Abstract
The Poisson regression model remains an important tool in the econometric analysis of count data. In a pioneering contribution to the econometric analysis of such models, Lung-Fei Lee presented a specification test for a Poisson model against a broad class of discrete distributions sometimes called the Katz family. Two members of this alternative class are the binomial and negative binomial distributions, which are commonly used with count data to allow for under- and over-dispersion, respectively. In this paper we explore the structure of other distributions within the class and their suitability as alternatives to the Poisson model. Potential difficulties with the Katz likelihood leads us to investigate a class of point optimal tests of the Poisson assumption against the alternative of over-dispersion in both the regression and intercept only cases. In a simulation study, we compare score tests of ‘Poisson-ness’ with various point optimal tests, based on the Katz family, and conclude that it is possible to choose a point optimal test which is better in the intercept only case, although the nuisance parameters arising in the regression case are problematic. One possible cause is poor choice of the point at which to optimize. Consequently, we explore the use of Hellinger distance to aid this choice. Ultimately we conclude that score tests remain the most practical approach to testing for over-dispersion in this context.
Highlights
The well-known Pearson family of continuous distributions, originally explored by Pearson (1895), is comprised of any solution to a particular differential equation
It provides a framework within which practitioners can extend simple Poisson models to models that allow for individual heterogeneity, using the Poisson regression model (PRM)
The PRM can, in turn, be extended to models that allow for either over-dispersion, using the negative binomial regression model (NBRM), or under-dispersion, using the binomial regression model
Summary
The well-known Pearson family of continuous distributions, originally explored by Pearson (1895), is comprised of any solution to a particular differential equation. Concurrently, papers by Cameron and Trivedi (1986); Lee (1986) and Lawless (1987ab) made substantial contributions to the literature on inference in the PRM, the NBRM, and testing for over-dispersion, with both Cameron and Trivedi (1986) and Lee (1986), in particular, couching substantial parts of their analysis within the context of the Katz family of distributions. This class of distributions is interesting because the binomial and negative binomial distributions are alternative specifications to the Poisson that allow under- and over-dispersion, respectively.
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