Inference for High-Dimensional Instrumental Variables Regression

Abstract

This thesis concerns statistical inference for the components of a high-dimensional regression parameter despite possible endogeneity of each regressor. Given a first-stage linear model for the endogenous regressors and a second-stage linear model for the response variable, we develop a novel adaptation of the parametric one-step update to a generic second-stage estimator. We provide high-level conditions under which the scaled update is asymptotically normal. We introduce a two-stage Lasso procedure and show that, under a Gaussian noise regime, the second-stage Lasso estimator satisfies the aforementioned conditions. Using these results, we construct asympotitically valid confidence intervals for the components of the second-stage regression vector. We complement our asymptotic theory with empirical studies, which demonstrate the relevance of our method in finite samples.