Simpler standard errors for two-stage optimization estimators

Aiming to lessen the analytic and computational burden faced by practitioners seeking to correct the standard errors of two-stage estimators, I offer a heretofore unexploited simplification of the conventional formulation for the most commonly encountered cases in empirical application—two-stage estimators that involve maximum likelihood or pseudomaximum likelihood estimation. With the applied researcher in mind, I focus on the two-stage residual inclusion estimator designed for nonlinear regression models involving endogeneity. I demonstrate the analytics and Stata and Mata code for implementing my simplified standard-error formula by applying the two-stage residual inclusion method to the birthweight model of Mullahy (1997, Review of Economics and Statistics 79: 586–593) using his original data.