The Minimax Estimator of the Average Treatment Effect, among Linear Combinations of Conditional Average Treatment Effects Estimators

I consider the estimation of the average treatment effect (ATE), in a population that can be divided into $G$ groups, and such that one has unbiased and uncorrelated estimators of the conditional average treatment effect (CATE) in each group. These conditions are for instance met in stratified randomized experiments. I first assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known constant $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This optimal estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the CATE's variance to the least precisely estimated CATEs. This optimal estimator is feasible: the weights only depend on known quantities. I then allow for heteroskedasticity and for positive correlations between the estimators. This latter condition is often met in differences-in-differences designs, where the CATEs are estimators of the effect of having been treated for a certain number of time periods. In that case, the optimal estimator is no longer feasible, as it depends on unknown quantities, but a feasible estimator can easily be constructed by replacing those unknown quantities by estimators.