High-dimensional Penalized Linear IV Estimation & Inference using BRIDGE and Adaptive LASSO

TL;DR

This paper demonstrates how BRIDGE and adaptive LASSO can be effectively used for high-dimensional linear IV estimation, ensuring model selection consistency and oracle efficiency even when parameters exceed sample size.

Contribution

It extends high-dimensional IV estimation methods to broader error distributions and compares the computational and theoretical advantages of BRIDGE and adaptive LASSO.

Findings

Both methods ensure model selection consistency and oracle efficiency.

BRIDGE works under weaker assumptions when p > n.

Adaptive LASSO is computationally faster.

Abstract

This paper is an exposition of how BRIDGE and adaptive LASSO can be used in a two-stage least squares problem, to estimate the second-stage coefficients when the number of parameters p in both stages is growing with the sample size n. Facing a larger class of problems compared to the usual analysis in the literature, i.e., replacing the assumption of normal with sub-Gaussian errors, I prove that both methods ensure model selection consistency and oracle efficiency even when the number of instruments and covariates exceeds the sample size. For BRIDGE, I also prove that if the former is growing but slower than the latter, the same properties hold even without sub-Gaussian errors. When p is greater than n, BRIDGE requires a slightly weaker set of assumptions to have the desirable properties, as adaptive LASSO requires a good initial estimator of the relevant weights. However, adaptive LASSO is expected to be much faster computationally, so the methods are competitive on different fronts and the one that is recommended depends on the researcher's resources.