ABSTRACT

I study the problem of identifying average partial effects (APEs), which are partial effects averaged across the population distribution of unobserved heterogeneity, under different assumptions. One possibility is that the unobserved heterogeneity is conditionally independent of the observed covariates. When the unobserved heterogeneity is independent of the original covariates, or conditional mean independent but heteroskedastic, the derivations of APEs provide a new view of traditional specification problems in widely used models such as probit and Tobit. In addition, the focus on average partial effects resolves scaling issues that arise in estimating the parameters of probit and Tobit models with endogenous explanatory variables.

INTRODUCTION

Econometric models, especially at the individual, family, or firm level, are often specified to depend on unobserved heterogeneity in addition to observable covariates. Models with unobserved heterogeneity are sometimes derived from economic theory; at other times they are based on introspection.

In nonlinear models, much has been made about the deleterious effects that ignoring heterogeneity can have on the estimation of parameters, even when the heterogeneity is assumed to be independent of the observed covariates. A leading case is the probit model with an omitted variable. Yatchew and Griliches (1985) show that when the omitted variable is independent of the explanatory variables and normally distributed, the probit estimators suffer from (asymptotic) attenuation bias. This result is sometimes cited to illustrate how a misspecification that is innocuous in linear models leads to problems in nonlinear models (see, for example, Greene [2000, p. 828]).