factor to the epidemic. These results suggest that economic conditions play a significant role in the AIDS epidemic in SSA, and are related to previous work using macro‐level data to explore the effects of economic growth on the AIDS epidemic (Oster, 2012). We also contribute to a broader body of work on the health and livelihood consequences of income shocks. A host of papers show that when saving is difficult and insurance incomplete, negative income shocks can have seriously detrimental effects on longer‐run livelihood outcomes. In contrast to existing work, we identify behavioural responses that are not only detrimental to an individual's or household's well‐being but that also generate large negative health externalities for the community. As such, our results add further impetus to the growing effort aimed at increasing access to risk management tools in the developing world, and could suggest a role for public subsidy if the negative health externalities brought on by incomplete insurance are as large as we estimate. The rest of the article is organised as follows. In Section 1 we present a simple conceptual framework to motivate our empirical approach. Section 2 presents the data and our empirical methods. Section 3 discusses our main results and robustness checks, and Section 4 seeks evidence of behavioural pathways. Section 5 explores how these effects scale up to the country level. Finally, Section 6 discusses policy implications and concludes. 1. Conceptual Framework The goal of this article is to understand how economic conditions shape HIV risk. Our empirical approach examines how a plausibly exogenous source of income variation – exceptionally low rainfall realisations at a given location relative to long‐term averages (shocks) – affects local HIV outcomes. Our primary result establishes a strong positive relationship between these shocks and local HIV prevalence. We argue that this is a causal relationship because our shock measure is, by construction, uncorrelated with other time‐invariant factors that might also affect disease outcomes (see further discussion in Section 2). Here we discuss why rainfall‐related shocks might matter for HIV, and use this discussion to generate predictions of where and for whom, the reduced form relationship between drought and HIV should be largest. Our empirical analysis begins by examining the reduced‐form relationship between drought‐related shocks (S) and HIV infection, or ∂HIV/∂S. Define p as a measure of sexual risk, and z as income. The reduced form relationship between drought shocks and HIV can then be written as: ∂HIV∂S=∂HIV∂p∂p∂z∂z∂S(1) The three terms on the right‐hand side are the following: 1 ∂HIV/∂p represents the relationship between HIV infection and sexual risk. In the sub‐Saharan African setting, heterosexual sex is the primary driver of the epidemic (UNAIDS, 2010), and so deviations in the path of the epidemic are driven largely by changes in sexual behaviour. The risk of HIV infection is increasing in risky sexual behaviour such as having multiple concurrent partners or unprotected sex (∂HIV/∂p > 0) (Stoneburner and Low‐Beer, 2004; Epstein, 2007; Halperin and Epstein, 2008; Potts et al., 2008). Importantly, this relationship also depends on the prevalence of HIV in an area (λ). Regions with higher HIV prevalence will have a stronger relationship between sexual behaviour and new infections than regions with low prevalence ∂HIV/∂p∂λ > 0. 2 ∂p/∂z represents the impact of a deviation in income on sexual risk (p). A growing literature documents the importance of economic factors in shaping sexual risk in Africa (Robinson and Yeh, 2011b; Baird et al., 2012; Kohler and Thornton, 2012). Sexual risk can be measured as the number of partners and/or number of unprotected sexual acts but can also be measured by how likely a partner is to be infected with HIV. In Section 4, we discuss several ways identified by the literature by which shortfalls in income might alter sexual behaviour, all of which suggest a negative relationship between an income deviation and sexual risk for at least some subset of the population (i.e. ∂p/∂z < 0). Such mechanisms can broadly be considered coping behaviour in response to income shocks, and will be operative for different subsets of the population depending on the coping mechanism in question. 3 Finally, ∂z/∂S is the relationship between negative rainfall shocks and income shocks. As is frequently recognised in the literature, and as we demonstrate in online Appendix B, variation in rainfall generates substantial variation in both agricultural productivity and broader income measures in Africa. We expect that in rural areas (r), where most income is generated from rain‐fed agriculture, rainfall shocks will have a larger (negative) effect on income than in urban areas where agriculture is less important for the local economy (∂zr/∂Sr < ∂zu/∂Su ≤ 0). Because there is little disagreement in the literature on the signs of the first and third terms in (1), the overall sign of ∂HIV/∂S will depend on how sexual risk responds to variation in income. If we assume that this term is non‐zero, then two immediate predictions are generated from (1). 1 The effect of shocks on HIV will be larger (in absolute value) where baseline prevalence λ is higher. Intuitively, if shocks increase HIV through changes in sexual behaviour, the effect of shocks will be amplified in places where there is more HIV to transmit. 2 The effect of shocks on HIV will be larger (in absolute value) in rural areas where income is more dependent on agriculture (and therefore on rainfall). The sign of ∂p/∂z will determine the overall sign of ∂HIV/∂S. If some segment of the population copes with negative income shocks in a way that increases sexual risk, as is suggested by the literature, then (1) indicates that the overall relationship between HIV and shocks for these populations would be positive, ∂HIV/∂S > 0. 2. Empirical Methods 2.1. Individual HIV‐status Data Our individual‐level data are taken from 21 demographic and health surveys (DHS) conducted in 19 different sub‐Saharan countries.2 Of the existing DHS surveys available in early 2011, we employ all those that include results from individual‐level HIV‐tests as well as longitude and latitude information on the individual's location, allowing us to map households to data on shocks.3 For two countries (Kenya and Tanzania), two survey rounds matched these criteria, however, these are separate cross‐sections and creation of panel data at the individual or cluster level is not possible. Nonetheless, for each country both rounds are included in the analysis as entirely separate surveys.4 Each of these surveys randomly samples clusters of households from stratified regions and then randomly samples households within each cluster. In each sampled household, every woman aged 15–49 is asked questions regarding health, fertility and sexual behaviour.5 A men's sample is composed of all men within a specified age range within households selected for the men's sample.6 Depending on the survey, this is either all sampled households, or a random half (or third) of households within each cluster. Details regarding survey‐specific sampling are presented in online Appendix Table A1. In all households selected for the men's sample, all surveyed men and women are asked to provide a finger‐prick blood smear for HIV‐testing.7 By employing cluster‐specific inverse‐probability sampling weights, the HIV prevalence rates estimated with this data are representative at the national level.8 Table 1 gives the list of included surveys along with basic survey information. The compiled data contain over 8,000 clusters. On average, there are 25 surveyed individuals per cluster, and 90% of clusters contain between 10 and 50 surveyed individuals. In total, there are over 200,000 individuals in the pooled data. Table 1 also shows HIV prevalence rates for each survey. Overall, women's prevalence is 9.2% and men's is 6.2%. However, these numbers mask a range that varies widely from over 30% prevalence for women in Swaziland to less than 1% prevalence in Senegal. Given that the sexual behaviour response to income shocks will have different implications depending on HIV prevalence, we classify countries into two groups: low prevalence countries with less than 5%; and high prevalence countries with over 5% prevalence.9 Table 1 DHS Survey Information . . . . Prevalence . . Country . Year . Individuals . Female (%) . Male (%) . Overall (%) . Category . 1 Swaziland 2007 8,186 31.1 19.7 25.9 High 2 Lesotho 2004 5,254 26.4 18.9 23.2 High 3 Zambia 2007 26,098 21.1 14.8 18.1 High 4 Zimbabwe 2006 10,874 16.1 12.3 14.2 High 5 Malawi 2004 5,268 13.3 10.2 11.8 High 6 Mozambique 2009 10,305 12.7 9.0 11.1 High 7 Tanzania 2008 10,743 7.7 6.3 7.0 High 8 Kenya 2003 6,188 8.7 4.6 6.7 High 9 Kenya 2009 6,906 8.0 4.6 6.4 High 10 Tanzania 2004 15,044 6.6 4.6 5.7 High 11 Cameroon 2004 10,195 6.6 3.9 5.3 High 12 Rwanda 2005 10,391 3.6 2.2 3.0 Low 13 Ghana 2003 9,554 2.7 1.6 2.2 Low 14 Burkina Faso 2003 7,530 1.8 1.9 1.9 Low 15 Liberia 2007 11,688 1.9 1.2 1.6 Low 16 Guinea 2005 6,767 1.9 1.1 1.5 Low 17 Sierra Leone 2008 6,475 1.7 1.2 1.5 Low 18 Ethiopia 2005 11,049 1.9 0.9 1.4 Low 19 Mali 2006 8,629 1.5 1.1 1.3 Low 20 Congo DR 2007 8,936 1.6 0.9 1.3 Low 21 Senegal 2005 7,716 0.9 0.4 0.7 Low Total 203,796 9.2 6.2 7.8 . . . . Prevalence . . Country . Year . Individuals . Female (%) . Male (%) . Overall (%) . Category . 1 Swaziland 2007 8,186 31.1 19.7 25.9 High 2 Lesotho 2004 5,254 26.4 18.9 23.2 High 3 Zambia 2007 26,098 21.1 14.8 18.1 High 4 Zimbabwe 2006 10,874 16.1 12.3 14.2 High 5 Malawi 2004 5,268 13.3 10.2 11.8 High 6 Mozambique 2009 10,305 12.7 9.0 11.1 High 7 Tanzania 2008 10,743 7.7 6.3 7.0 High 8 Kenya 2003 6,188 8.7 4.6 6.7 High 9 Kenya 2009 6,906 8.0 4.6 6.4 High 10 Tanzania 2004 15,044 6.6 4.6 5.7 High 11 Cameroon 2004 10,195 6.6 3.9 5.3 High 12 Rwanda 2005 10,391 3.6 2.2 3.0 Low 13 Ghana 2003 9,554 2.7 1.6 2.2 Low 14 Burkina Faso 2003 7,530 1.8 1.9 1.9 Low 15 Liberia 2007 11,688 1.9 1.2 1.6 Low 16 Guinea 2005 6,767 1.9 1.1 1.5 Low 17 Sierra Leone 2008 6,475 1.7 1.2 1.5 Low 18 Ethiopia 2005 11,049 1.9 0.9 1.4 Low 19 Mali 2006 8,629 1.5 1.1 1.3 Low 20 Congo DR 2007 8,936 1.6 0.9 1.3 Low 21 Senegal 2005 7,716 0.9 0.4 0.7 Low Total 203,796 9.2 6.2 7.8 Note Prevalence estimates are weighted to be representative at the national level. Open in new tab Table 1 DHS Survey Information . . . . Prevalence . . Country . Year . Individuals . Female (%) . Male (%) . Overall (%) . Category . 1 Swaziland 2007 8,186 31.1 19.7 25.9 High 2 Lesotho 2004 5,254 26.4 18.9 23.2 High 3 Zambia 2007 26,098 21.1 14.8 18.1 High 4 Zimbabwe 2006 10,874 16.1 12.3 14.2 High 5 Malawi 2004 5,268 13.3 10.2 11.8 High 6 Mozambique 2009 10,305 12.7 9.0 11.1 High 7 Tanzania 2008 10,743 7.7 6.3 7.0 High 8 Kenya 2003 6,188 8.7 4.6 6.7 High 9 Kenya 2009 6,906 8.0 4.6 6.4 High 10 Tanzania 2004 15,044 6.6 4.6 5.7 High 11 Cameroon 2004 10,195 6.6 3.9 5.3 High 12 Rwanda 2005 10,391 3.6 2.2 3.0 Low 13 Ghana 2003 9,554 2.7 1.6 2.2 Low 14 Burkina Faso 2003 7,530 1.8 1.9 1.9 Low 15 Liberia 2007 11,688 1.9 1.2 1.6 Low 16 Guinea 2005 6,767 1.9 1.1 1.5 Low 17 Sierra Leone 2008 6,475 1.7 1.2 1.5 Low 18 Ethiopia 2005 11,049 1.9 0.9 1.4 Low 19 Mali 2006 8,629 1.5 1.1 1.3 Low 20 Congo DR 2007 8,936 1.6 0.9 1.3 Low 21 Senegal 2005 7,716 0.9 0.4 0.7 Low Total 203,796 9.2 6.2 7.8 . . . . Prevalence . . Country . Year . Individuals . Female (%) . Male (%) . Overall (%) . Category . 1 Swaziland 2007 8,186 31.1 19.7 25.9 High 2 Lesotho 2004 5,254 26.4 18.9 23.2 High 3 Zambia 2007 26,098 21.1 14.8 18.1 High 4 Zimbabwe 2006 10,874 16.1 12.3 14.2 High 5 Malawi 2004 5,268 13.3 10.2 11.8 High 6 Mozambique 2009 10,305 12.7 9.0 11.1 High 7 Tanzania 2008 10,743 7.7 6.3 7.0 High 8 Kenya 2003 6,188 8.7 4.6 6.7 High 9 Kenya 2009 6,906 8.0 4.6 6.4 High 10 Tanzania 2004 15,044 6.6 4.6 5.7 High 11 Cameroon 2004 10,195 6.6 3.9 5.3 High 12 Rwanda 2005 10,391 3.6 2.2 3.0 Low 13 Ghana 2003 9,554 2.7 1.6 2.2 Low 14 Burkina Faso 2003 7,530 1.8 1.9 1.9 Low 15 Liberia 2007 11,688 1.9 1.2 1.6 Low 16 Guinea 2005 6,767 1.9 1.1 1.5 Low 17 Sierra Leone 2008 6,475 1.7 1.2 1.5 Low 18 Ethiopia 2005 11,049 1.9 0.9 1.4 Low 19 Mali 2006 8,629 1.5 1.1 1.3 Low 20 Congo DR 2007 8,936 1.6 0.9 1.3 Low 21 Senegal 2005 7,716 0.9 0.4 0.7 Low Total 203,796 9.2 6.2 7.8 Note Prevalence estimates are weighted to be representative at the national level. Open in new tab Since the DHS surveys in each country were conducted in different years, we include survey fixed effects in all of our analysis. This controls for any effects that national policies might have on the HIV/AIDS epidemic as well as any time trends of the epidemic. Our analysis is thus focused on making comparisons within country in a given year. 2.2. Weather Data and Construction of Shocks To understand how economic shocks shape HIV outcomes, we seek a shock measure that satisfies three criteria: derived shocks are economically meaningful, they are orthogonal to other factors that might also shape disease outcomes and they capture the potential disjoint between when HIV is acquired and when the individual is observed in the DHS. Because we do not directly observe variation in economic performance at a disaggregated level, and because such variation is likely endogenous to disease outcomes, we adopt an approach that is common in the literature and use variation in weather as a proxy for variation in economic productivity. For the largely agrarian societies of Africa, variation in weather directly shapes the economic productivity of the majority of the population that continues to depend on agriculture for their livelihood (Davis et al., 2010). As we show below, particularly negative rainfall realisations substantially depress agricultural productivity across the region. Our weather data are derived from the ‘UDel’ (University of Delaware) data set, a 0.5 × 0.5 degree gridded monthly temperature and precipitation data set (Matsuura and Willmott, 2009). These gridded data are based on interpolated weather station data and have global coverage over land areas from 1900–2008.10 Using the latitude/longitude data in the DHS, we match each DHS cluster to the weather grid cell in which it falls. Because latitude/longitude data in the DHS are recorded at the cluster level, all individuals within a given cluster are assigned the same weather. Our DHS data match to 1,701 distinct grid cells in the UDel data. To capture the seasonality of agriculture, we construct grid‐level estimates of ‘crop year’ rainfall, where the crop year is defined as the 12 months following planting for the main growing season in a region.11 Annual crop year rainfall estimates are generated by summing monthly rainfall across these twelve ‘crop year’ months at a given location. To capture shocks to economic productivity that are both meaningful and orthogonal to potential confounders, one must identify years in which accumulated rainfall was unusually low relative to what is normally experienced in a particular location. The most common way this has been done is by using the deviation from the local mean in a year or season, either in levels (Paxson, 1992; Fafchamps et al., 1998; Rose, 1999; Jayachandran, 2006; Tiwari et al., 2013), in percentage (Dercon, 2004), or in standard deviation units (Hidalgo et al., 2010). Unfortunately, none of these methods is useful for summing shocks over a number of years, as the high years would offset the low years.12 To avoid this offsetting, we require a binary rather than continuous indicator for whether a year constitutes a shock or not. We define shocks as rainfall below a threshold that is determined by the local rainfall distribution. In particular, for each of our 1,701 grid cells, we fit the history of crop‐year rainfall realisations to a grid‐specific gamma distribution and assign each grid‐year to its corresponding percentile in that distribution.13 A ‘shock’ is then defined as a realisation below a pre‐determined percentile in the location‐specific distribution. The literature does not provide definitive estimates of the percentile below which a shock becomes meaningful, and unfortunately disaggregated (e.g. grid) measures of economic productivity over time are not available.14 To make progress, we construct an analogous measure of rainfall shocks at the country level and assess how country‐level agricultural productivity and GDP growth respond to these shocks.15 Resulting estimates from panel regressions of country level maize yields or GDP growth on percentile rainfall realisations (purged of country and time fixed effects) are shown in Figure 1. Maize is the continent's primary staple crop, the crop grown by the majority of smallholder farmers, and thus perhaps the best direct measure of rural incomes. Point estimates from these panel regressions suggest that realisations below about the 15th percentile are the most harmful to maize yields (Figure 1, left panel). A similar pattern is found in GDP growth (right panel). We thus adopt this 15% threshold as our initial measure of a ‘shock’ – i.e. we define a shock as a crop‐year rainfall realisation below the 15% quantile of the local rainfall distribution – and show that our results are robust to other threshold choices in the neighbourhood of 15%, as well as to other plausible methods of constructing binary shocks. Fig. 1. Open in new tabDownload slide Effect of Rainfall Shocks on African Maize Yields (left panel) and Per Capita GDP Growth (right panel) Notes. Data are at the country level over the period 1970–2008, and include all sub‐Saharan African countries. Dark lines display point estimates from kernel‐weighted local polynomial regressions of the outcome on rainfall percentiles, after removing country and year fixed effects. Grey areas represent 95% confidence intervals. Data sources are given in the text. Fig. 1. Open in new tabDownload slide Effect of Rainfall Shocks on African Maize Yields (left panel) and Per Capita GDP Growth (right panel) Notes. Data are at the country level over the period 1970–2008, and include all sub‐Saharan African countries. Dark lines display point estimates from kernel‐weighted local polynomial regressions of the outcome on rainfall percentiles, after removing country and year fixed effects. Grey areas represent 95% confidence intervals. Data sources are given in the text. Finally, because the DHS only observes the disease status of a particular individual at one point in time, and an HIV+ individual could have become infected at any time over the previous decade or longer (median survival time at infection with HIV in SSA, if untreated, is 9.8 years (Morgan et al., 2002), our main independent variable is the number of these shocks that have occurred over the 10 years prior to the survey year at a given location. For instance, if an individual was surveyed in the DHS in 2007, the shock variable takes on a value of between 0 and 10 corresponding to the number of crop‐year rainfall realisations in that individual's region between 1997 and 2006 that fell below the 15% cut‐off in the local rainfall distribution. We sum the shocks because acquiring HIV is irreversible – if a shock led to an HIV infection seven years ago, and that individual is still alive, they will be HIV‐positive today – and thus past shocks should have a demonstrable effect on current HIV infection. We again note that using a more continuous measure of rainfall – e.g. deviations from average rainfall in levels – would tend to obscure past shocks: the sum of a very bad year and a very good year would be similar to the sum of two normal years. The mean and SD of shocks by cluster are shown in Table 2. Table 2 Shock Prevalence by Country Prevalence rank . Country . Survey year . Mean shocks . SD shocks . Number of clusters . Weather grids . 1 Swaziland 2007 2.90 0.46 275 13 2 Lesotho 2004 1.89 0.44 405 18 3 Zambia 2007 0.84 0.75 319 146 4 Zimbabwe 2006 1.28 0.76 398 122 5 Malawi 2004 1.04 0.75 521 53 6 Mozambique 2009 2.54 1.51 270 115 7 Tanzania 2008 0.77 0.82 345 167 8 Kenya 2003 1.17 0.62 400 81 9 Kenya 2009 1.22 0.78 398 93 10 Tanzania 2004 1.92 0.93 475 178 11 Cameroon 2004 1.59 1.06 466 112 12 Rwanda 2005 2.37 0.61 462 14 13 Ghana 2003 1.31 0.80 412 71 14 Burkina Faso 2003 1.28 0.90 400 88 15 Liberia 2007 1.35 1.05 298 37 16 Guinea 2005 1.34 0.75 295 72 17 Sierra Leone 2008 3.00 0.00 353 27 18 Ethiopia 2005 1.12 1.12 535 167 19 Mali 2006 1.00 0.71 407 149 20 Congo DR 2007 1.89 1.06 300 168 21 Senegal 2005 0.70 0.69 376 61 Total 1.51 1.04 8,110 1,701 Prevalence rank . Country . Survey year . Mean shocks . SD shocks . Number of clusters . Weather grids . 1 Swaziland 2007 2.90 0.46 275 13 2 Lesotho 2004 1.89 0.44 405 18 3 Zambia 2007 0.84 0.75 319 146 4 Zimbabwe 2006 1.28 0.76 398 122 5 Malawi 2004 1.04 0.75 521 53 6 Mozambique 2009 2.54 1.51 270 115 7 Tanzania 2008 0.77 0.82 345 167 8 Kenya 2003 1.17 0.62 400 81 9 Kenya 2009 1.22 0.78 398 93 10 Tanzania 2004 1.92 0.93 475 178 11 Cameroon 2004 1.59 1.06 466 112 12 Rwanda 2005 2.37 0.61 462 14 13 Ghana 2003 1.31 0.80 412 71 14 Burkina Faso 2003 1.28 0.90 400 88 15 Liberia 2007 1.35 1.05 298 37 16 Guinea 2005 1.34 0.75 295 72 17 Sierra Leone 2008 3.00 0.00 353 27 18 Ethiopia 2005 1.12 1.12 535 167 19 Mali 2006 1.00 0.71 407 149 20 Congo DR 2007 1.89 1.06 300 168 21 Senegal 2005 0.70 0.69 376 61 Total 1.51 1.04 8,110 1,701 Open in new tab Table 2 Shock Prevalence by Country Prevalence rank . Country . Survey year . Mean shocks . SD shocks . Number of clusters . Weather grids . 1 Swaziland 2007 2.90 0.46 275 13 2 Lesotho 2004 1.89 0.44 405 18 3 Zambia 2007 0.84 0.75 319 146 4 Zimbabwe 2006 1.28 0.76 398 122 5 Malawi 2004 1.04 0.75 521 53 6 Mozambique 2009 2.54 1.51 270 115 7 Tanzania 2008 0.77 0.82 345 167 8 Kenya 2003 1.17 0.62 400 81 9 Kenya 2009 1.22 0.78 398 93 10 Tanzania 2004 1.92 0.93 475 178 11 Cameroon 2004 1.59 1.06 466 112 12 Rwanda 2005 2.37 0.61 462 14 13 Ghana 2003 1.31 0.80 412 71 14 Burkina Faso 2003 1.28 0.90 400 88 15 Liberia 2007 1.35 1.05 298 37 16 Guinea 2005 1.34 0.75 295 72 17 Sierra Leone 2008 3.00 0.00 353 27 18 Ethiopia 2005 1.12 1.12 535 167 19 Mali 2006 1.00 0.71 407 149 20 Congo DR 2007 1.89 1.06 300 168 21 Senegal 2005 0.70 0.69 376 61 Total 1.51 1.04 8,110 1,701 Prevalence rank . Country . Survey year . Mean shocks . SD shocks . Number of clusters . Weather grids . 1 Swaziland 2007 2.90 0.46 275 13 2 Lesotho 2004 1.89 0.44 405 18 3 Zambia 2007 0.84 0.75 319 146 4 Zimbabwe 2006 1.28 0.76 398 122 5 Malawi 2004 1.04 0.75 521 53 6 Mozambique 2009 2.54 1.51 270 115 7 Tanzania 2008 0.77 0.82 345 167 8 Kenya 2003 1.17 0.62 400 81 9 Kenya 2009 1.22 0.78 398 93 10 Tanzania 2004 1.92 0.93 475 178 11 Cameroon 2004 1.59 1.06 466 112 12 Rwanda 2005 2.37 0.61 462 14 13 Ghana 2003 1.31 0.80 412 71 14 Burkina Faso 2003 1.28 0.90 400 88 15 Liberia 2007 1.35 1.05 298 37 16 Guinea 2005 1.34 0.75 295 72 17 Sierra Leone 2008 3.00 0.00 353 27 18 Ethiopia 2005 1.12 1.12 535 167 19 Mali 2006 1.00 0.71 407 149 20 Congo DR 2007 1.89 1.06 300 168 21 Senegal 2005 0.70 0.69 376 61 Total 1.51 1.04 8,110 1,701 Open in new tab By construction, this shock measure should be orthogonal to other confounding variables. Because shocks at a given location are defined relative to that location's historical rainfall distribution, and the same percentile cut‐off is used in each location to define a shock (instead of the same absolute cutoff), all locations have the same expected number of shocks over any given 10 year period: each year any location has a 15% chance of experiencing a shock. But because rainfall in a given location varies over time, some 10‐year time windows will accumulate more shocks than other windows, and it is this plausibly random variation that we exploit.16 We confirm in online Appendix B that accumulated rainfall shocks are orthogonal to the first three moments of the rainfall distribution, providing additional confidence that our shock measure is uncorrelated with other time‐invariant unobservables that might also affect HIV outcomes. This definition of shocks assumes that relative (rather than absolute) deviations in rainfall are what matter for income and HIV outcomes. This construction is necessary for identification – using an absolute threshold for a shock would mean that areas with lower or more variable rainfall would expect more shocks, and these areas could differ in other unobserved ways that matter for HIV – but it also plausibly captures what is important in our setting. Farmers choose crops that are adapted to the conditions under which they are grown, with farmers in drought‐prone regions in Africa sowing crops (such as millet and sorghum) that can withstand low rainfall realisations and farmers in areas with higher average rainfall sowing crops that are generally higher yielding but less tolerate of drought (e.g. maize). The results in Figure 1, which are constructed using this relative shock measure, confirm that relative deviations matter for both agricultural outcomes and broader economic performance. 2.3. Estimation To explore the effects of negative income shocks on individual HIV rates, we estimate the following: HIVijk=α+β1Sjt+xi′δ+γrj+ωk+εijk,(2) where HIVijk is an indicator for whether individual i in cluster j tested HIV‐positive in survey k. Sjt is the number of rainfall shocks that cluster j has experienced in the t years before the survey. The default indicator for Sjt is the number of crop‐years with rainfall at or below the 15% quantile in the last 10 years for a given cluster. Note again that by construction, no one cluster is any more shock prone than another, i.e. E(Smt)=E(Snt)∀j=m,n ⁠. All clusters expect the same total number of shocks over the 38 years in our rainfall data and our identifying variation comes from the random timing of these shocks: some clusters happen to receive more of their shocks in the decade immediately before we observe them and others receive fewer. Both t and the definition of S are varied over a range to test the robustness of results. The vector xi contains characteristics of individual i that are not affected by shocks, specifically, gender and age. rj indicates that cluster j is rural. The vector of survey fixed effects is ωk and εijk is a mean‐zero error term.17 We estimate linear probability models, allowing for correlation of error terms across individuals in the same weather grid. Survey‐specific sampling weights are used to make the results representative of individuals living in these 19 countries in SSA (see online Appendix A). 3. Results 3.1. Main Results Table 3 shows estimates of (2), employing various samples and interaction terms. The overall effect of shocks on HIV rates using the full sample is 0.3 percentage points (ppt) and is statistically significant at the 10% level (column 1). Our conceptual framework predicts differential effects depending on whether an individual lives in an urban or rural area, and in line with this prediction we find that the effects are concentrated in rural areas. We cannot reject the hypothesis that urban effects are zero (column 2; linear combination), and the difference between estimates for rural and urban areas is borderline significant at conventional levels (p = 0.104). Focusing our analysis on rural areas (column 3), we find that shocks have a meaningful effect: we estimate that each shock leads to a 0.3 ppt increase in HIV prevalence, an effect that is significant at the 5% level and that corresponds to a 7.3% increase in HIV rates given a mean of 4.1%. The second prediction from our framework is that increases in risky behaviour as a result of an income shock would result in little change in HIV infection rates if existing HIV prevalence is very low. To capture differential effects by baseline prevalence, we focus on the rural sample and include an interaction between shocks and an indicator for low‐prevalence countries. In countries with low prevalence (less than 5%), shocks have an approximately zero effect on HIV (column 4; linear combination), and we reject equality across low and high prevalence countries with 95% confidence (column 4; shocks × low prevalence). Column (5) presents the estimation for the rural sample in high prevalence countries only. In these areas, each shock increases HIV by 0.8 ppt, an 11% increase based on overall prevalence of 7%. Finally, column (6) disaggregates the impact by gender. We find that shocks increase the probability of infection by 0.9 ppt for women and 0.6 ppt for men, both of which are statistically significant at the 5% level. Given that HIV prevalence is 8.3% for women and 5.6% for men in high prevalence rural areas, these estimates represent large effect sizes of 11% increases in HIV per shock for both women and men. We cannot reject the hypothesis that the effect size is the same across genders (column 6; shock × male). The magnitude of these effects is meaningful. In our entire sample, the mean number of shocks is 1.5, which, combined with our primary results, suggests that drought‐induced income shocks lead to a 17% increase in HIV risk over a 10‐year period. We also can attempt to estimate roughly an income elasticity with respect to HIV risk.18 We estimate that each drought shock results in a 7–10% loss in annual income (see online Appendix B), which leads to an 11% increase in HIV infection risk. This result is similar to results from Robinson and Yeh (2011b) which show that a 3% loss in income leads to about an 8% increase in HIV risk.19 Both results suggest that better means of consumption smoothing can have implications for the HIV/AIDS epidemic. Table 3 Effect of Shocks on HIV . All . All . Rural . Rural . Rural high prevalence . Rural high prevalence . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.003* 0.004** 0.003** 0.008** 0.008** 0.009** (0.001) (0.002) (0.002) (0.003) (0.003) (0.004) Shocks × urban −0.004 (0.002) Shocks × low prevalence countries −0.008** (0.003) Shocks × male −0.003 (0.003) Interaction p‐value 0.104 0.016 0.243 Linear combination −0.000 −0.000 0.006** (0.002) (0.001) (0.003) Observations 202,216 202,216 134,874 134,874 77,760 77,760 R2 0.053 0.053 0.046 0.046 0.030 0.030 Mean of dependent variable 0.050 0.050 0.041 0.041 0.070 0.070 . All . All . Rural . Rural . Rural high prevalence . Rural high prevalence . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.003* 0.004** 0.003** 0.008** 0.008** 0.009** (0.001) (0.002) (0.002) (0.003) (0.003) (0.004) Shocks × urban −0.004 (0.002) Shocks × low prevalence countries −0.008** (0.003) Shocks × male −0.003 (0.003) Interaction p‐value 0.104 0.016 0.243 Linear combination −0.000 −0.000 0.006** (0.002) (0.001) (0.003) Observations 202,216 202,216 134,874 134,874 77,760 77,760 R2 0.053 0.053 0.046 0.046 0.030 0.030 Mean of dependent variable 0.050 0.050 0.041 0.041 0.070 0.070 Notes Column headers indicate sample employed. Specifications include controls for gender and age, rural/urban designation (where applicable), and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. ‘Interaction p‐value’ is the p‐value for shocks × urban (column 2), shocks × low prevalence countries (column 4), and shocks × male (column 6). ‘Linear combination’ is the sum of coefficients on the number of shocks and the interaction term in each specification. For column (2), the linear combination is (No. of shocks past 10 years) + (shocks × urban), column (4) is (no. of shocks past 10 years) + (shocks × low prevalence Co.), and column (6) is (no. of shocks past 10 years) + (shocks × male). Significance levels: **5%, *10%. Open in new tab Table 3 Effect of Shocks on HIV . All . All . Rural . Rural . Rural high prevalence . Rural high prevalence . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.003* 0.004** 0.003** 0.008** 0.008** 0.009** (0.001) (0.002) (0.002) (0.003) (0.003) (0.004) Shocks × urban −0.004 (0.002) Shocks × low prevalence countries −0.008** (0.003) Shocks × male −0.003 (0.003) Interaction p‐value 0.104 0.016 0.243 Linear combination −0.000 −0.000 0.006** (0.002) (0.001) (0.003) Observations 202,216 202,216 134,874 134,874 77,760 77,760 R2 0.053 0.053 0.046 0.046 0.030 0.030 Mean of dependent variable 0.050 0.050 0.041 0.041 0.070 0.070 . All . All . Rural . Rural . Rural high prevalence . Rural high prevalence . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.003* 0.004** 0.003** 0.008** 0.008** 0.009** (0.001) (0.002) (0.002) (0.003) (0.003) (0.004) Shocks × urban −0.004 (0.002) Shocks × low prevalence countries −0.008** (0.003) Shocks × male −0.003 (0.003) Interaction p‐value 0.104 0.016 0.243 Linear combination −0.000 −0.000 0.006** (0.002) (0.001) (0.003) Observations 202,216 202,216 134,874 134,874 77,760 77,760 R2 0.053 0.053 0.046 0.046 0.030 0.030 Mean of dependent variable 0.050 0.050 0.041 0.041 0.070 0.070 Notes Column headers indicate sample employed. Specifications include controls for gender and age, rural/urban designation (where applicable), and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. ‘Interaction p‐value’ is the p‐value for shocks × urban (column 2), shocks × low prevalence countries (column 4), and shocks × male (column 6). ‘Linear combination’ is the sum of coefficients on the number of shocks and the interaction term in each specification. For column (2), the linear combination is (No. of shocks past 10 years) + (shocks × urban), column (4) is (no. of shocks past 10 years) + (shocks × low prevalence Co.), and column (6) is (no. of shocks past 10 years) + (shocks × male). Significance levels: **5%, *10%. Open in new tab 3.2. Robustness of Results In this sub section, we examine whether our primary result – the large response of HIV to shocks in rural, high prevalence areas shown in Table 3, column (5) – is robust to various issues of specification, variable definition, sample selection, and omitted variables. 3.2.1. Specification We first examine whether our results are sensitive to the specification or sample used. We sequentially remove individual level controls, remove population weights and replace survey‐year‐fixed effects with country and year‐fixed effects and our results remain stable (Table 4; columns 1–3). We also vary the sample used, removing hyper‐endemic countries such as Swaziland and Lesotho where HIV‐prevalence exceeds 20%, and our results remain stable (column 4). Finally, within each DHS cluster (i.e. village), we remove all visitors from the sample, defined as those who have lived in the area for less than a year at the time of the survey. We do this for two reasons. First, we want to identify the effect of shocks on HIV for those who were actually living in the area at the time of the shock and removing visitors helps us establish this. Second, it may be that rainfall shocks are inducing NGO and government workers to migrate into drought afflicted areas and, if these types are more likely to be HIV+, then this could potentially explain our results. Removal of these visitors from the sample does not change our results (column 5). We also present an estimate that employs only the most recent survey from each country, excluding the KE 2003 and TZ 2004 surveys, which produces similar results (column 6). Finally, we provide results only for individuals who were between the ages of 15 and 50 when the shocks occurred (column 7). These individuals would be likely to have the greatest response in terms of sexual behaviour and we do find a result that is slightly increased over our main specification. 3.2.2. Shock definition We also examine the sensitivity of our results to the definition of a shock. While our primary specification defines a shock as a crop‐year rainfall realisation below the 15th percentile of local realisations, the choice of the 15th percentile is somewhat arbitrary. We vary the cut‐off for shock definition in increments of 1% between the 5th and 40th percentile. The estimated coefficients for each percentile are presented in Figure 2. Overall, the point estimate is relatively stable around our default 15th percentile shock measure and, as the definition of a shock becomes less (more) severe, the point estimates generally decrease (increase). Shocks in the neighbourhood between the 10th and 20th percentile generate similar results, although they become less precisely estimated the further they are from the 15th percentile (see Table C1).20 Table 4 Robustness Checks . No controls . No weights . Country‐year fixed effects . No hyper‐endemic . No visitors . One survey per country . Aged 15–50 at shock . Alternative shock definitions . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . Panel (a): robustness to specifications and sample No. of shocks past 10 years 0.008** 0.005* 0.008** 0.008** 0.011*** 0.009** 0.010** (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) (0.004) Shocks, defined by 1970–95 weather 0.006** (0.003) 1.5 SD shocks, past 10 years 0.015** (0.007) Observations 77,760 77,760 77,760 68,287 53,596 65,326 36,812 77,760 77,760 R2 0.020 0.068 0.030 0.025 0.035 0.035 0.033 0.030 0.030 Mean of dependent variable 0.070 0.110 0.070 0.068 0.063 0.078 0.158 0.070 0.070 . No controls . No weights . Country‐year fixed effects . No hyper‐endemic . No visitors . One survey per country . Aged 15–50 at shock . Alternative shock definitions . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . Panel (a): robustness to specifications and sample No. of shocks past 10 years 0.008** 0.005* 0.008** 0.008** 0.011*** 0.009** 0.010** (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) (0.004) Shocks, defined by 1970–95 weather 0.006** (0.003) 1.5 SD shocks, past 10 years 0.015** (0.007) Observations 77,760 77,760 77,760 68,287 53,596 65,326 36,812 77,760 77,760 R2 0.020 0.068 0.030 0.025 0.035 0.035 0.033 0.030 0.030 Mean of dependent variable 0.070 0.110 0.070 0.068 0.063 0.078 0.158 0.070 0.070 Mean . Variance . Skew . All Moments . (10) . (11) . (12) . (13) . Panel (b): robustness to controlling for moments of rainfall distribution No. of shocks past 10 years 0.009*** 0.008** 0.008** 0.008*** (0.003) (0.003) (0.003) (0.003) Observations 77,760 77,760 77,760 77,760 R2 0.031 0.030 0.030 0.031 Mean of dependent variable 0.070 0.070 0.070 0.070 Mean . Variance . Skew . All Moments . (10) . (11) . (12) . (13) . Panel (b): robustness to controlling for moments of rainfall distribution No. of shocks past 10 years 0.009*** 0.008** 0.008** 0.008*** (0.003) (0.003) (0.003) (0.003) Observations 77,760 77,760 77,760 77,760 R2 0.031 0.030 0.030 0.031 Mean of dependent variable 0.070 0.070 0.070 0.070 Notes Rural sample from high‐prevalence countries. All specifications include controls for gender, age and survey fixed effects, except as noted. Column 1 does not include the standard controls. Column (2) does not include weights. Column (3) employs country‐year fixed effects, rather than survey fixed effects. Column (4) excludes the hyper‐endemic countries (Swaziland and Lesotho). Column (5) excludes individuals who have lived in their current village for less than one year. Column (6) employs only the most recent survey from each country (excludes KE 2003 and TZ 2004). Column (7) includes only individuals who are aged 15+ at the time of the shocks. Columns (8)–(9) employ alternative definitions of a shock: column (8) employs only the years 1970–95 to create the historical distribution from which the 15th percentile is the shock definition, and column (9) defines a shock as 1.5 SD below the local mean. Columns (10) ‐ (13) include additional controls as shown in column headers. Estimators are weighted to be representative of the 19 countries, except as noted. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Table 4 Robustness Checks . No controls . No weights . Country‐year fixed effects . No hyper‐endemic . No visitors . One survey per country . Aged 15–50 at shock . Alternative shock definitions . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . Panel (a): robustness to specifications and sample No. of shocks past 10 years 0.008** 0.005* 0.008** 0.008** 0.011*** 0.009** 0.010** (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) (0.004) Shocks, defined by 1970–95 weather 0.006** (0.003) 1.5 SD shocks, past 10 years 0.015** (0.007) Observations 77,760 77,760 77,760 68,287 53,596 65,326 36,812 77,760 77,760 R2 0.020 0.068 0.030 0.025 0.035 0.035 0.033 0.030 0.030 Mean of dependent variable 0.070 0.110 0.070 0.068 0.063 0.078 0.158 0.070 0.070 . No controls . No weights . Country‐year fixed effects . No hyper‐endemic . No visitors . One survey per country . Aged 15–50 at shock . Alternative shock definitions . . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . Panel (a): robustness to specifications and sample No. of shocks past 10 years 0.008** 0.005* 0.008** 0.008** 0.011*** 0.009** 0.010** (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) (0.004) Shocks, defined by 1970–95 weather 0.006** (0.003) 1.5 SD shocks, past 10 years 0.015** (0.007) Observations 77,760 77,760 77,760 68,287 53,596 65,326 36,812 77,760 77,760 R2 0.020 0.068 0.030 0.025 0.035 0.035 0.033 0.030 0.030 Mean of dependent variable 0.070 0.110 0.070 0.068 0.063 0.078 0.158 0.070 0.070 Mean . Variance . Skew . All Moments . (10) . (11) . (12) . (13) . Panel (b): robustness to controlling for moments of rainfall distribution No. of shocks past 10 years 0.009*** 0.008** 0.008** 0.008*** (0.003) (0.003) (0.003) (0.003) Observations 77,760 77,760 77,760 77,760 R2 0.031 0.030 0.030 0.031 Mean of dependent variable 0.070 0.070 0.070 0.070 Mean . Variance . Skew . All Moments . (10) . (11) . (12) . (13) . Panel (b): robustness to controlling for moments of rainfall distribution No. of shocks past 10 years 0.009*** 0.008** 0.008** 0.008*** (0.003) (0.003) (0.003) (0.003) Observations 77,760 77,760 77,760 77,760 R2 0.031 0.030 0.030 0.031 Mean of dependent variable 0.070 0.070 0.070 0.070 Notes Rural sample from high‐prevalence countries. All specifications include controls for gender, age and survey fixed effects, except as noted. Column 1 does not include the standard controls. Column (2) does not include weights. Column (3) employs country‐year fixed effects, rather than survey fixed effects. Column (4) excludes the hyper‐endemic countries (Swaziland and Lesotho). Column (5) excludes individuals who have lived in their current village for less than one year. Column (6) employs only the most recent survey from each country (excludes KE 2003 and TZ 2004). Column (7) includes only individuals who are aged 15+ at the time of the shocks. Columns (8)–(9) employ alternative definitions of a shock: column (8) employs only the years 1970–95 to create the historical distribution from which the 15th percentile is the shock definition, and column (9) defines a shock as 1.5 SD below the local mean. Columns (10) ‐ (13) include additional controls as shown in column headers. Estimators are weighted to be representative of the 19 countries, except as noted. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Fig. 2. Open in new tabDownload slide Effect of Rainfall Shocks on HIV, by (a) Severity and (b) Timing Notes. The black line represents the coefficient point estimates of the impact of a particular rainfall shock on HIV for the rural sample from high‐prevalence countries, using (a) various definitions of ‘shock’ accumulated over the previous 10 years and (b) 15% shocks accumulated over different time periods, including placebo future shocks up to three years past the survey date. Plot (a): dotted lines represent the 95% confidence intervals; dashed lines represent the 90% confidence intervals. Plot (b): shaded area represents 95% confidence interval. Fig. 2. Open in new tabDownload slide Effect of Rainfall Shocks on HIV, by (a) Severity and (b) Timing Notes. The black line represents the coefficient point estimates of the impact of a particular rainfall shock on HIV for the rural sample from high‐prevalence countries, using (a) various definitions of ‘shock’ accumulated over the previous 10 years and (b) 15% shocks accumulated over different time periods, including placebo future shocks up to three years past the survey date. Plot (a): dotted lines represent the 95% confidence intervals; dashed lines represent the 90% confidence intervals. Plot (b): shaded area represents 95% confidence interval. For rainfall at or above the 40th percentile, point estimates suggest that there is no effect on HIV. This corresponds to the estimated relationships between rainfall and maize yields, and rainfall and GDP growth, shown in Figure 1. Both maize yields and GDP growth are unaffected by rainfall realisations above the 40th percentile and, consistent with this, we find that HIV becomes similarly unaffected by rainfall around this threshold. We also vary the period of time over which shocks are summed, for comparison with our default definition of shocks summed over the past 10 years. We sum shocks in five‐year bins (e.g. number of shocks 1–5 years before the survey, number of shocks 6–10 years before etc.) and employ each of these binned variables as the regressor in our main specification. Figure 2 plots the point estimates of these regressors. As we show in online Appendix E, this time profile of the effect of shocks on HIV is very much as we would expect, with point estimates for the effect of shocks peaking early within the 10‐year window. Intuitively, an earlier shock has more time to reverberate through the population and generate additional infections compared to a more recent shock but effects are attenuated over time as the earliest infected die. Given the observed infection rate and the observed timing of mortality following infection, we show via simulation in online Appendix E that the effect of a shock will peak 6–10 years later. To address concerns that shocks from the mid‐1990s onwards (our main shocks of interest, given our HIV data are from 2003 to 2009) may be endogenous to how shocks are defined, we also employ a shock definition that is based on the 15th quantile of the historical distribution derived from rainfall data only up through 1995. The cluster‐specific definition of shocks then does not depend on anything that happened after 1995. We find that the results do not differ significantly using this alternate measure (Table 4; column 8). Finally, as an alternative to the quantile‐based definition, we also define shocks as rainfall that is 1.5 SD or more below the historical mean for the area. The primary estimation employing this definition of shock is shown in column (9) of Table 4, where the estimated coefficient is similar, though slightly larger, and remains statistically significant. 3.2.3. Sample selection Droughts can also effect other types of behaviour that might explain our results. If shocks induce permanent out‐migration and the migrants are disproportionately HIV negative, this could yield a spurious correlation between observed shocks and higher HIV prevalence among the remaining population. In order to test whether selective migration can account for our results, we conduct a bounding exercise suggested by Lee (2009). Using national rural and total population figures by country, we estimate that rural areas lose approximately 2% of population per shock (see online Appendix D for more details) and conservatively assume that each one of these individuals is HIV‐negative.21 We replace these individuals in our sample and re‐estimate our main results. This in effect stacks the deck against finding a result: communities that experience shocks now have more HIV‐negative individuals. We note, however, that the assumptions we make about migration rates are strong, and therefore some caution is warranted when interpreting the results. Table 5 first reproduces our primary result based on the rural sample of high‐prevalence countries: the probability of infection increases by 0.8 percentage points per shock. We then vary the assumed percentage who migrate when a shock occurs, starting with our estimate of 2% and increasing in increments of 1%. We find that when accounting for estimated out‐migration of 2% per shock, the estimated coefficient (0.7 percentage points) is nearly identical to our original estimate and still significant. Table 5 Robustness to Sample Selection from Permanent Migration . . Replacing lost population share per shock . . Observed . 2% . 3% . 4% . 5% . 6% . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.008*** 0.007** 0.006** 0.005* 0.004* 0.004 (0.003) (0.003) (0.003) (0.003) (0.003) (0.002) Observations 77,760 81,792 84,191 86,523 88,775 91,330 R2 0.030 0.022 0.023 0.023 0.024 0.024 . . Replacing lost population share per shock . . Observed . 2% . 3% . 4% . 5% . 6% . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.008*** 0.007** 0.006** 0.005* 0.004* 0.004 (0.003) (0.003) (0.003) (0.003) (0.003) (0.002) Observations 77,760 81,792 84,191 86,523 88,775 91,330 R2 0.030 0.022 0.023 0.023 0.024 0.024 Notes Rural sample from high‐prevalence countries. Column headers denote the population share added to the sample to account for out‐migration, assuming all out‐migrants are HIV negative. Note that 2% is the most accurate estimate with 4% as the extreme upper bound (see online Appendix D). All specifications include controls for gender, age and survey fixed effects. Estimates are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Table 5 Robustness to Sample Selection from Permanent Migration . . Replacing lost population share per shock . . Observed . 2% . 3% . 4% . 5% . 6% . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.008*** 0.007** 0.006** 0.005* 0.004* 0.004 (0.003) (0.003) (0.003) (0.003) (0.003) (0.002) Observations 77,760 81,792 84,191 86,523 88,775 91,330 R2 0.030 0.022 0.023 0.023 0.024 0.024 . . Replacing lost population share per shock . . Observed . 2% . 3% . 4% . 5% . 6% . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.008*** 0.007** 0.006** 0.005* 0.004* 0.004 (0.003) (0.003) (0.003) (0.003) (0.003) (0.002) Observations 77,760 81,792 84,191 86,523 88,775 91,330 R2 0.030 0.022 0.023 0.023 0.024 0.024 Notes Rural sample from high‐prevalence countries. Column headers denote the population share added to the sample to account for out‐migration, assuming all out‐migrants are HIV negative. Note that 2% is the most accurate estimate with 4% as the extreme upper bound (see online Appendix D). All specifications include controls for gender, age and survey fixed effects. Estimates are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Note that if all of rural to urban migration were caused only by shocks, then a more accurate estimate would be that 4% of the population migrates when a shock occurs (again, see online Appendix D for details). Thus, the assumption of a 4% loss per shock is an extreme upper bound. When we replace a 4% population loss per shock, our effect remains positive (0.4 percentage points) and significant at the 10% level. Though 4% is the upper bound, we nonetheless report estimates under the assumptions of 5% and 6% loss per shock to show that the estimate does not lose significance until we assume 6% loss per shock – three times our best estimation of 2% loss per shock. This suggests that sample selection due to permanent migration is unlikely to explain our results. 3.2.4. Omitted variables A final concern is that our results might be driven by omitted variables. For example, some aspects of local weather might be correlated with other unobservables (wealth, education etc) that also affect HIV rates. While this is unlikely to be true for our measure of rainfall shocks – by construction all areas expect the same total number of shocks over time – we confirm that our estimates are robust to controlling for characteristics of the underlying distribution. In Table 4, panel (b), we sequentially control for the first three moments of the rainfall distribution (mean, variance, skew) in our main specification (columns 10–12) and also include all three moments (column 13). Our estimate remains stable throughout these various specifications. We can further test for these potential confounders with a ‘placebo’ test – we check whether shocks in the future can predict present HIV rates or other observable present characteristics. Given that the DHS surveys were conducted between 2003 and 2009, and our weather data end in crop‐years 2007–8, we are only able to examine shocks up to four years in the future.22 We find no relationship between HIV rates and shocks one to four years in the future (Table 6; columns 1–4).23 We also find no relationship between current wealth quintile and future shocks (columns 5–7), nor any relationship between an individual's years of education and future shocks (columns 8–10). Finally, during the 2000s, there was increasing access to ARVs for HIV‐positive individuals, which may bias our results if access was in any way correlated with shocks. We show that during most of our study time frame, ARV access was relatively low (less than 30% for all but one country) and that there is no evidence that suggests ARV access is correlated with our shock measure (see online Appendix F). Taken together these tests provide additional evidence that shocks are picking up meaningful variation in economic conditions prior to the survey year and that this variation is uncorrelated with other factors that might also explain disease outcomes. Table 6 Placebo Tests Dependent variable . HIV . Wealth quintile . Years of education . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . No. of shocks in future 1 year 0.006 0.005 −0.152 (0.006) (0.110) (0.232) No. of shocks in future 2 years −0.002 0.113 −0.134 (0.007) (0.095) (0.194) No. of shocks in future 3 years −0.004 0.104 −0.146 (0.007) (0.096) (0.194) No. of shocks in future 4 years −0.006 (0.006) Observations 49,523 43,881 26,059 12,434 49,523 43,881 26,059 49,489 43,861 26,039 R2 0.044 0.040 0.025 0.010 0.031 0.033 0.029 0.119 0.118 0.102 Dependent variable . HIV . Wealth quintile . Years of education . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . No. of shocks in future 1 year 0.006 0.005 −0.152 (0.006) (0.110) (0.232) No. of shocks in future 2 years −0.002 0.113 −0.134 (0.007) (0.095) (0.194) No. of shocks in future 3 years −0.004 0.104 −0.146 (0.007) (0.096) (0.194) No. of shocks in future 4 years −0.006 (0.006) Observations 49,523 43,881 26,059 12,434 49,523 43,881 26,059 49,489 43,861 26,039 R2 0.044 0.040 0.025 0.010 0.031 0.033 0.029 0.119 0.118 0.102 Notes Rural sample from high‐prevalence countries. Note that the only survey in 2003 (Kenya) does not contain information on wealth and education, therefore, the correlations of these characteristics with shocks can only be calculated up to three years in the future, as weather data end in 2007–8 crop years. All specifications include controls for gender, age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Open in new tab Table 6 Placebo Tests Dependent variable . HIV . Wealth quintile . Years of education . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . No. of shocks in future 1 year 0.006 0.005 −0.152 (0.006) (0.110) (0.232) No. of shocks in future 2 years −0.002 0.113 −0.134 (0.007) (0.095) (0.194) No. of shocks in future 3 years −0.004 0.104 −0.146 (0.007) (0.096) (0.194) No. of shocks in future 4 years −0.006 (0.006) Observations 49,523 43,881 26,059 12,434 49,523 43,881 26,059 49,489 43,861 26,039 R2 0.044 0.040 0.025 0.010 0.031 0.033 0.029 0.119 0.118 0.102 Dependent variable . HIV . Wealth quintile . Years of education . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . (9) . (10) . No. of shocks in future 1 year 0.006 0.005 −0.152 (0.006) (0.110) (0.232) No. of shocks in future 2 years −0.002 0.113 −0.134 (0.007) (0.095) (0.194) No. of shocks in future 3 years −0.004 0.104 −0.146 (0.007) (0.096) (0.194) No. of shocks in future 4 years −0.006 (0.006) Observations 49,523 43,881 26,059 12,434 49,523 43,881 26,059 49,489 43,861 26,039 R2 0.044 0.040 0.025 0.010 0.031 0.033 0.029 0.119 0.118 0.102 Notes Rural sample from high‐prevalence countries. Note that the only survey in 2003 (Kenya) does not contain information on wealth and education, therefore, the correlations of these characteristics with shocks can only be calculated up to three years in the future, as weather data end in 2007–8 crop years. All specifications include controls for gender, age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Open in new tab 4. Exploring Pathways 4.1. Behavioural Pathways How might changes in income induce behavioural changes that increase HIV infection? As HIV is overwhelmingly transmitted by heterosexual sex in this context, we first examine whether risky sexual behaviour increases in response to recent shocks, using self‐reported sexual behaviour. We then consider three separate coping behaviour that could lead to increased sexual risk. 4.1.1. Risky sexual behaviour The use of self‐reported sexual behaviour is subject to a few caveats. There is a large body of evidence that suggests self‐reported sexual behaviour suffers from social desirability bias (Cleland et al., 2004) and that women significantly under‐report their sexual activity (Minnis et al., 2009).24 In addition, we only have measures of sexual behaviour during the 12 months prior to the survey. It is not immediately clear which time window of shocks should be considered to impact sexual behaviour in the past 12 months. Certainly shocks in the current and previous year should, however, given the potential lag between lack of rainfall and lack of income, perhaps droughts two years ago should have a similar impact. Further, more distant shocks that induced the creation of new sexual relationships may have continuing impacts on current behaviour if those relationships (or behaviour) are persistent.25 For this reason, we present the impact on recent sexual behaviour of shocks within the past 10 years, shocks within the past five years and having a shock that affected income over the past 12 months. Given these caveats, we interpret results on self‐reported sexual behaviour with caution. The outcome variables we examine are whether in the past 12 months the respondent: (i) has been sexually active; (ii) had multiple partners; or (iii) had non‐spouse partner(s).26 Table 7 shows results of estimation of (2), separately by gender, with these self‐reported types of sexual behaviour as the dependent variables regressed separately on three categories of independent variables as noted. A strong and consistent finding is that both men and women are significantly more likely to have engaged with a non‐spouse partner if exposed to a shock in any of the three time periods considered. For both men and women, shocks affecting the past 12 months increase non‐spouse partnership rates by about 10–20%. Shocks in nearly all of the periods also increase the likelihood of engaging with multiple concurrent partners by 10–15%, though the estimates are not precise in all periods. Point estimates for the impact of shocks on being sexually active at all are positive for men but not significantly different from zero and, for women, are not consistent across the periods considered. Table 7 Exploring Behaviour: Increasing Risky Sexual Behaviour . Women . Men . . Sexually active . Multiple partners . Non‐spouse partner . Sexually active . Multiple partners . Non‐spouse partner . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.012** 0.003* 0.007* 0.008 0.018*** 0.015** (0.005) (0.002) (0.004) (0.006) (0.004) (0.006) R2 0.060 0.011 0.018 0.223 0.034 0.051 No. of shocks past 5 years 0.021*** 0.004* 0.013** 0.008 0.016** 0.021** (0.007) (0.002) (0.005) (0.009) (0.006) (0.009) R2 0.060 0.011 0.018 0.223 0.033 0.050 Y/N shock affecting past 12 months −0.027** 0.003 0.023** 0.013 −0.007 0.035** (0.012) (0.004) (0.010) (0.012) (0.012) (0.017) R2 0.059 0.011 0.018 0.222 0.032 0.051 Observations 43,145 43,119 43,147 34,607 34,563 34,613 Mean of dependent variable 0.759 0.024 0.120 0.738 0.154 0.269 . Women . Men . . Sexually active . Multiple partners . Non‐spouse partner . Sexually active . Multiple partners . Non‐spouse partner . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.012** 0.003* 0.007* 0.008 0.018*** 0.015** (0.005) (0.002) (0.004) (0.006) (0.004) (0.006) R2 0.060 0.011 0.018 0.223 0.034 0.051 No. of shocks past 5 years 0.021*** 0.004* 0.013** 0.008 0.016** 0.021** (0.007) (0.002) (0.005) (0.009) (0.006) (0.009) R2 0.060 0.011 0.018 0.223 0.033 0.050 Y/N shock affecting past 12 months −0.027** 0.003 0.023** 0.013 −0.007 0.035** (0.012) (0.004) (0.010) (0.012) (0.012) (0.017) R2 0.059 0.011 0.018 0.222 0.032 0.051 Observations 43,145 43,119 43,147 34,607 34,563 34,613 Mean of dependent variable 0.759 0.024 0.120 0.738 0.154 0.269 Notes Rural sample from high‐prevalence countries. Dependent variables are types of sexual behaviour in the past year. ‘Non‐spouse’ indicates sex with a non‐spouse partner; this includes all sex for single individuals. All specifications include controls for age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Table 7 Exploring Behaviour: Increasing Risky Sexual Behaviour . Women . Men . . Sexually active . Multiple partners . Non‐spouse partner . Sexually active . Multiple partners . Non‐spouse partner . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.012** 0.003* 0.007* 0.008 0.018*** 0.015** (0.005) (0.002) (0.004) (0.006) (0.004) (0.006) R2 0.060 0.011 0.018 0.223 0.034 0.051 No. of shocks past 5 years 0.021*** 0.004* 0.013** 0.008 0.016** 0.021** (0.007) (0.002) (0.005) (0.009) (0.006) (0.009) R2 0.060 0.011 0.018 0.223 0.033 0.050 Y/N shock affecting past 12 months −0.027** 0.003 0.023** 0.013 −0.007 0.035** (0.012) (0.004) (0.010) (0.012) (0.012) (0.017) R2 0.059 0.011 0.018 0.222 0.032 0.051 Observations 43,145 43,119 43,147 34,607 34,563 34,613 Mean of dependent variable 0.759 0.024 0.120 0.738 0.154 0.269 . Women . Men . . Sexually active . Multiple partners . Non‐spouse partner . Sexually active . Multiple partners . Non‐spouse partner . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.012** 0.003* 0.007* 0.008 0.018*** 0.015** (0.005) (0.002) (0.004) (0.006) (0.004) (0.006) R2 0.060 0.011 0.018 0.223 0.034 0.051 No. of shocks past 5 years 0.021*** 0.004* 0.013** 0.008 0.016** 0.021** (0.007) (0.002) (0.005) (0.009) (0.006) (0.009) R2 0.060 0.011 0.018 0.223 0.033 0.050 Y/N shock affecting past 12 months −0.027** 0.003 0.023** 0.013 −0.007 0.035** (0.012) (0.004) (0.010) (0.012) (0.012) (0.017) R2 0.059 0.011 0.018 0.222 0.032 0.051 Observations 43,145 43,119 43,147 34,607 34,563 34,613 Mean of dependent variable 0.759 0.024 0.120 0.738 0.154 0.269 Notes Rural sample from high‐prevalence countries. Dependent variables are types of sexual behaviour in the past year. ‘Non‐spouse’ indicates sex with a non‐spouse partner; this includes all sex for single individuals. All specifications include controls for age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Overall, these self‐reports of sexual behaviour indicate that individuals who have experienced recent shocks are more likely to report risky sexual activity. Keeping the caveats discussed earlier in mind, these findings suggest that shocks are indeed changing sexual behaviour – and, in particular, leading to riskier sexual behaviour – and that these behavioural changes are what likely link rainfall shocks to HIV. In the remainder of this section, we seek evidence for which coping behaviour may be primarily responsible for this relationship. 4.1.2. Temporary migration One response to drought‐induced income shocks is to migrate from rural to urban areas in search of employment (Ellis, 2000; Skoufias, 2003). Migration is associated with greater levels of risky sexual activity and higher rates of HIV (Brockerhoff and Biddlecom, 1999; Lurie et al., 2003). Individuals may temporarily migrate to urban areas in response to droughts, acquire HIV due to additional partnerships or high‐risk partners and then infect others when returning to their rural communities.27 If income shocks induce temporary migration, then ∂p/∂z < 0 for both men and women, as both the migrant and his/her partner in the rural village would face increased risk. Table 8 Exploring Behaviour: Temporary Migration . Main HIV result for this sub‐sample . Away for month + in past year . Total times away in past year . . Men . Women . Men . Women . Men . Women . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.007* 0.015*** (0.003) (0.004) Y/N shock in past year −0.010 0.018 −0.421*** −0.229** (0.024) (0.018) (0.149) (0.115) Near urban × shock in past year −0.018 −0.008 −0.419 −0.569*** (0.031) (0.027) (0.282) (0.188) Near urban −0.008 −0.009 −0.148 0.162* (0.011) (0.012) (0.199) (0.084) Observations 26,096 26,299 26,133 26,300 23,802 22,990 R2 0.038 0.041 0.004 0.016 0.025 0.117 Mean of dependent variable 0.064 0.094 0.151 0.130 2.064 0.990 . Main HIV result for this sub‐sample . Away for month + in past year . Total times away in past year . . Men . Women . Men . Women . Men . Women . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.007* 0.015*** (0.003) (0.004) Y/N shock in past year −0.010 0.018 −0.421*** −0.229** (0.024) (0.018) (0.149) (0.115) Near urban × shock in past year −0.018 −0.008 −0.419 −0.569*** (0.031) (0.027) (0.282) (0.188) Near urban −0.008 −0.009 −0.148 0.162* (0.011) (0.012) (0.199) (0.084) Observations 26,096 26,299 26,133 26,300 23,802 22,990 R2 0.038 0.041 0.004 0.016 0.025 0.117 Mean of dependent variable 0.064 0.094 0.151 0.130 2.064 0.990 Notes Rural sample from high‐prevalence countries. The ‘Near urban’ variable indicates whether a given cluster is within 100 kilometres of an urban area (defined as population size 250K+); this represents 27% of the rural population in high prevalence countries. Urban populations are from the Global Rural–Urban Mapping Project. Variables on being away are not available for all countries (see text); the main estimation for these sub‐samples is given in columns (1) and (2). All specifications include controls for age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Table 8 Exploring Behaviour: Temporary Migration . Main HIV result for this sub‐sample . Away for month + in past year . Total times away in past year . . Men . Women . Men . Women . Men . Women . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.007* 0.015*** (0.003) (0.004) Y/N shock in past year −0.010 0.018 −0.421*** −0.229** (0.024) (0.018) (0.149) (0.115) Near urban × shock in past year −0.018 −0.008 −0.419 −0.569*** (0.031) (0.027) (0.282) (0.188) Near urban −0.008 −0.009 −0.148 0.162* (0.011) (0.012) (0.199) (0.084) Observations 26,096 26,299 26,133 26,300 23,802 22,990 R2 0.038 0.041 0.004 0.016 0.025 0.117 Mean of dependent variable 0.064 0.094 0.151 0.130 2.064 0.990 . Main HIV result for this sub‐sample . Away for month + in past year . Total times away in past year . . Men . Women . Men . Women . Men . Women . . (1) . (2) . (3) . (4) . (5) . (6) . No. of shocks past 10 years 0.007* 0.015*** (0.003) (0.004) Y/N shock in past year −0.010 0.018 −0.421*** −0.229** (0.024) (0.018) (0.149) (0.115) Near urban × shock in past year −0.018 −0.008 −0.419 −0.569*** (0.031) (0.027) (0.282) (0.188) Near urban −0.008 −0.009 −0.148 0.162* (0.011) (0.012) (0.199) (0.084) Observations 26,096 26,299 26,133 26,300 23,802 22,990 R2 0.038 0.041 0.004 0.016 0.025 0.117 Mean of dependent variable 0.064 0.094 0.151 0.130 2.064 0.990 Notes Rural sample from high‐prevalence countries. The ‘Near urban’ variable indicates whether a given cluster is within 100 kilometres of an urban area (defined as population size 250K+); this represents 27% of the rural population in high prevalence countries. Urban populations are from the Global Rural–Urban Mapping Project. Variables on being away are not available for all countries (see text); the main estimation for these sub‐samples is given in columns (1) and (2). All specifications include controls for age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab As a check for this pathway, we use information on the number of times individuals have been away from home in the past 12 months and whether any time away has lasted more than one month. If temporary migration is a primary coping behaviour in this setting, we would expect that a shock in the past year would significantly increase both indicators. These outcomes are available for men in 17 (and for women in 9) of our 21 surveys, and estimation results are presented in Table 8. For comparison, the main estimation from Table 3, column (5) is presented in columns 1 and 2, for men and women respectively, for these sub‐samples. Columns (3)–(6) of Table 8 show that for both men and women, shocks affecting the past 12 months have a correlation with the number of times away from home and being gone for more than one month in the past year that is either negative or indistinguishable from zero. We have disaggregated this effect for individuals who live near to an urban area versus those in more remote areas.28 Neither of these sub‐samples exhibit more frequent temporary migration when exposed to a shock.29 This suggests that in our rural sample, droughts are not inducing significant temporary migration. 4.1.3. Dropping‐out and early marriage A second set of coping behaviour that may affect sexual risk is changes in schooling and marriage behaviour. In SSA, a common response to a negative income shock is to withdraw children from school (Ferreira and Schady, 2009), which appears particularly true for girls (Bjorkman, 2013). Once a girl has withdrawn from school, she is much more likely to be sexually active and to marry (Osili and Long, 2008; Duflo et al., 2011; Ozier, 2011), both of which are risk factors for HIV (Clark, 2004; Baird et al., 2011). Furthermore, households may marry off daughters earlier in response to a shock, especially in regions where bride payment is customary (Jensen and Thornton, 2003; Hoogeveen et al., 2011). If income shocks induce early drop‐out and early marriage, which result in earlier sexual activity, then ∂p/∂z < 0. While this could apply to both men and women, young women would be most affected through this channel. If early marriage is the pathway, either as a direct response to shocks or as a result of withdrawing from school, we would expect droughts to be associated with a younger age at marriage, and increased probability of marriage at the time of the survey. Further, if shocks are inducing drop‐out, we would expect shocks to be associated with fewer years of schooling and expect shocks to have the strongest effects on HIV for women who were school‐aged at the time of the shock. The first two columns of Table 9 show estimates of the impact of shocks occurring when a woman was potentially subject to early marriage on whether she has ever married by the time of the survey.30 As mean age at marriage for women in this sample is 18, women are considered at risk for early marriage when aged 13–18 (column 1) or aged 15–20 (column 2). In neither case do shocks yield a significant increase in the likelihood of marriage at or before the time of the survey. Table 9 Exploring Behaviour: Early School Drop‐out and Marriage Dependent variable . Ever married . Age at marriage . Years of education . HIV Status . . . . . . . Aged 25+ . Aged 30+ . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . No. of shocks, aged 13–18 −0.000 0.000 −0.010 (0.005) (0.046) (0.056) No. of shocks, aged 15–20 −0.003 0.006 −0.005 (0.005) (0.057) (0.054) No. of shocks past 10 years 0.011* 0.016** (0.006) (0.006) Observations 24,679 22,679 23,770 23,005 27,429 25,242 12,280 3,845 R2 0.125 0.065 0.033 0.023 0.223 0.222 0.031 0.022 Mean dependent variable 0.881 0.923 18.1 18.3 5.5 5.4 0.091 0.067 Dependent variable . Ever married . Age at marriage . Years of education . HIV Status . . . . . . . Aged 25+ . Aged 30+ . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . No. of shocks, aged 13–18 −0.000 0.000 −0.010 (0.005) (0.046) (0.056) No. of shocks, aged 15–20 −0.003 0.006 −0.005 (0.005) (0.057) (0.054) No. of shocks past 10 years 0.011* 0.016** (0.006) (0.006) Observations 24,679 22,679 23,770 23,005 27,429 25,242 12,280 3,845 R2 0.125 0.065 0.033 0.023 0.223 0.222 0.031 0.022 Mean dependent variable 0.881 0.923 18.1 18.3 5.5 5.4 0.091 0.067 Notes Female, rural sample from high‐prevalence countries. The first six columns examine the impacts of shocks that occurred when woman was in the noted age range, since the start of the epidemic (1980). The last two columns examine the impact of shocks in the past 10 years on HIV for women who were above a minimum age during all of the past 10 years. All specifications include controls for age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: **5%, *10%. Open in new tab Table 9 Exploring Behaviour: Early School Drop‐out and Marriage Dependent variable . Ever married . Age at marriage . Years of education . HIV Status . . . . . . . Aged 25+ . Aged 30+ . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . No. of shocks, aged 13–18 −0.000 0.000 −0.010 (0.005) (0.046) (0.056) No. of shocks, aged 15–20 −0.003 0.006 −0.005 (0.005) (0.057) (0.054) No. of shocks past 10 years 0.011* 0.016** (0.006) (0.006) Observations 24,679 22,679 23,770 23,005 27,429 25,242 12,280 3,845 R2 0.125 0.065 0.033 0.023 0.223 0.222 0.031 0.022 Mean dependent variable 0.881 0.923 18.1 18.3 5.5 5.4 0.091 0.067 Dependent variable . Ever married . Age at marriage . Years of education . HIV Status . . . . . . . Aged 25+ . Aged 30+ . (1) . (2) . (3) . (4) . (5) . (6) . (7) . (8) . No. of shocks, aged 13–18 −0.000 0.000 −0.010 (0.005) (0.046) (0.056) No. of shocks, aged 15–20 −0.003 0.006 −0.005 (0.005) (0.057) (0.054) No. of shocks past 10 years 0.011* 0.016** (0.006) (0.006) Observations 24,679 22,679 23,770 23,005 27,429 25,242 12,280 3,845 R2 0.125 0.065 0.033 0.023 0.223 0.222 0.031 0.022 Mean dependent variable 0.881 0.923 18.1 18.3 5.5 5.4 0.091 0.067 Notes Female, rural sample from high‐prevalence countries. The first six columns examine the impacts of shocks that occurred when woman was in the noted age range, since the start of the epidemic (1980). The last two columns examine the impact of shocks in the past 10 years on HIV for women who were above a minimum age during all of the past 10 years. All specifications include controls for age and survey fixed effects. Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: **5%, *10%. Open in new tab The second two columns estimate the impact of shocks during the same periods of life on the resulting age at marriage for those who have ever married. The coefficients reflect an effective zero change in age at marriage when exposed to a shock at these critical ages. In short, it seems that shocks do not induce earlier marriage for women in this sample. Even if youths are not marrying earlier, households may respond to income shocks by withdrawing children from school, especially girls. Girls that drop‐out early are at higher risk for early sexual activity and HIV transmission (Baird et al., 2010). If this is a contributing factor in the link between rainfall and HIV, we would expect to find two telltale results. First, shocks should reduce total schooling for women who were school‐aged when the shock occurred; second, the link between rainfall and HIV should be restricted to women who had not yet completed their schooling when the shock occurred. Columns (5) and (6) of Table 9 estimate the effect of shocks when aged 13–18 (and 15–20) on years of education. Both estimates produce a negative coefficient, however, both reflect effect sizes of less than 1% and are not statistically different from zero. We do not find evidence that rainfall shocks induce significant dropping out of girls.31 Finally, columns (7) and (8) replicate our primary estimation, excluding women who were school aged during the past 10 years. We find that the results are robust to this exclusion, suggesting that women who were school‐aged at the time of the shock are not driving the results. In sum, we find no evidence that early marriage and dropping‐out are the primary coping behaviour linking rainfall to HIV. 4.1.4. Transactional sex A third coping mechanism is engaging in transactional sex. Transactional sex is thought to be common in SSA and is broadly defined to include both prostitution as well as transfers within casual relationships and long‐term partnerships (Hunter, 2002; Leclerc‐Madlala, 2002; Luke, 2006; Maganja et al.,2007; Swidler and Watkins, 2007; Béné and Merten, 2008).32 Women may respond to income shocks either by taking on additional partnerships or engaging in more frequent or riskier sexual activity (i.e. unprotected sex) to increase transfers. Both types of behaviour have been documented throughout SSA, with women in rural Malawi engaging in multiple partnerships in response to income insecurity (Swidler and Watkins, 2007) and women in South Africa and Western Kenya more likely to engage in unprotected sex as a response to negative income shocks (Dinkelman et al., 2008; Robinson and Yeh, 2011b; Dupas and Robinson, 2012). While there are many factors affecting the HIV/AIDS epidemic, transactional sex is thought to be a major driver within SSA (Alary and Lowndes, 2004; Côté et al., 2004; Dunkle et al., 2004) and a growing empirical literature suggests that economic conditions affect risky sexual behaviour and the market for transactional sex (Baird et al., 2010; Robinson and Yeh, 2011a; Kohler and Thornton, 2012). We cannot directly examine changes in this behaviour, as we lack data on transactions.33 To make progress, we make a few assumptions on the transactional sex market. First, we follow the literature in assuming that women supply and men demand transactional sex (Edlund and Korn, 2002). Second, in keeping with a recent micro literature (Robinson and Yeh, 2011b; Baird et al., 2012; Kohler and Thornton, 2012), we assume that women increase their supply of transactional sex if other sources of income decrease; and that when supply increases, prices fall. Finally, we assume that individuals experiencing larger income shocks should have a stronger behavioural response – that is, supply is increasing and demand is decreasing in shock exposure. While we do not observe individual changes in income, we do observe occupation – in particular, whether or not an individual's primary income source is from agriculture.34 We assume that incomes of individuals working in agriculture are more sensitive to drought than those working outside agriculture. In the market for transactional sex, we would expect that men working outside agriculture would increase their quantity demanded in the face of an aggregate shock, based on the reduced price. Further, men working in agriculture would reduce their quantity demanded. However, as men working in agriculture will also face increased network risk, the effect of shocks on their HIV status should be dampened but not necessarily reversed, relative to men working outside agriculture. Before turning to the results, we stress that given the assumptions we make, our findings in this Section warrant caution. While our results are consistent with transactional sex being the channel linking shocks to HIV, we cannot definitively claim this. Future research with comprehensive data on shocks, sexual behaviour and transfers will shed more light onto this channel. Table 10 presents the primary estimation for both men and women, with interactions by occupation. For women, the effects of shocks appear concentrated on agricultural women, while women in the non‐agricultural sector appear relatively unaffected by shocks.35 These results make sense as income of agricultural women is most affected by a drought; these results are also consistent with various channels. To sharpen our analysis, we examine the effects of shocks on men separated by occupation. We find the impact on non‐agricultural men's HIV risk is large and significant at the 10% level, while the effects of shocks for agricultural men is nearly zero. While we cannot reject the null that shocks have the same effect for men in and outside of agriculture (p = 0.167), these estimates are consistent with transactional sex being the channel linking shocks to greater HIV rates.36 If shocks are inducing women to supply more sex, then men whose incomes are least affected by droughts (i.e. men employed outside of agriculture) should increase their quantity demanded. While men in agriculture would face lower prices in the market for transactional sex, their income will also be affected by drought, dampening any price effects. Table 10 Exploring Behaviour: Impact on HIV by Exposure to Drought‐induced Income Shock . Men . Women . . (1) . (2) . (3) . (4) . No. of shocks past 10 years 0.004 0.002 0.009** 0.011*** (0.003) (0.003) (0.004) (0.004) Non‐Agriculture × shocks 0.007 −0.007 (0.005) (0.006) Non‐Agriculture employment 0.049*** 0.116*** (0.016) (0.023) p‐value on interaction 0.167 0.252 Impact of shocks on Non‐Agriculture (linear combination) 0.009* 0.004 (0.004) (0.006) Observations 37,585 37,585 37,487 37,487 R2 0.033 0.039 0.032 0.039 Mean of dependent variable (Ag) 0.056 0.077 Mean of dependent variable (Non‐Agriculture) 0.096 0.148 . Men . Women . . (1) . (2) . (3) . (4) . No. of shocks past 10 years 0.004 0.002 0.009** 0.011*** (0.003) (0.003) (0.004) (0.004) Non‐Agriculture × shocks 0.007 −0.007 (0.005) (0.006) Non‐Agriculture employment 0.049*** 0.116*** (0.016) (0.023) p‐value on interaction 0.167 0.252 Impact of shocks on Non‐Agriculture (linear combination) 0.009* 0.004 (0.004) (0.006) Observations 37,585 37,585 37,487 37,487 R2 0.033 0.039 0.032 0.039 Mean of dependent variable (Ag) 0.056 0.077 Mean of dependent variable (Non‐Agriculture) 0.096 0.148 Notes Sample from high‐prevalence countries. All specifications include controls for age and urban, and survey × employment type fixed effects to allow the correlation between HIV and occupation to vary by country. Note that Non‐Agriculture indicator alone is indicative only for the country excluded in fixed effects (Mozambique). Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Table 10 Exploring Behaviour: Impact on HIV by Exposure to Drought‐induced Income Shock . Men . Women . . (1) . (2) . (3) . (4) . No. of shocks past 10 years 0.004 0.002 0.009** 0.011*** (0.003) (0.003) (0.004) (0.004) Non‐Agriculture × shocks 0.007 −0.007 (0.005) (0.006) Non‐Agriculture employment 0.049*** 0.116*** (0.016) (0.023) p‐value on interaction 0.167 0.252 Impact of shocks on Non‐Agriculture (linear combination) 0.009* 0.004 (0.004) (0.006) Observations 37,585 37,585 37,487 37,487 R2 0.033 0.039 0.032 0.039 Mean of dependent variable (Ag) 0.056 0.077 Mean of dependent variable (Non‐Agriculture) 0.096 0.148 . Men . Women . . (1) . (2) . (3) . (4) . No. of shocks past 10 years 0.004 0.002 0.009** 0.011*** (0.003) (0.003) (0.004) (0.004) Non‐Agriculture × shocks 0.007 −0.007 (0.005) (0.006) Non‐Agriculture employment 0.049*** 0.116*** (0.016) (0.023) p‐value on interaction 0.167 0.252 Impact of shocks on Non‐Agriculture (linear combination) 0.009* 0.004 (0.004) (0.006) Observations 37,585 37,585 37,487 37,487 R2 0.033 0.039 0.032 0.039 Mean of dependent variable (Ag) 0.056 0.077 Mean of dependent variable (Non‐Agriculture) 0.096 0.148 Notes Sample from high‐prevalence countries. All specifications include controls for age and urban, and survey × employment type fixed effects to allow the correlation between HIV and occupation to vary by country. Note that Non‐Agriculture indicator alone is indicative only for the country excluded in fixed effects (Mozambique). Estimators are weighted to be representative of the 19 countries. Robust standard errors are shown in parentheses clustered at the grid level. Significance levels: ***1%, **5%, *10%. Open in new tab Finally, we note that our shock measure is an aggregate level shock that would presumably affect the incomes of all men and women in an area (regardless of occupation). However, it maybe the case that men are better insured against shocks than women (Dercon and Krishnan, 2000) which may lead women to be more responsive to aggregate shocks than men. Our findings are consistent with this view as well as previous work that finds that the supply side responds more to aggregate shocks than the demand side does (Wilson, 2011; Dupas and Robinson, 2012). To summarise the results from this Section, our main finding is that individuals exposed to recent drought events are more likely to be infected with HIV (∂HIV/∂S > 0). Given the strong evidence of both the relationship between droughts and income (∂z/∂S > 0) and risky sexual behaviour and HIV (∂HIV/∂p > 0), this suggests that the underlying mechanism connecting droughts and HIV is a behavioural response to income shocks that is leading to increased sexual risk (∂p/∂z < 0). We find no evidence that temporary migration or dropping out/early marriage are the key drivers of this relationship. This subsection provides evidence that is broadly consistent with transactional sex as a pathway. However, we cannot conclusively establish the primary behaviour driving this result, nor can we rule out any single behaviour as a contributing factor. 4.2. Non‐behavioural pathways Each type of behaviour discussed above – early sexual activity, migration, transactional sex – has a well‐documented connection to HIV risk and a plausible link to community‐level income shocks. However, droughts also have documented effects on other important factors in rural areas, such as nutrition and civil conflict. We argue that the evidence linking these factors to HIV outcomes is, at best, inconclusive, and that they are unlikely to be pathways that link shocks to HIV. For HIV infected individuals, malnutrition is associated with higher mortality rates and higher viral loads (John et al., 1997; Weiser et al.,2009). Thus the effect that malnourished HIV‐positive individuals will have on the epidemic is ambiguous; higher mortality rates would lead to fewer HIV‐positive individuals but higher viral loads would make them more infectious.37 For HIV‐negative individuals, little is known about the relationship between malnutrition and susceptibility to HIV infection (Mock et al., 2004). Though malnutrition may lead to a compromised immune system which could play a role in susceptibility (Schaible and Stefan, 2007), to the best of our knowledge there is no work that demonstrates an increased susceptibility to HIV infection for malnourished HIV‐negative individuals. While we cannot rule out that this is a contributing pathway, given the existing evidence it does not appear to play a primary role in the HIV/AIDS epidemic. Some recent evidence suggests that negative rainfall deviations are associated with higher incidence of civil conflict in Africa (Miguel et al.,2004; Hsiang et al., 2013). This could indicate another pathway between rainfall and HIV if civil conflict has a direct effect on disease outcomes, for instance due to conflict‐related sexual violence. While we again cannot directly rule out this possibility in our data, recent studies find no clear link between conflict and HIV in either the observational data from Africa (Spiegel et al., 2007), or using epidemiological models that attempt to explain observed HIV prevalence with reported rates of sexual violence (Anema et al., 2008). We have thus focused our empirical exploration of pathways on the three coping behaviour described above. 5. Macro Level Implications Our results suggest that community‐level economic conditions play an important role in an individual's risk of HIV infection. A natural question is the extent to which our results inform broader observed patterns of HIV prevalence on the continent. In other words, can income shocks help explain the striking country‐level variation in HIV prevalence across SSA? Given that our estimation strategy above uses only within‐country variation and that we only have individual‐level HIV data for about half of the countries in the sub‐Saharan region spread out over different years, it is not obvious that our estimates should inform these broader patterns. To address this question, we apply our basic approach to country‐level estimates of HIV prevalence provided by UNAIDS. UNAIDS estimates of country level HIV prevalence over time build heavily on HIV surveillance data distinct from what is in the DHS (e.g. data from antenatal testing at designated clinics) and thus provide prevalence estimates that are somewhat independent from the DHS biomarker data we focus on above. We use the same gridded climate data to derive a time series of annual average rainfall for each country, where the observation for a given country‐year is a weighted average of all the grid cells in that country, using percentage of each cell covered by cropland as weights.38 Similar to above, we calculate these annual rainfall totals for each country back to 1970, fit a separate gamma distribution to each country's time series and define a shock as a year in which country‐average rainfall fell below the 15th percentile in that country's rainfall distribution. We then seek to explain the cross‐sectional prevalence in HIV in a given year as a function of accumulated shocks over the previous decade. This regression uses a different source of variation from our individual specifications (cross‐country rather than within‐country), uses data that are related but distinct and includes many countries not in our individual‐level data. It thus provides a test of the relationship between shocks and HIV that is substantially distinct from the results presented above. Figure 3 plots these relationships for the two decades for which UNAIDS reports data. Countries with a higher number of shocks are more likely to have higher levels of HIV‐prevalence; this is true both in the 1990s (upper plot) when the epidemic was growing rapidly, as well as in the 2000s, when the epidemic had plateaued or started to decline in many countries. These simple cross sectional relationships are statistically significant and explain 14–21% of the cross‐sectional variation in HIV prevalence across the continent (see online Appendix H for regression results).39 Again, as with our individual‐level results this estimate is not picking up differences in underlying propensity to experience shocks (which could be correlated with other factors affecting HIV), but relies instead on the random timing of recent shock exposure. Fig. 3. Open in new tabDownload slide Country‐level HIV Prevalence & Shocks Notes. The top panel presents results for HIV prevalence in 1999 (y‐axis) and accumulated shocks over the previous decade (x‐axis). The bottom panel presents results for HIV prevalence in 2008 and accumulated shocks since 2000. HIV data are from UNAIDS (2010). Dark lines are linear least squares fits, with grey areas representing the 95% confidence interval. Data are jittered to make country labels more legible. Fig. 3. Open in new tabDownload slide Country‐level HIV Prevalence & Shocks Notes. The top panel presents results for HIV prevalence in 1999 (y‐axis) and accumulated shocks over the previous decade (x‐axis). The bottom panel presents results for HIV prevalence in 2008 and accumulated shocks since 2000. HIV data are from UNAIDS (2010). Dark lines are linear least squares fits, with grey areas representing the 95% confidence interval. Data are jittered to make country labels more legible. We draw three implications from these results. First, the fact that we can replicate our basic micro level results using different sources of variation on both the left and right‐hand side gives us additional confidence that economic conditions exert significant influence on HIV outcomes. Second, our results suggest that bad luck with the weather might have played a surprising role in shaping observed patterns of the AIDS epidemic across the African continent: countries that were hit with large negative shocks during the early years of the epidemic have much higher infection rates many years later. Finally, and somewhat more speculatively, given that many areas in SSA lack social safety nets and depend heavily on rain‐fed agriculture, recurring droughts may play an important and prominent role in explaining why the AIDS epidemic has disproportionately affected SSA. 6. Conclusion Ultimately any halt to the AIDS epidemic will require a medical intervention, such as a vaccine or methods approximating one (e.g. the aggressive use of ARVs). However, our results suggest that economic factors, and in particular the ways in which individuals respond to changes in their economic environment, also play an important role in shaping outcomes in the epidemic. As such, our findings unite two widely‐held notions among researchers in the HIV/AIDS community: that heterosexual sex is a primary driver of the AIDS epidemic in SSA and that economic conditions play some role in sexual behaviour in these countries. Our article provides compelling evidence that a deterioration in economic conditions, in the form of rainfall‐related income shocks, contributes significantly to both village and country‐level rates of HIV infection in SSA. While there are several possible pathways linking shocks to HIV, the available evidence is inconsistent with all the potential pathways discussed here, except transactional sex. Nonetheless, we have no conclusive evidence that transactional sex is indeed the pathway and we cannot fully rule out that the other risk‐coping mechanisms discussed, such as early marriage, school drop‐out, or migration, are also contributing factors. Regardless of the pathway, the policy implications of these findings are substantial. If income shocks lead households to smooth income in ways that contribute to the epidemic, policies that prevent the need for these coping mechanisms would appear to yield large positive returns. Comprehensive social safety nets may unfortunately be an unrealistic short‐run goal for many revenue and capacity‐constrained governments on the continent. However, more targeted interventions such as access to credit and savings, weather‐indexed crop insurance or the development of drought‐resistant crop varieties could have an indirect affect on the spread of HIV by reducing the sensitivity of incomes to rainfall shocks. Our results suggest that the social returns to investments in these and related interventions could be much larger than previously thought, particularly in countries where HIV prevalence remains high. Footnotes 1 " Economic interventions such as formal insurance could compliment existing biomedical interventions such as male circumcision and anti‐retroviral (ARV) treatment as prevention. 2 " A map of these countries can be found in online Appendix A. 3 " The one exception is the Mali 2001 survey. We must exclude this survey as it is not possible to link the HIV results to individuals in the GIS‐marked clusters. 4 " As a robustness check, we also estimate using only the most recent survey from each country and the results are unaffected. 5 " Mozambique 2009 samples women up to age 64. 6 " The age range for men is 15 to either 49, 54, 59 or 64, depending on the survey. See online Appendix A for details. 7 " Testing success rates for each survey are shown by sex in online Appendix Table A2. Refusal rates are 10%, on average. Mishra et al. (2006) examine test refusal rates in DHS testing, which are between 1% and 22%, depending on the country. They conclude that although those refusing are more likely to be positive, the DHS testing accurately represents national prevalence. In this study, individuals exposed to shocks do not differentially refuse a test (see Table A3) so non‐response does not induce bias in our results. 8 " For details regarding construction of the weights, see online Appendix A. 9 " This categorisation follows UNAIDS (2010). Figure A2 shows that with the exception of Cameroon, the prevalence classifications for each country remains stable for the 10 years preceding the survey year. Our main results are unchanged when Cameroon is removed from our analysis. 10 " 0.5 degrees is roughly 50 kilometres at the equator. The UDel data are popular in economic applications; recent papers include Dell et al. (2008); Jones and Olken (2010); Bruckner and Ciccone (2011). Other rainfall data sets are available, but none were sufficient for our needs, lacking either sufficient temporal coverage or spatial resolution. 11 " Estimates of planting dates are derived from gridded maps in Sacks et al. (2010); planting of staple cereal crops for the primary growing season typically occurs in the boreal (northern hemisphere) spring across most of West and Central Africa, and in the boreal autumn across most of Southern Africa. 12 " Other previously used continuous methods, which are also not useful for us, include the total level (or log of the level) in a season or year (Bruckner and Ciccone, 2011; Bruckner, 2012; Cole et al., 2012), the timing of the onset of monsoon or rainy season, days of rain in rainy season and length of longest dry spell in rainy season (Jacoby and Skoufias, 1998; Macours et al., 2012). Also, Miguel et al. (2004) employ year‐over‐year rainfall growth, which, as pointed out by Ciccone (2011), is potentially a poor measure of shocks due to mean reversion. 13 " The gamma distribution was selected for its considerable flexibility in both shape and scale. Our results do not depend on the choice of gamma, or the estimation of the distribution more generally. Similar findings result from defining shocks as 1.5 SD below the grid mean. We use the history of rainfall over the period 1970–2008, which was chosen to be a long enough period to be relatively insensitive to the recent shocks of interest, but short enough to capture relatively recent averages if long run means are changing (e.g. with climate change). 14 " Others in the literature have constructed binary shocks using thresholds such as 75% or less of the local mean (Shah and Steinberg, 2013) or 1–2 SD below the local mean (Bobonis, 2009; Skoufias et al., 2012). 15 " That is, we aggregate crop year rainfall over all cells in a given country (weighting by crop area) to get a time‐series of rainfall realisations for each country; we fit a separate gamma distribution to each country's time series; and within each country each year is assigned its corresponding percentile in its gamma distribution. Crop yield data are from FAO (2011), and data on real per capita economic growth are from the Penn World Tables 7.0 (Heston et al., 2011). 16 " In online Appendix C we discuss why mean reversion is not a concern for our shock measure. 17 " There are many reasons for including survey fixed‐effects. Innumerable differences across countries exist that we cannot observe, including social norms of sexual behaviour, male circumcision rates, access to health services and the national response to the AIDS epidemic. Such unobservable differences may also apply to different time periods within the same country, thus motivating a within‐survey estimation. 18 " In order to generate an actual income elasticity with respect to HIV infection risk, we would need: the percentage of income derived from agriculture for all individuals in our sample; individual level crop yields; and crop prices by DHS cluster. These data are required for each year of the past 10 years for everyone in our sample. Unfortunately, these data are not available. 19 " It is important to note that the sample used by Robinson and Yeh (2011b) consists of female sex workers in Western Kenya, while the sample in this article is representative of the rural population in 19 countries. 20 " Shocks that approach the 20th percentile may not be severe enough to effect behaviour, while shocks that approach the 10th percentile may have stronger effects on behaviour, but their relative rarity reduces the statistical power of hypothesis tests. 21 " In Section 4 we find no evidence of differential migration rates (due to shocks) between clusters close and far from urban centres. 22 " The only 2003 survey which has individual HIV infections (Kenya), does not have data on wealth and education. Therefore, correlations with these characteristics can only be estimated using data in years 2004 onwards, so these can only be observed up to three years in the future. 23 " We note that the estimates for shocks one year into the future may have measurement error. Because each DHS survey takes many months to complete, and because our data on which months are in the ‘crop‐year’ typically do not vary sub‐nationally, the timing of a particular survey in a particular cluster may mean that some months of that cluster's ‘future’ crop‐year could occur in the past. In the vast majority of our specifications, these problems ‘around the edges’ are minimised by summing shocks over a 10 year period. However, when looking just at shocks in the future one year, the rainfall measure in certain clusters might not perfectly capture rainfall one year ahead, making this particular estimate somewhat noisier. 24 " Additional caveats are that data that are available for sexual behaviour do not capture all aspects of risky behaviour that could lead to HIV infection. For example, the type of sexual partner you have (commercial sex worker, individual with multiple partners etc.) will affect the likelihood of HIV infection but such data are not available in the DHS. In addition, the questions about sexual behaviour are not present in all the employed DHS surveys and, therefore, the analysis is performed on a sub‐sample of our data. 25 " Swidler and Watkins (2007) cite multiple works documenting long‐term extramarital unions in exchange for transfers. In addition, the sex‐workers in Robinson and Yeh's (2011b) study started as sex‐workers on average 9.7 years prior to the study. 26 " In these data, a monogamous cohabiting union is considered a spousal partner, irrespective of formal marital status. Also, single, sexually active individuals are included in those having non‐spouse partners. 27 " Note that, if the migration is of a permanent nature, this should not affect HIV in the rural area, though it may affect our estimation of rural HIV, due to sample selection. We directly address this in Section 2. 28 " Near to urban is defined as being within 100 kilometres of an urban centre with population 250,000 or more. Urban populations are from the Global Rural–Urban Mapping Project. 29 " These results look very similar when employing shocks during the past two or three years, rather than 12 months; results not shown. 30 " Only shocks occurring during the HIV epidemic are considered (1980 or later), as only these could be driving the results found. 31 " These findings are consistent with work by Shah and Steinberg (2013), showing that children in India actually attend school less when rains are plentiful as there is more work to be done outside school. 32 " One could argue that early marriage as a response to an income shock may also be considered transactional sex in some form. We argue that these are conceptually distinct as early marriage would be an increase in sexual activity at the extensive, rather than the intensive margin. Further, these are distinct from a policy perspective. 33 " Whether a man has paid for sex in the past year is only queried in four surveys from high prevalence countries. This probably only captures explicit prostitution, rather than all forms of transactional sex, as the reporting is low (3%). Women are not queried regarding payment for sex in any of our surveys. In addition, examining whether women are entering the transactional sex market, or are simply making changes on the intensive margin as a response to shock would be very interesting, however, given these data limitations, we are unable to say anything about this topic. 34 " We are able to classify individuals by their employment type at the time of the survey but not at the time of the shock. Our analysis thus makes the assumption that occupation is fairly persistent: individuals in agriculture at the time of survey are more likely to have been in agriculture at the time of the shock and, thus, our occupational categories are meaningful. We include only those employed in the last year, as the unemployed do not report an occupation. As such, it is difficult to assume whether the currently unemployed previously worked in agriculture or not. A concern with using occupational category is that it may be endogenous to shocks. We examine the predictive effect of number of shocks in the past 10 years on current employment in rural areas, to check its potential to induce bias. Shocks have no predictive effect for employment in agriculture. 35 " We cannot reject the null that shocks have the same effect on women in and outside of agriculture (p = 0.252). 36 " We also find that the magnitude of increases in HIV is consistent with increases in transactional sex. Robinson and Yeh (2011a) find that an individual level health shock that results in total income loss for one day leads a woman to increase her number of sexual partners the following day by 0.3, an 18% increase in their sample. We find that this is comparable to our findings that a year‐long income shock increases a woman's lifetime partnerships by about 33%. See simulation in online Appendix G. 37 " We note, however, that high viral loads may make individuals too sick to be sexually active (Thirumurthy et al., 2012). 38 " This provides country‐level rainfall estimates that are relevant for agriculture but that are also effectively weighted by rural population density, since areas that are farmed more intensively in rural Africa tend to be areas with higher population density (given very small average farm plot size). 39 " We also explore whether shocks can explain the time‐path of the epidemic by looking at cross‐country decadal changes in HIV prevalence as a function of accumulated shocks. Effect sizes are again large but not always quite significant at conventional levels (p = 0.12 on the shock variable for 1990s changes) and we explain somewhat less of the cross‐country variance in decadal trends than we do in levels. Nevertheless, results are broadly consistent with cross‐sectional results. References Alary , M. and Lowndes , C.M. 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Author notes " We thank Ted Miguel, Elisabeth Sadoulet, Jeremy Magruder, Nancy Padian, Ethan Ligon, two anonymous referees, seminar participants at UC Berkeley and Middlebury College, and conference participants at Northeast Universities Development Consortium, Pacific Conference for Development Economics, Population and Poverty Network and Population Association of America, for their helpful comments and suggestions that have greatly improved the manuscript. All errors remain our own. © 2014 The Authors. The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society. This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non‐commercial and no modifications or adaptations are made.