🤖 AI Summary
This paper addresses estimation and inference in high-dimensional instrumental variable (IV) regression, where both covariates and instruments have dimension $p$ that may greatly exceed the sample size $n$. For estimating second-stage coefficients, we propose embedding either the BRIDGE or adaptive LASSO as penalty functions within the two-stage least squares (2SLS) framework. Theoretically, we establish, for the first time under sub-Gaussian errors, model selection consistency and oracle efficiency of both methods in high-dimensional IV settings. BRIDGE relaxes distributional assumptions—its consistency holds even without sub-Gaussianity when $p > n$, yielding weaker theoretical conditions. Adaptive LASSO achieves comparable asymptotic properties with superior computational efficiency. Together, the two methods offer complementary advantages: BRIDGE provides enhanced robustness, while adaptive LASSO delivers practical scalability. Our work thus furnishes a theoretically grounded yet implementable solution for sparse high-dimensional IV regression.
📝 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.