INTRODUCTION

In this chapter we consider regression models for an m-dimensional vector of jointly distributed and, in general, correlated random variables y = (y1, y2, …, ym), a subset of which are event counts. One special case of interest is that of m seemingly unrelated count regressions denoted as y∣x = (y1|x1, y2|x2, …, ym|xm), where x = (x1, …, xm) are observed exogenous covariates and the counts are conditionally correlated. In econometric terminology this model is a multivariate reduced-form model in which multivariate dependence is not causal. Most of this chapter deals with such reduced-form dependence. Causal dependence, such as y1 depending explicitly on y2, is covered elsewhere, most notably in Chapter 10.

Depending on the multivariate model, ignoring multivariate dependence may or may not affect the consistency of the univariate model estimator. In either case, joint modeling of y1, …, ym leads to improved efficiency of estimation and the ability to make inferences about the dependence structure. A joint model can also support probability statements about the conditional distribution of a subset of variables, say y1, given realization of another subset, say y2.

Multivariate nonlinear, non-Gaussian models are used much less often than multivariate linear Gaussian models, and there is no model with the universality of the linear Gaussian model. Fully parametric approaches based on the joint distribution of non-Gaussian vector y, given a set of covariates x, are difficult to apply because analytically and computationally tractable expressions for such joint distributions are available for special cases only.