On the Power Properties of Confidence Sets for Parameters with Interval Identified Sets

Abstract: 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 to construct the CIs proposed by Stoye (2009), referred to as $CI_{\alpha}^{1}$, $CI_{\alpha}^{2}$, and $CI_{\alpha}^{3}$. We also assume that these estimators are ``ordered’': the lower bound estimator is less than or equal to the upper bound estimator.

Under these conditions, we establish two results. First, we show that $CI_{\alpha}^{1}$ and $CI_{\alpha}^{2}$ are equally powerful, and both dominate $CI_{\alpha}^{3}$. 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 $CI_{\alpha}^{3}$.