Moment Estimation of the Probit Model with an Endogenous Continuous Regressor

We propose a generalized method of moments (GMM) estimator with optimal instruments for a probit model that includes a continuous endogenous regressor. This GMM estimator incorporates the probit error and the heteroscedasticity of the error term in the first-stage equation in order to construct the optimal instruments. The estimator estimates the structural equation and the first-stage equation jointly and, based on this joint moment condition, is efficient within the class of GMM estimators. To estimate the heteroscedasticity of the error term of the first-stage equation, we use the k-nearest neighbour (k-nn) non-parametric estimation procedure. Our Monte Carlo simulation shows that in the presence of heteroscedasticity and endogeneity, our GMM estimator outperforms the two-stage conditional maximum likelihood estimator. Our results suggest that in the presence of heteroscedasticity in the first-stage equation, the proposed GMM estimator with optimal instruments is a useful option for researchers.