Endogeneity in high dimensions

TL;DR

This paper addresses the challenge of endogeneity in high-dimensional regression, proposing a novel FGMM method that achieves consistency and efficiency even with endogenous regressors.

Contribution

It introduces a focused GMM approach for high-dimensional settings, ensuring model selection consistency and oracle properties despite endogeneity.

Findings

FGMM achieves oracle property with endogenous regressors

Proves penalized regression methods can be inconsistent due to endogeneity

Two-step approach attains semi-parametric efficiency

Abstract

Most papers on high-dimensional statistics are based on the assumption that none of the regressors are correlated with the regression error, namely, they are exogenous. Yet, endogeneity can arise incidentally from a large pool of regressors in a high-dimensional regression. This causes the inconsistency of the penalized least-squares method and possible false scientific discoveries. A necessary condition for model selection consistency of a general class of penalized regression methods is given, which allows us to prove formally the inconsistency claim. To cope with the incidental endogeneity, we construct a novel penalized focused generalized method of moments (FGMM) criterion function. The FGMM effectively achieves the dimension reduction and applies the instrumental variable methods. We show that it possesses the oracle property even in the presence of endogenous predictors, and that the solution is also near global minimum under the over-identification assumption. Finally, we also show how the semi-parametric efficiency of estimation can be achieved via a two-step approach.