Pseudo Maximum Likelihood Methods: Applications to Poisson Models

Abstract

Pseudo maximum likelihood techniques are applied to basic Poisson models and to Poisson models with specification errors. In the latter case it is shown that consistent and asymptotically normal estimators can be obtained without specifying the p.d.f. of the disturbances. These estimators are compared both from the finite sample and the asymptotic point of view. Quasi generalized PML estimators, which asymptotically dominate all PML estimators, are also proposed. Finally, bivariate and panel data Poisson models are discussed. THE ANALYSIS OF ECONOMIC BEHAVIOR often leads to the study of characteristics taking a small number of positive values. The classical linear model is not adapted to explain how such discrete variables depend on other quantitative or qualitative variables. The reasons are similar to those usually given in the case of an endogenous qualitative variable: the shape of the observation set does not correspond to a linear model, the assumption of normality of the disturbances cannot be made, since the endogenous variables take a small number of values with strictly positive probabilities, and the prediction formulae which are deduced from a linear model give impossible values. In the models considered in the literature to describe discrete variables (Cox and Lewis [2], El. Sayyad [3], Frome, Kutner, and Beauchamp [4], Gilbert [5], Hausman, Hall, and Griliches [8]; see also Lancaster [12]) the endogenous variable is assumed to have a Poisson distribution conditional upon the exogenous variables. The parameter of this distribution is a function of the values of the exogenous variables. The choice of such a model is justified if the dependent variable counts the occurrence of a given event during a fixed period and if the usual assumptions of the Poisson process are satisfied. For instance, the model is adapted to describe daily numbers of oil tankers' arrivals in a port, the number of accidents at work by factory, or the number of patents applied for and received by firms (Hausman, Hall, and Griliches [8]).