A Unified Theory of Regression Adjustment for Design-based Inference

Under the Neyman causal model, it is well-known that OLS with treatment-by-covariate interactions cannot harm asymptotic precision of estimated treatment effects in completely randomized experiments. But do such guarantees extend to experiments with more complex designs? This paper proposes a general framework for addressing this question and defines a class of generalized regression estimators that are applicable to experiments of any design. The class subsumes common estimators (e.g., OLS). Within that class, two novel estimators are proposed that are applicable to arbitrary designs and asymptotically optimal. The first is composed of three Horvitz-Thompson estimators. The second recursively applies the principle of generalized regression estimation to obtain regression-adjusted regression adjustment. Additionally, variance bounds are derived that are tighter than those existing in the literature for arbitrary designs. Finally, a simulation study illustrates the potential for MSE improvements.