Regression Adjustment, Cross-Fitting, and Randomized Experiments with Many Controls

This paper studies estimation and inference for average treatment effects in randomized experiments with many covariates, under a design-based framework with a deterministic number of treated units. We show that a simple yet powerful cross-fitted regression adjustment achieves bias-correction and leads to sharper asymptotic properties than existing alternatives. Specifically, we derive higher-order stochastic expansions, analyze associated inference procedures, and propose a modified HC3 variance estimator that accounts for up to second-order. Our analysis reveals that cross-fitting permits substantially faster growth in the covariate dimension $p$ relative to sample size $n$, with asymptotic normality holding under favorable designs when $p = o(n^{3/4}/(\log n)^{1/2})$, improving on standard rates. We also explain and address the poor size performance of conventional variance estimators. The methodology extends naturally to stratified experiments with many strata. Simulations confirm that the cross-fitted estimator, combined with the modified HC3, delivers accurate estimation and reliable inference across diverse designs.