On the power properties of inference for parameters with interval identified sets

This paper studies the power properties of confidence intervals (CIs) for a partially-identified parameter of interest with an interval identified set. We assume the researcher has bounds estimators needed to construct the CIs proposed by Imbens and Manski (2004), Stoye (2009), and Stoye (2020), denoted by CI_alpha^1, CI_alpha^2, CI_alpha^3, and CI_alpha^4. We also assume these bounds estimators are ``ordered'': the lower bound estimator is less than or equal to the upper bound estimator. This setup arises in economic applications involving missing data and treatment effects.
Under these conditions, we establish two results. First, we show that CI_alpha^1 and CI_alpha^2 are equally powerful, and both dominate CI_alpha^3 and CI_alpha^4. Second, we consider a favorable situation in which there are two possible bounds estimators to construct these CIs, and one is more efficient than the other. One would expect that the more efficient bounds estimator yields more powerful inference. We prove that this desirable result holds for CI_alpha^1 and CI_alpha^2, but not necessarily for CI_alpha^3 or CI_alpha^4. In summary, within the class of models considered, CI_alpha^1 and CI_alpha^2 have identical power properties, and both compare favorably to CI_alpha^3 or CI_alpha^4.