  This paper studies covariate adjusted estimation of the average treatment
effect in stratified experiments. We work in a general framework that includes
matched tuples designs, coarse stratification, and complete randomization as
special cases. Regression adjustment with treatment-covariate interactions is
known to weakly improve efficiency for completely randomized designs. By
contrast, we show that for stratified designs such regression estimators are
generically inefficient, potentially even increasing estimator variance
relative to the unadjusted benchmark. Motivated by this result, we derive the
asymptotically optimal linear covariate adjustment for a given stratification.
We construct several feasible estimators that implement this efficient
adjustment in large samples. In the special case of matched pairs, for example,
the regression including treatment, covariates, and pair fixed effects is
asymptotically optimal. We also provide novel asymptotically exact inference
methods that allow researchers to report smaller confidence intervals, fully
reflecting the efficiency gains from both stratification and adjustment.
Simulations and an empirical application demonstrate the value of our proposed
methods.
