INTRODUCTION

The most commonly used models for count regression, Poisson and negative binomial, were presented in Chapter 3. In this chapter we introduce richer models for count regression using cross-section data. For some of these models the conditional mean retains the exponential functional form. Then the Poisson QMLE and NB2 ML estimators remain consistent, although they may be inefficient and may not be suitable for predicting probabilities, rather than the conditional mean. For many of these models, however, the Poisson and NB2 estimators are inconsistent. Then alternative methods are used, ones that generally rely heavily on parametric assumptions.

One reason for the failure of the Poisson regression is that the Poisson process has unobserved heterogeneity that contributes additional randomness. This leads to mixture models, the negative binomial being only one example. A second reason is the failure of the Poisson process assumption and its replacement by a more general stochastic process.

Some common departures from the standard Poisson regression are as follows.

1. Failure of the mean-equals-variance restriction: Frequently the conditional variance of data exceeds the conditional mean, which is usually referred to as extra-Poisson variation or overdispersion relative to the Poisson model. Overdispersion may result from neglected or unobserved heterogeneity that is inadequately captured by the covariates in the conditional mean function. It is common to allow for random variation in the Poisson conditional mean by introducing a multiplicative error term. This leads to families of mixed Poisson models.

2. […]