Distributionally Robust Treatment Effect

Using only retrospective data, we propose an estimator for predicting the treatment effect for the same treatment/policy to be implemented in another location or time period, which requires no input from the target population. More specifically, we minimize the worst-case mean square error for the prediction of treatment effect within a class of distributions inside the Wasserstein ball centered on the source distribution. Since the joint distribution of potential outcomes is not identified, we pick the best and worst copulas of the marginal distributions of two potential outcomes as our optimistic and pessimistic optimization objects for partial identification. As a result, we can attain the upper and lower bounds of the minimax optimizer. The minimax solution differs depending on whether treatment effects are homogeneous or heterogeneous. We derive the consistency and asymptotic distribution of the bound estimators, provide a two-step inference procedure, and discuss the choice of the Wasserstein ball radius.