Local Semi-Parametric Efficiency of the Poisson Fixed Effects Estimator

Abstract Hausman, Hall, and Griliches [Hausman, J., H. B. Hall, and Z. Griliches. 1984. “Econometric Models for Count Data with an Application to the Patents-R & D Relationship.” Econometrica 52 (4): 909–938.] have defined the Poisson fixed effects (PFE) estimator to estimate models of panel data with count dependent variables under distributional assumptions conditional on covariates and unobserved heterogeneity, but without any restriction on the distribution of unobserved heterogeneity conditional on covariates. Wooldridge [Wooldridge, J. M. 1999. “Distribution-Free Estimation of some Nonlinear Panel Data Models.” Journal of Econometrics 90 (1): 77–97.] showed that the PFE estimator is actually consistent even if the distributional assumptions of the PFE model are violated, as long as the restrictions imposed on the conditional mean of the dependent variable are satisfied. In this note I study the efficiency of the PFE estimator in the absence of distributional assumptions. I show that the PFE estimator corresponds to the optimal estimator for random coefficients models of Chamberlain [Chamberlain, G. 1992. “Efficiency Bounds for Semiparametric Regression.” Econometrica 60 (3): 567–596.] in the particular case where the assumptions of equal conditional mean and variance and zero conditional serial correlation are satisfied, regardless of whether the distributional assumptions of the PFE model hold. For instance the dependent variable does not need to be a count variable. This local efficiency result, combined with the simplicity and robustness of the PFE estimator, should provide a useful additional justification for its use to estimate conditional mean models of panel data.