Multiple Fixed Effects in Nonlinear Panel Data Models Theory and Evidence

This paper considers the adaptability of nonlinear panel data models to multiple fixed effects. It is motivated by the gravity equation used in international trade, where influential papers such as Santos Silva and Tenreyro (2006) use nonlinear models with fixed effects for both importing and exporting countries. It is also relevant for other areas of microeconomics such as labor economics, where a wage equation might contain both worker and firm fixed effects, or industrial organization, where knowledge diffusion equations using patent data can include citing and cited country fixed effects. Econometric theory has mostly focused on the estimation of single fixed effects models. This paper investigates whether existing methods can be modified to eliminate multiple fixed effects for some specific models in which the incidental parameter problem has already been solved in the presence of a single fixed effect. We find that it is possible to generalize the conditional maximum likelihood approach of Rasch (1960, 1961) to include two fixed effects for the logit, the Poisson and the Negative binomial regression models considered by Hausman, Hall and Griliches (1984) as well as for the Gamma model. Surprisingly, Manski’s (1987) maximum score estimator for binary response models cannot be adapted to the presence of two fixed effects. We also look at a multiplicative form model. In that case, it is possible to consistently estimate the parameters when there are two fixed effects with the use of a moment condition. Monte Carlo simulations show that the conditional logit estimator presented in this paper is less biased than other logit estimators without sacrificing on the precision. This superiority is emphasized in small samples. An application on trade data using both the logit and Poisson estimators further highlights the importance of properly accounting for two fixed effects. Indeed, estimates of the gravity model parameters produced by the method presented in this paper differ significantly from those obtained with the various estimators used in the trade literature.