Computing Correct Standard Errors for Multi-Stage Regression-Based Estimators

With a view towards lessening the analytic and computational burden faced by practitioners seeking to correct the standard errors of two-stage estimators, we offer a heretofore unnoticed simplification of the conventional formulation for the most commonly encountered cases in empirical application – two-stage estimators involving maximum likelihood estimation or nonlinear least squares in either stage. Also with the applied researcher in mind, we cast the discussion in the context of nonlinear regression models involving endogeneity – a sampling problem whose solution often requires two-stage estimation. We detail our simplified standard error formulations for three very useful estimators in applied contexts involving endogeneity in a nonlinear setting (endogenous regressors, endogenous sample selection, and causal effects). The analytics and Stata/Mata code for implementing our simplified formulae are demonstrated with illustrative real-world examples and simulated data.