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Open AccessFeature PaperArticle

Distributions You Can Count On …But What’s the Point?

School of Management, University of Liverpoo, Liverpool L69 7ZH, UK
Department of Economics, The University of Melbourne, Carlton VIC 3053, Australia
Author to whom correspondence should be addressed.
An early version of this paper was originally prepared for the Fest in Celebration of the 65th Birthday of Professor Maxwell King, hosted by Monash University. It was started while Skeels was visiting the Department of Economics at the University of Bristol. He would like to thank them for their hospitality and, in particular, Ken Binmore for a very helpful discussion. We would also like to thank David Dickson and David Harris for useful comments along the way.
Econometrics 2020, 8(1), 9;
Received: 2 September 2019 / Revised: 25 February 2020 / Accepted: 25 February 2020 / Published: 4 March 2020
(This article belongs to the Special Issue Discrete-Valued Time Series: Modelling, Estimation and Forecasting)
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. View Full-Text
Keywords: Katz family of distributions; binomial distribution; negative binomial distribution; point optimal test; regression; score test; Hellinger distance Katz family of distributions; binomial distribution; negative binomial distribution; point optimal test; regression; score test; Hellinger distance
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McCabe, B.P.M.; Skeels, C.L. Distributions You Can Count On …But What’s the Point? Econometrics 2020, 8, 9.

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