On the Lower Confidence Band for the Optimal Welfare in Policy Learning

TL;DR

This paper develops methods for constructing reliable lower confidence bands for optimal welfare in policy learning, especially when standard methods fail due to the treatment policy's proximity to others, with empirical validation on a real study.

Contribution

It introduces a novel approach for constructing lower confidence bands using moment-inequality tests, applicable even when standard debiased methods are invalid.

Findings

Proposed LCBs achieve reliable coverage in empirical tests.

Standard methods can be dominated by alternative LCBs under certain conditions.

The approach is applicable when the optimal policy is not well-separated from others.

Abstract

We study inference on the optimal welfare in a policy learning problem and propose reporting a lower confidence band (LCB). A natural approach to constructing an LCB is to invert a one-sided t-test based on an efficient estimator for the optimal welfare. However, we show that for an empirically relevant class of DGPs, such an LCB can be first-order dominated by an LCB based on a welfare estimate for a suitable suboptimal treatment policy. We show that such first-order dominance is possible if and only if the optimal treatment policy is not ``well-separated'' from the rest, in the sense of the commonly imposed margin condition. When this condition fails, standard debiased inference methods are not applicable. We show that uniformly valid and easy-to-compute LCBs can be constructed analytically by inverting moment-inequality tests with the maximum and quasi-likelihood-ratio test statistics. As an empirical illustration, we revisit the National JTPA study and find that the proposed LCBs achieve reliable coverage and competitive length.