18 - Endogeneity in Probit Response Models

Summary

Abstract. Endogeneity bias is an issue in regression models, including linear and probit models. Conventional methods for removing the bias have their own problems. The usual Heckman two-step procedure should not be used in the probit model: From a theoretical perspective, this procedure is unsatisfactory, and likelihood methods are superior. However, serious numerical problems occur when standard software packages try to maximize the biprobit likelihood function, even if the number of covariates is small. The log-likelihood surface may be nearly flat or may have saddle points with one small positive eigenvalue and several large negative eigenvalues. The conditions under which parameters in the model are identifiable are described; this produces novel results.

Introduction

Suppose a linear regression model describes responses to treatment and to covariates. If subjects self-select into treatment, the process being dependent on the error term in the model, endogeneity bias is likely. Similarly, we may have a linear model that is to be estimated on sample data; if subjects self-select into the sample, endogeneity becomes an issue.

Heckman (1978, 1979) suggested a simple and ingenious two-step method for taking care of endogeneity, which works under the conditions described in those papers. This method is widely used. Some researchers have applied the method to probit response models. However, the extension is unsatisfactory.