Introduction

When individual data are used to estimate the parameters of a model, a persistent question that has to be addressed is whether the sample of individuals has been randomly selected. If their presence in the sample is a consequence of some choice made, and selection has been a function of some economic variables, then it is possible that a failure to recognize the endogeneity of inclusion in the sample can lead to inconsistent parameter estimators. The presence of such endogenous sample-selection bias has been recognized for many years, for example Roy (1951), but its implication for estimation theory was left until the papers by Gronau (1974) and Heckman (1974). Since that time it has been recognized that the problems caused by sample selection are pervasive in the analysis of microeconomic data and some adjustment for its effects needs to be made.

Conceptually, the effects of sample selection are simple; selection transforms a linear model with errors having a zero conditional mean and being homoskedastic into one where the errors have a nonzero conditional mean and are heteroskedastic. Somehow an adjustment needs to be performed to compensate for these changes to the error term. Parametric approaches, in particular that of Heckman (1976), have proceeded by finding expressions for the conditional means and variances of the error term under self-selection, and then making corrections to compensate for these effects. Section 8.2 outlines this estimator. It involves a two-step procedure that is simple to use and is available in many computer packages. An alternative, also discussed in this section, is to do maximum likelihood estimation, and this is also a standard option in programs such as LIMDEP.