Dogyoon Song : Regression adjustment with high-dimensional covariates

Posted by Kyunghee Han , part of the Statistics and Data Science Seminar.

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Nov. 19, 2025, 4:15 p.m.

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Abstract

Regression adjustment is a classical technique in causal inference that leverages covariates to improve precision of estimators in randomized controlled trials (RCTs) and to adjust for confounding in observational studies. While well-understood in low-dimensional settings, its behavior in modern high-dimensional regimes---where the number of covariates may be comparable to or even exceed the number of observations---remains underexplored. In particular, existing theoretical results are largely asymptotic, often rely on residual-based arguments, and provide limited insights into finite-sample inference especially when $p>n$. In this talk, we revisit regression adjustment for the average treatment effecting (ATE) estimation under complete randomization with many covariates, in a design-based, finite-population framework, via two vignettes. First, we introduce a novel theoretical perspective on the asymptotic properties of regression adjustment through a Neumann-series decomposition, yielding a refined analysis in the $p<n$ regime. Specifically, for ordinary least squares (OLS) regression adjustment, we show that the degree-$d$ Neumann-corrected estimator is asymptotically normal when $p^{d+3}(\log p)^{d+1}=o(n^{d+2})$. This result strictly enlarges the previously reported admissible growth of $p = o(n^{1/2})$ or $p = o(n^{2/3})$ with a single de-biasing step. Second, we present a non-asymptotic analysis of the regression-adjusted ATE estimators that is valid in both $p<n$ and $p>n$ settings. Leveraging concentration of measure tools, we quantify uncertainty without relying on classical asymptotic variance estimation, and further control the design bias of estimators via Stein's method of exchangeable pairs. Time permitting, we will discuss potential extensions and ongoing work.