LASSO Methods for Gaussian Instrumental Variables Models

TL;DR

This paper introduces sparse LASSO-based methods for estimating optimal instruments in high-dimensional Gaussian IV models, demonstrating their theoretical properties and empirical performance.

Contribution

It develops and analyzes LASSO and related methods for IV estimation with many instruments, providing asymptotic theory and practical guidance.

Findings

Sparse IV estimators are asymptotically oracle-efficient under certain conditions.

Data-driven penalty methods perform well in simulations.

Application to real data illustrates practical usefulness.

Abstract

In this note, we propose to use sparse methods (e.g. LASSO, Post-LASSO, sqrt-LASSO, and Post-sqrt-LASSO) to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments in the canonical Gaussian case. The methods apply even when the number of instruments is much larger than the sample size. We derive asymptotic distributions for the resulting IV estimators and provide conditions under which these sparsity-based IV estimators are asymptotically oracle-efficient. In simulation experiments, a sparsity-based IV estimator with a data-driven penalty performs well compared to recently advocated many-instrument-robust procedures. We illustrate the procedure in an empirical example using the Angrist and Krueger (1991) schooling data.