GLM estimation of trade gravity models with fixed effects

Abstract

Many empirical gravity models are now based on generalized linear models (GLM), of which the poisson pseudo-maximum likelihood estimator is a prominent example and the most frequently used estimator. Previous literature on the performance of these estimators has primarily focussed on the role of the variance function for the estimators’ behavior. We add to this literature by studying the small sample performance of estimators in a data-generating process that is fully consistent with general equilibrium economic models of international trade. Economic theory suggests that (1) importer- and exporter-specific effects need to be accounted for in estimation, and (2) that they are correlated with bilateral trade costs through general equilibrium (or balance-of-payments) restrictions. We compare the performance of structural estimators, fixed effects estimators, and quasi-differences estimators in such settings, using the GLM approach as a unifying framework.

Appendix

1.1 First-order conditions for fixed effects

The fixed effects $e_i$ and $m_j$ are estimated as the coefficients on a set of exporter and importer dummy variables, respectively:

$$\begin{aligned} e_i=\sum _{k=2}^Ce_{k}D_{ki}, \quad m_j=\sum _{k=2}^Cm_{k}D_{kj}, \end{aligned}$$

Fig. 3

Iterative structural estimators: Predictions and Pearson residuals. Notes: The figure plots predictions of trade flows against Pearson residuals for iterative structural GLM estimates of the trade gravity model (21), corresponding to columns 6–9 of Table 9

where $D_{ki}=\varvec{1} (i=k)$, $k=2,\ldots ,C$ are the exporter indicator variables; and $D_{kj}=\varvec{1} (j=k)$, $k=2,\ldots ,C$, the importer indicator variables. The constant $\beta _0$ absorbs $e_1$ and $m_1$. Then the $2\times (C-1)$ first-order conditions for the parameters $e_k$ and $m_k$, for $k=2,\ldots ,C$, are

$$\begin{aligned}&\sum _{i=1}^C\sum _{j=1}^C \frac{\left[ X_{ij} - \exp \left( \sum _{k=2}^Ce_{k}D_{ki} + \sum _{k=2}^Cm_{k}D_{kj} + d_{ij}'\beta \right) \right] }{V(X_{ij})}\\&\qquad \times \exp \left( \sum _{k=2}^Ce_{k}D_{ki} + \sum _{k=2}^Cm_{k}D_{kj} + d_{ij}'\beta \right) D_{ki}=0, \\&\sum _{i=1}^C\sum _{j=1}^C \frac{\left[ X_{ij} - \exp \left( \sum _{k=2}^Ce_{k}D_{ki} + \sum _{k=2}^Cm_{k}D_{kj} + d_{ij}'\beta \right) \right] }{V(X_{ij})}\\&\qquad \times \exp \left( \sum _{k=2}^Ce_{k}D_{ki} + \sum _{k=2}^Cm_{k}D_{kj} + d_{ij}'\beta \right) D_{kj}=0. \end{aligned}$$

Fig. 4

Iterative-structural estimators: Density of deviance residuals. Notes: The figure plots kernel density of deviance residuals for iterative-structural GLM estimates of the trade gravity model (21), corresponding to columns 6–9 of Table 9

Table 10 Alternative scenario $\alpha =-2$ (1,000 replications)

Table 11 $t$-statistics for iterative-structural estimators (baseline specification, $C=10$), 1,000 replications

Table 12 Negative Binomial Quasi-generalized pseudo-maximum likelihood estimation (Baseline specification, 1,000 replications)