Lasso Methods for Gaussian Instrumental Variables Models

In this note, we propose to use sparse methods (e.g. LASSO, Post-LASSO, √ LASSO, and Post√ LASSO) to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p, in the canonical Gaussian case. The methods apply even when p is much larger than the sample size, n. 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.