On the Lower Confidence Band for the Optimal Welfare

This article addresses the question of reporting a lower confidence band (LCB) for optimal welfare in policy learning problems. A straightforward procedure inverts a one-sided t-test based on an efficient estimator of the optimal welfare. We argue that in an empirically relevant class of data-generating processes, a LCB corresponding to suboptimal welfare may exceed the straightforward LCB, with the average difference of order N-{1/2}. We relate this result to a lack of uniformity in the so-called margin assumption, commonly imposed in policy learning and debiased inference. We advocate for using uniformly valid asymptotic approximations and show how existing methods for inference in moment inequality models can be used to construct valid and tight LCBs for the optimal welfare. We illustrate our findings in the context of the National JTPA study.