Gravity Models, PPML Estimation and the Bias of the Robust Standard Errors Michael Pfaffermayr August 23, 2018 Abstract In gravity models with exporter and importer dummies the robust standard errors of the slope parameters tend to be severely downward biased when estimated by PPML. The coverage rate of confidence intervals of the estimated slope parameters may thus be much too small in cross-sections of the size typically used in empirical research. Keywords: Poisson Pseudo Maximum Likelihood Estimation; Heteroskedasticity-robust inference; Gravity Equation JEL: F10, F15, C13, C50 Department of Economic Theory, -Policy and -History, University of Innsbruck, Universitaetsstr. 15, A-6020 Innsbruck, Austria and Austrian Institute of Economic Research, E-mail: michael.pfaffermayr@uibk.ac.at.
1 Introduction In cross-sections the Poisson pseudo maximum likelihood estimator (PPML, Santos Silva and Trenreyro, 2006) is routinely used to estimate gravity models with exporter and importer dummies. Fernandez-Val and Weidner (2017) show that the PPML estimator for two-dimensional panels with large n and large t asymptotics is among the few, which allow to estimate the slope parameters (but not the coefficients of the dummies) consistently and without asymptotic bias. This note demonstrates that under PPML the heteroskedasticity robust standard errors of the slope parameters tend to be severely downward biased in crosssections of the size typically used in empirical research. The reason is twofold. On the one hand, the distribution of trade flows is highly skewed and the PPMLprojection matrix used to estimate the residuals typically exhibits excessive leverage. On the other hand, due the inclusion of the exporter and importer dummies the convergence rate of the leverage to zero is of the order of the number of the countries, rather than its square. 2 Robust standard errors in a cross-section gravity model and their bias Formally, in a cross-section of C countries the DGP of the gravity model in levels generates bilateral trade flows, y ij, as y ij = e z ij α+β i+γ j η ij. (1) Bilateral trade flows depend on trade frictions collected in the the K 1 vector z ij with corresponding vector of slope parameter vector α. Exporter and importer specific effects are denoted by β i and γ j, respectively. The disturbances η ij are assumed to be independently distributed with E[η ij z ij ] = 1, but possibly heteroskedastic. For estimation, the gravity model can be reformulated with additive disturbances y ij = e z ij α+β i+γ j + ε ij, ε ij = e z ij α+β i+γ j (η ij 1), (2) Observations on the explanatory variables are collected in the C 2 K matrix Z and exporter and importer dummies (depending on the dummy design possibly also the constant) in the C 2 2C 1 design matrix D. W = [Z, D] contains all right hand side variables including exporter and importer dummies. For missing values one may define the selection matrix V that is derived from the identity matrix by setting all ones in the main diagonal to zero if the corresponding observation 1
is missing. Following Fernandez-Val and Weidner (2017) the PPML estimator for the slope parameters can be written as 1 ( Z) 1 α = Z V Q V D V Z V Q V Dỹ, (3) with M = diag(e z ij α+ β i + γ j j ), ỹ = M 1 2 y, Z = M 1 2 Z, D = M 1 2 D and Q V D = I V D ( D V D) 1 D V. Under a set of standard regularity conditions, the limit distribution of α can be derived as C ( α α 0 ) d N (0, V α ), where V α = B0 1 A 0 Ω ε A 0B0 1 1 with B 0 = p lim Z C V Q C D V Z 2 V is assumed to be invertible, A 0 Ω ε A 1 0 = p lim Z C V Q C D M 1 2 2 V εε M 1 2 Q D V Z V. 2 and Ω ε = E[εε ]. Plugging in the estimated residuals ε, one can use 1 V C 2 α = C2 1B( α) 1 C 4 A( α) diag( ε ε )A( α) B( α) 1 for inference in finite samples. Following Chesher and Jewitt (1987) the bias of V α is given by ] E [ Vα V α = B0 1 A 0 M 0 diag(p W,ij (Ω η 2ω ij I C 2) p W,ij )M 0 A 0B0 1 + o(1). (4) p W,ij denotes the ij-th column of P W = M 1/2 0 V W ( W V W ) 1 W V M 1/2 0 and ω ij the ij, ij-th main diagonal element of Ω η. Under homoskedasticity of η the term diag(p W,ij (Ω η 2ω ij I C 2) p W,ij ) reverts to σ 2 diag(p W,ij,ij ), where p W,ij,ij denotes the leverage of observation ij (the diagonal element of P W referring to observation ij). In this case, V α is always downward biased of order O(C 1 ) rather than O(C 2 ), since the leverage is bounded in [0, 1] and rank(p W [ ) = ] K + 2C 1. v The proportionate bias (pb) can be defined as E ( V α V α)v for some vector v V αv ν and, as Chesher and Jewitt (1987) show, may be bounded by [ ] [( ) ] σ 2 sup pb( V α ) max η 1 v ij σ 2 η p W,ij,ij (1 p W,ij,ij ) p W,ij,ij = O(C 1 ) (5) ( ) ( ) ( ) inf pb( V α ) max v ij p W,ij,ij max ij (p W,ij,ij ) 2 = O(C 1 ) (6) 1 An Appendix provided as a supplement provides details on all derivations of the paper. https://eeecon.uibk.ac.at/~pfaffermayr/pfaffermayr_gravity_econometrics_ PPML_standard_errors_Appendix.pdf 2 The star indicates that these matrices are evaluated at parameter values lying in between the estimated and true ones element by element. 2
If heteroskedastiy of η is not too severe ( σ2 η < 2), the proportionate bias of the σ 2 η ] ] V α is always negative, since in this case sup v [pb( V α ) max ij [ p 2 W,ij,ij. More importantly, the lower bound does not depend on the nature of heteroscedasticity, but only on the leverage. Using the cross-section data from WIOD (Timmer et al. 2015) covering 42 economies and estimating a standard gravity model specified as Model 3 in the next section for a random sample of 21 economies (441 country pairs) exhibits a maximum leverage of 0.995 and the minimum leverage amounts to 0.004. This translates in a lower bound of the proportionate bias of 99.997%. The maximum and minimum leverage changes only marginally to 0.983 and 0.001, respectively, when using all 42 countries (1764 country pairs), as does the lower bound of the proportionate bias. Similar numbers are observed using trade data from OECD- STAN or GTAP. The estimated robust standard errors of the slope parameters are thus expected to be severely downward biased. 3 Monte Carlo simulations In each Monte Carlo run a simplified structural gravity model (Model 1) is generated as s ij = e 1,66z ij,1 0.90z ij,2 +β i +γ j η ij (7) C κ i = e 1,66z ij,1 0.90z ij,2 +β i +γ j (8) θ j = j=1 C e 1,66z ij,1 0.90z ij,2 +β i +γ j, (9) i=1 where the disturbances η ij are independent with E[η ij ] = 1 and enter in multiplicative form. Aggregate sales κ i,c and expenditure θ j,c (both measured as share of GDP in world GDP ) come from GTAP. β i and γ j, are derived as equilibrium solutions to the system of multilateral resistances (8) and (9) at true parameter values α 0. Data are sorted such that the sample always includes the C largest ones. z ij,1 is a border dummy taking the value 1 for i = j and zero else. z ij,2 refers to log weighted distance and is taken from CEPII s database (Model 1). To illustrate the impact of the leverage on the estimated robust standard errors, in Model 2 z i1 and z i2 are generated from a uniform distribution. Model 3 mimics a more realistic standard gravity model using dummies for contiguity, common language, colonial relationships and regional trade agreements as explanatory variables in addition to border and ln distance, all entering with coefficients as estimated by 3
PPML ( 1.66, 0.90, 0.35, 0.49,.04, 0.41). Models 2 and 3 keep the exporter and importer effects as in Model 1. Lastly, Model 4 uses the explanatory variables of Model 2, but additionally generates the exporter and importer effects from a uniform distribution as well. In all four models the explanatory variables including the exporter and importer effects remain fixed in repeated samples. The Monte Carlo experiments consider C {20, 60} and generate independent disturbances from three different DGPs. DGP 1 generates the homoskedastic disturbances from a truncated normal that guarantees positive trade flows. DGP exp(0.1z 2 multiplies the truncated normal disturbances of DGP 1 by ij,2 ) C C i=1 j=1 exp(0.1z ij,2) to induce heteroskedasticity in addition to that stemming from the additive disturbances. The third DGP is based on a skewed distribution of the disturbances assuming that η is generated as χ 2 (10). In all three cases the disturbances are transformed to obtain E[η] = 1 and (an average) variance is 0.2. Under these assumptions the standard errors of the estimated parameters are similar to those found in the literature. Figure 1 delivers a clear message. The estimated standard errors of ˆα 1 under Models 1 and 3 are heavily downward biased and exhibit considerable variation. In Model 2 with uniformly distributed explanatory variables the bias is substantially reduced. However, comparing the results of Model 2 and Model 4 shows that it is mainly the skewness of the fixed exporter and importer effects that leads to the large variation of the estimated standard errors. Moreover, the bias disappears in Model 4 as one would expect. Table 1 exhibits the average of the simulated standard errors of α 1 along with its estimated counterpart, the simulated coverage rates of a 95%-confidence intervals and the simulated proportionate bias. 3 The simulated standard errors are calculated as the standard deviation of α 1 in 10000 Monte Carlo runs, while the estimated ones are based on the means of the estimated standard errors. In addition, the table reports the asymptotic lower and upper bound of the proportionate bias as derived in (5) and (6). 3 Results for α 2 are rather similar and available upon request. 4
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Table 1: Monte Carlo simulation results: Leverage, simulated standard errors, estimated standard errors and 95% coverage rates of the slope parameters under dummy PPML for α 1 Model Leverage in % σ 2 η/σ 2 η sim. sd est. sd sim. bias in % cr lower b. upper b. C=20 Truncated normal 1 99.86 0.608 1.00 0.084 0.046 70.43 0.705 2 91.34 0.380 1.00 0.049 0.042 27.48 0.907 3 99.98 0.318 1.00 0.179 0.088 75.99 0.657 4 50.95 1.675 1.00 0.028 0.027 5.13 0.944 Truncated normal, heteroskedastic 1 99.86 13.679 3.06 0.072 0.040 69.03 0.719 2 91.34 0.297 1.22 0.048 0.041 26.03 0.909 3 99.98 13.679 3.06 0.160 0.079 75.33 0.662 4 50.95 1.311 1.22 0.012 0.012 4.78 0.944 χ(10) 1 99.86 0.608 1.00 0.082 0.044 70.74 0.695 2 91.34 0.380 1.00 0.049 0.041 28.29 0.903 3 99.98 0.318 1.00 0.173 0.084 76.40 0.641 4 50.95 1.675 1.00 0.028 0.027 6.42 0.941 C=60 Truncated normal 1 99.80 0.022 1.00 0.078 0.040 74.07 0.681 2 78.41 0.002 1.00 0.043 0.038 20.09 0.918 3 99.96 0.021 1.00 0.132 0.062 78.14 0.638 4 18.54 0.360 1.00 0.009 0.009 0.66 0.949 Truncated normal, heteroskedastic 1 99.80 33.440 4.00 0.066 0.034 72.72 0.692 2 78.41 0.001 1.22 0.042 0.038 19.14 0.918 3 99.96 33.440 4.00 0.118 0.055 77.86 0.642 4 18.54 0.281 1.22 0.004 0.004 0.55 0.950 χ(10) 1 99.80 0.022 1.00 0.074 0.038 73.45 0.677 2 78.41 0.002 1.00 0.043 0.038 23.86 0.913 3 99.96 0.021 1.00 0.125 0.059 77.51 0.633 4 18.54 0.360 1.00 0.009 0.009 2.00 0.953 Notes: 10000 Monte Carlo runs. cr refers to the coverage rate of a nominal 95%-confidence interval using the normal distribution. 6
In the sample of the 20 biggest countries, Model 1 implies a simulated proportionate bias of around minus 70% and a coverage rate of a 95%-confidence interval between 69.5% and 71.9% under the three considered DPGs. It turns out that the leverage is excessively high in bilateral trade data. Under DGP 1 and 3 the lower bound of the proportionate bias amounts to 99.86% at C = 20 and to 99.80% at C = 60, while its upper bound are 0.61% and 0.02%, respectively. Under DGP 2 the ratio of maximum and minimum variances of η ij amounts to σ 2 /σ 2 = 3.06 at C = 20 and 4.00 at C = 60. In Model 1 the upper bound is now at 13.68 at C = 20 (33.40 at C = 60). The proportionate bias of the estimated standard error of α 1 decreases just marginally. Only very pronounced heteroskedasticty of η would reduce it substantially. Results for Model 2 generating the explanatory variables from a uniform distribution, indicate a considerable lower proportionate bias of the estimated standard error of α 1 and, at C = 20, a coverage ratio of the 95%-confidence interval between 0.903 and 0.909. Model 3 looks at the richer gravity model. With a simulated downward bias of as much as 77.86% it exhibits an even worse performance. In Model 4 all variables and the exporter and importer fixed effects come from the uniform distribution. The proportionate bias is negligible and lies between 6.48% and 4.79% at C = 20 and the coverage ratio is correct. Overall, these results suggest that the downward bias of the estimated standard error of α 1 is mainly determined by the large leverage induced by the trade friction indicators. Increasing the sample size to the 60 largest countries does not reduce the bias in Models 1 and 3, while for Models 2 and 4 the bias becomes smaller. The nature of the DGP seems not to make a big difference. Using a degrees of freedom correction reduces the bias and improves the coverage rate only marginally. Simulation results for this case are available upon request. 4 Conclusions PPML with dummies is now widely used for estimation of gravity models in levels. However, for calculating standard errors and confidence intervals of the estimated slope parameters it is of limited use. Typically, bilateral trade flows data are characterized by excessively high leverage with a slow convergence rate due to the increase of the number of parameters in sample size. As a result, the coverage rates of confidence intervals of the estimated slope parameters are far to small and parameter tests severely oversized. The present available statistical software such as stata and R does not account for this problem and there is an urgent need for improving the finite sample properties of PPML estimation in this respect. Approaches similar to those available for linear models and surveyed by MacKinnon (2013) could possibly bring improve- 7
ments. References Chesher A. and I. Jewitt (1987), The Bias of a Heteroskedasticity Consistent Covariance Matrix Estimator, Econometrica 55(5), 1217 1222. Fernandez-Val, I. and M. Weidner (2016), Individual and Time Effects in Nonlinear Panel Models with large N, T, Journal of Econometrics 192(1), 291 312. MacKinnon J.G. (2013), Thirty Years of Heteroskedasticity-Robust Inference. In: Chen X., Swanson N. (eds) Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis. Springer, New York, NY. Santos Silva, J.M.C. and S. Tenreyro (2006), The Log of Gravity, Review of Economics and Statistics 88(4), 641 658. Timmer, M. P., Dietzenbacher, E., Los, B., Stehrer, R. and de Vries, G. J. (2015), An Illustrated User Guide to the World Input Output Database: the Case of Global Automotive Production, Review of International Economics 23(3), 575 605. 8