Introduction

Longitudinal data or panel data are observations on a cross-section of individual units such as persons, households, firms, and regions that are observed over several time periods. The data structure is similar to that of multivariate data considered in Chapter 8. Analysis is simpler than for multivariate data because for each individual unit the same outcome variable is observed, rather than several different outcome variables. Analysis is more complex because this same outcome variable is observed at different points in time, introducing time series data considerations presented in Chapter 7.

In this chapter we consider longitudinal data analysis if the dependent variable is a count variable. Remarkably, many count regression applications are to longitudinal data rather than simpler cross-section data. Econometrics examples include the number of patents awarded to each of many individual firms over several years, the number of accidents in each of several regions, and the number of days of absence for each of many persons over several years. A political science example is the number of protests in each of several different countries over many years. A biological and health science example is the number of occurrences of a specific health event, such as seizure, for each of many patients in each of several time periods.

A key advantage of longitudinal data over cross-section data is that they permit more general types of individual heterogeneity. Excellent motivation was provided by Neyman (1965), who pointed out that panel data enable one to control for heterogeneity and thereby distinguish between true and apparent contagion.

AIC: Akaike information criterion

ARMA: autoregressive moving average

BHHH: Berndt-Hall-Hall-Hausman algorithm

BIC: Bayes information criterion

BP: binary Poisson

Boot: bootstrap

CAIC: consistent Akaike information criterion

CB: correlated binomial

cdf: cumulative distribution function

CFMNB: slope-constrained finite mixture of negative binomials

CFMP: slope-constrained finite mixture of Poissons

CM: conditional moment (function or test)

Many widely used regression packages, including LIMDEP, STATA, TSP, and GAUSS, support maximum likelihood estimation of standard Poisson and negative binomial regressions, the latter of these in a separate count module. LIMDEP also supports the QGPML versions of the standard models, maximum-likelihood estimation of truncated or censored Poisson, geometric and negative binomial models, and ZIP and sample selection models. STATA also supports the generalized negative binomial regression in which the overdispersion parameter is further parameterized as a function of additional covariates. In addition, any statistical package with a generalized linear models component will include maximum likelihood and QGPML estimation of the Poisson, although not necessarily negative binomial. Thus, regression packages cover the models in Chapter 3 and roughly half of those in Chapter 4. The packages vary somewhat in the provision of diagnostics such as overdispersion tests and goodness-of-fit measures.

At the time of writing (late 1997) there is virtually no specialized software for the models presented in Chapters 7 through 12. A notable exception is estimation of basic panel count data models, which is provided by both LIMDEP and TSP. For models for which off-the-shelf software is not available, one needs to provide at least the likelihood function, for maximum likelihood estimation, or the moment conditions and weighting matrix, for GMM estimation. In principle this can be done using many regression packages, or using matrix programming languages such as GAUSS, MATLAB, S-PLUS, or SAS/IML. In practice numerical problems can be encountered if models are quite nonlinear.