Distributionally Robust Instrumental Variables Estimation

Instrumental variables (IV) estimation is a fundamental method in econometrics and statistics for estimating causal effects in the presence of unobserved confounding. However, challenges such as untestable model assumptions and poor finite sample properties have undermined its reliability in practice. Viewing common issues in IV estimation as distributional uncertainties, we propose DRIVE, a distributionally robust framework of the classical IV estimation method. When the ambiguity set is based on a Wasserstein distance, DRIVE minimizes a square root ridge regularized variant of the two stage least squares (TSLS) objective. We develop a novel asymptotic theory for this regularized regression estimator based on the square root ridge, showing that it achieves consistency without requiring the regularization parameter to vanish. This result follows from a fundamental property of the square root ridge, which we call ``delayed shrinkage''. This novel property, which also holds for a class of generalized method of moments (GMM) estimators, ensures that the estimator is robust to distributional uncertainties that persist in large samples. We further derive the asymptotic distribution of Wasserstein DRIVE and propose data-driven procedures to select the regularization parameter based on theoretical results. Simulation studies confirm the superior finite sample performance of Wasserstein DRIVE. Thanks to its regularization and robustness properties, Wasserstein DRIVE could be preferable in practice, particularly when the practitioner is uncertain about model assumptions or distributional shifts in data.