Endogenous Heteroskedasticity in Linear Models

Linear regressions with endogeneity are widely used to estimate causal effects. This paper studies a framework that involves two common practical issues: endogeneity of the regressors and heteroskedasticity that depends on endogenous regressors, i.e., endogenous heteroskedasticity. To address the inconsistency of the two-stage least squares estimator in this scenario, and recover the causal parameters of interest, we develop a framework for practical estimation and inference based on the control function approach allowing for discrete and continuous regressors. In particular, we suggest a simple two-step estimation procedure. We establish the limiting properties of the estimator, namely, consistency and asymptotic normality. In addition, we develop practical valid inference methods by proposing an estimator for the asymptotic variance-covariance matrix, and formally establishing its consistency. Monte Carlo simulations provide evidence on the finite-sample performance of the proposed methods and evaluate different implementation strategies. We revisit an empirical application on job training to illustrate the methods.