Simpler Standard Errors for Multi-Stage Regression-Based Estimators: Illustrations in Health Economics

Abstract

With a view towards lessening the analytic and computational burden faced by researchers in empirical health economics who seek an alternative to bootstrapping for the standard errors of two-stage estimators, we offer heretofore unexploited simplifications of the typical, but somewhat daunting, textbook approach. For the most commonly encountered cases in empirical health economics – two-stage estimators that, in either stage, involve maximum likelihood estimation or the nonlinear least squares method – we show that: 1) the usual textbook formulation of the relevant asymptotic covariance can be substantially reduced in complexity; and 2) nearly all components of our simplified formulation can be retrieved as outputs from packaged regression routines (e.g., in Stata). With the applied researcher in mind, we illustrate these points with two examples in empirical health economics that involve the estimation of causal effects in the presence of endogeneity – a sampling problem that can often be solved via two-stage estimation. As a by-product of this illustrative discussion, we detail four very useful two-stage estimators (and their asymptotic standard errors) that are consistent for the model parameters in such settings, along with their corresponding multi-stage causal effect estimators (and their asymptotic standard errors).