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

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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 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 $CI_{\alpha}^{4}$.