Sparse Linear Models and L1-Regularized 2SLS with High-Dimensional Endogenous Regressors and Instruments

Abstract

We explore the validity of the 2-stage least squares estimator with L1-regularization in both stages, for linear models where the numbers of endogenous regressors in the main equation and instruments in the first-stage equations can exceed the sample size, and the regression coefficients are exactly or approximately sparse. We establish finite-sample performance bounds with conditions for consistency, and provide a simple practical procedure for choosing the regularization parameters with theoretical guarantees. We also illustrate a uniform inference strategy built upon the L1-regularized 2-stage least squares estimator.