Foundations and Trends® in Econometrics > Vol 1 > Issue 2

Functional Form and Heterogeneity in Models for Count Data

By William Greene, Department of Economics, Stern School of Business, New York University, USA, wgreene@stern.nyu.edu

 
Suggested Citation
William Greene (2007), "Functional Form and Heterogeneity in Models for Count Data", Foundations and Trends® in Econometrics: Vol. 1: No. 2, pp 113-218. http://dx.doi.org/10.1561/0800000008

Publication Date: 08 Aug 2007
© 2007 W. Greene
 
Subjects
Models for count data
 
Keywords
Poisson regression, Negative binomial, Panel data, Heterogeneity, Lognormal, Bivariate Poisson, Zero inflation, Two part model, Hurdle model
 

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In this article:
1. Introduction 
2. Basic Functional Forms for Count Data Models 
3. Two Part Models 
4. Models for Panel Data 
5. The Bivariate Poisson Model 
6. Applications 
7. Conclusions 
Appendix A. Log Likelihood and Gradient for NBP model 
Appendix B. Derivatives of Partial Effects in the Poisson Model with Sample Selection 
Appendix C. Derivatives of Partial Effects of ZIP Model with Endogenous Zero Inflation 
Appendix D. Derivatives of Partial Effects in Hurdle Models 
Appendix E. Partial Effects and Derivatives of Partial Effects in Hurdle Models with Endogenous Participation 
References 

Abstract

This study presents several extensions of the most familiar models for count data, the Poisson and negative binomial models. We develop an encompassing model for two well-known variants of the negative binomial model (the NB1 and NB2 forms). We then analyze some alternative approaches to the standard log gamma model for introducing heterogeneity into the loglinear conditional means for these models. The lognormal model provides a versatile alternative specification that is more flexible (and more natural) than the log gamma form, and provides a platform for several "two part" extensions, including zero inflation, hurdle, and sample selection models. (We briefly present some alternative approaches to modeling heterogeneity.) We also resolve some features in Hausman, Hall and Griliches (1984, Economic models for count data with an application to the patents–R&D relationship, Econometrica52, 909–938) widely used panel data treatments for the Poisson and negative binomial models that appear to conflict with more familiar models of fixed and random effects. Finally, we consider a bivariate Poisson model that is also based on the lognormal heterogeneity model. Two recent applications have used this model. We suggest that the correlation estimated in their model frameworks is an ambiguous measure of the correlation of the variables of interest, and may substantially overstate it. We conclude with a detailed application of the proposed methods using the data employed in one of the two aforementioned bivariate Poisson studies.

DOI:10.1561/0800000008

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Table of contents:
1. Introduction
2. Basic Function Forms of Count Data Models
3. The Two Part Models
4. Models for Panel Data
5. The Bivariate Poisson Model
6. Applications
7. Conclusions
Appendix A. Log Likelihood and Gradient for NBP model
Appendix B. Derivatives of Partial Effects in the Poisson Model with Sample Selection
Appendix C. Derivatives of Partial Effects of ZIP Model with Endogenous Zero Inflation
Appendix D. Derivatives of Partial Effects in Hurdle Models
Appendix E. Partial Effects and Derivatives of Partial Effects in Hurdle Models with Endogenous Participation
References

Functional Form and Heterogeneity in Models for Count Data

Functional Form and Heterogeneity in Models for Count Data surveys practical extensions of the Poisson and negative binomial (NB) models that practitioners can employ to refine the specifications or broaden their reach into new situations. The author resolves some inconsistencies of the panel data models with other more familiar results for the linear regression model.

Functional Form and Heterogeneity in Models for Count Data is focused on two large issues: the accommodation of overdispersion and heterogeneity in the basic count framework and the functional form of the conditional mean and the extension of models of heterogeneity to models for panel data and sources of correlation across outcomes. The first is more straightforward since, in principle, these are elements of the conditional variance of the distribution of counts that can be analyzed apart from the conditional mean. Robust inference methods for basic models can be relied upon to preserve the validity of estimation and inference procedures. The second feature motivates the development of more intricate models such as the two part, panel and bivariate models presented in the text.

 
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