Simultaneous Confidence Intervals for High-dimensional Linear Models with Many Endogenous Variables

TL;DR

This paper develops methods for constructing valid simultaneous confidence intervals in high-dimensional linear models with many endogenous variables and instruments, ensuring correct coverage despite high dimensionality.

Contribution

It introduces new estimators and confidence regions that achieve asymptotically correct coverage in high-dimensional endogenous models using orthogonal moment conditions and pivotal procedures.

Findings

Proposes estimators with valid coverage in high-dimensional endogenous models.

Uses a multiplier bootstrap procedure to compute critical values.

Establishes the validity of the bootstrap method for confidence regions.

Abstract

High-dimensional linear models with endogenous variables play an increasingly important role in recent econometric literature. In this work we allow for models with many endogenous variables and many instrument variables to achieve identification. Because of the high-dimensionality in the second stage, constructing honest confidence regions with asymptotically correct coverage is non-trivial. Our main contribution is to propose estimators and confidence regions that would achieve that. The approach relies on moment conditions that have an additional orthogonal property with respect to nuisance parameters. Moreover, estimation of high-dimension nuisance parameters is carried out via new pivotal procedures. In order to achieve simultaneously valid confidence regions we use a multiplier bootstrap procedure to compute critical values and establish its validity.