Adjustment with Many Regressors Under Covariate-Adaptive Randomizations

Our paper identifies a trade-off when using regression adjustments (RAs) in causal inference under covariate-adaptive randomizations (CARs). On one hand, RAs can improve the efficiency of causal estimators by incorporating information from covariates that are not used in the randomization. On the other hand, RAs can degrade estimation efficiency due to their estimation errors, which are not asymptotically negligible when the number of regressors is of the same order as the sample size. Failure to account for the cost of RAs can result in over-rejection of causal inference under the null hypothesis. To address this issue, we develop a unified inference theory for the regression-adjusted average treatment effect (ATE) estimator under CARs. Our theory has two key features: (1) it ensures the exact asymptotic size under the null hypothesis, regardless of whether the number of covariates is fixed or diverges at most at the rate of the sample size, and (2) it guarantees weak efficiency improvement over the ATE estimator with no adjustments.