INTRODUCTION

Count regressions with endogenous regressors occur frequently. Ignoring the feedback from the response variable to the endogenous regressor, and simply conditioning the outcome on variables with which it is jointly determined, leads in general to inconsistent parameter estimates. The estimation procedure should instead allow for stochastic dependence between the response variable and endogenous regressors. In considering this issue the existing literature on simultaneous equation estimation in nonlinear models is of direct relevance (T. Amemiya, 1985).

The empirical example of Chapter 3 models doctor visits as depending in part on the individual's type of health insurance. In Chapter 3 the health insurance indicator variables were treated as exogenous, but health insurance is frequently a choice variable rather than exogenously assigned. A richer model is a simultaneous model with a count outcome depending on endogenous variable(s) that may be binary (two insurance plans), multinomial (more than two insurance plans), or simply continuous.

This chapter deals with several classes of models with endogenous regressors, tailored to the outcome of interest being a count. It discusses estimation and inference for both fully parametric full-information methods and less parametric limited-information methods. These approaches are based on a multiple equation model in which that for the count outcome is of central interest, but there is also an auxiliary model for the endogenous regressor, sometimes called the first-stage or reduced-form equation. Estimation methods differ according to the detail in which the reduced form is specified and exploited in estimation.