Estimating the gravity model without gravity using panel data

Abstract

This article examines the effects of zero trade on the estimation of the gravity model using both simulated and real data with a panel structure, which is different from the more conventional cross-sectional structure. We begin by showing that the usual log-linear estimation method can result in highly deceptive inference when some observations are zero. As an alternative approach, we suggest using the poisson fixed effects estimator. This approach eliminates the problems of zero trade, controls for heterogeneity across countries, and is shown to perform well in small samples.

Acknowledgements

Previous versions of this article were presented at the 2006 spring meeting of the Midwest International Economics Group and at a seminar at Lund University. The authors would like to thank conference and seminar participants, and in particular Yves Bourdet, Joakim Gullstrand, Mark Taylor and one anonymous referee for many valuable comments and suggestions. J. Westerlund gratefully acknowledges financial support from the Jan Wallander and Tom Hedelius Foundation, research grant number W2006–0068:1. F. Wilhelmsson gratefully acknowledges financial support from Stiftelsen för främjande av ekonomisk forskning vid Lunds Universitet and Sparbanksstiftelsen Fars & Frosta.

Notes

¹ Note that since each country is both an exporter and an importer in a bilateral trade relation, each country pair is observed twice. The number of observations is therefore twice the number of country pairs.

² As long as Equation 4 holds the poisson estimator works, see for example, Wooldridge (2002) and Winkelman (2008). In fact, neither Equation 4 nor the maximization of the log-likelihood function require that the dependent variable is a count. It could be a binary variable or, as in our case, a nonnegative continuous variable. This property of the estimator has been used by Silva and Tenereyro (2006). The interpretation of the estimated coefficients is similar to the interpretation of the coefficients in the log-linear model. That is, the estimated coefficient reflects the elasticity of the dependent variable with respect to the relevant independent variable. In the case of an dummy variable, the estimated coefficient provides a reasonable approximation for small estimated values (see Winkelman, 2008, for a more elaborative discussion).

³ Another possibility is to use the Zero Inflated Poisson (ZIP) model. But, so far it seems that the estimation of this model with fixed effects has not yet been analysed in the literature. In fact, Winkelman (2008) points out that the properties of the fixed effects poisson ML estimator does not carry over to the ZIP model, and that the estimation of this model is still an open issue.

⁴ Other values of σ _ij ² produced very similar results and are thus not reported.

⁵ The poisson ML estimator is implemented using the GAUSS optimization library OPTMUM. We use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) gradient algorithm with numerical derivatives. The SEs of the estimated parameters are computed based on the conventional Hessian method, which generally worked best in the simulations. The truncated LS is used to start up the estimation.

⁶ We also simulated the power of the t-tests. However, since the size of the LS-based tests turned out to be heavily distorted, with rejection frequencies close to 100% in most experiments, power is not very interesting, and the results are therefore not reported.

⁷ The quasi-ML estimator only requires that the conditional mean in Equation 4 is correctly specified, and does not make use of Equation 5 (see, e.g. Gourieroux et al., 1984; Wooldridge, 2002).

⁸ We used the T = 2 version of the Kyriazidou (1997) estimator, which is relatively easy to compute, but preliminary results suggest that the poor performance extends also to the case when T > 2. Also, for this experiment, the data-generating process was adapted so as to fit the sample selection setting of Kyriazidou (1997).

⁹ Another possibility is to use the wild bootstrap, see Cameron and Trivedi (1998) for a discussion.