  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.
