ON THE POWER PROPERTIES OF INFERENCE FOR PARAMETERS WITH INTERVAL IDENTIFIED SETS

This article studies the power properties of confidence intervals (CIs) for a partially-identified parameter of interest with an interval identified set. We assume that the researcher has bounds estimators needed to construct the CIs proposed by Imbens and Manski (2004), Stoye (2009), and Stoye (2020), denoted by

,

C I α 2 $CI_{\alpha }^{2}$ upper C upper I Subscript alpha Superscript 2

,

C I α 3 $CI_{\alpha }^{3}$ upper C upper I Subscript alpha Superscript 3

, and

C I α 4 $CI_{\alpha }^{4}$ upper C upper I Subscript alpha Superscript 4

. We also assume that 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

C I α 1 $CI_{\alpha }^{1}$ upper C upper I Subscript alpha Superscript 1

and

C I α 2 $CI_{\alpha }^{2}$ upper C upper I Subscript alpha Superscript 2

are equally powerful, and both dominate

C I α 3 $CI_{\alpha }^{3}$ upper C upper I Subscript alpha Superscript 3

and

C I α 4 $CI_{\alpha }^{4}$ upper C upper I Subscript alpha Superscript 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

C I α 1 $CI_{\alpha }^{1}$ upper C upper I Subscript alpha Superscript 1

and

C I α 2 $CI_{\alpha }^{2}$ upper C upper I Subscript alpha Superscript 2

, but not necessarily for

C I α 3 $CI_{\alpha }^{3}$ upper C upper I Subscript alpha Superscript 3

or

C I α 4 $CI_{\alpha }^{4}$ upper C upper I Subscript alpha Superscript 4

. In summary, within the class of models considered,

C I α 1 $CI_{\alpha }^{1}$ upper C upper I Subscript alpha Superscript 1

and

C I α 2 $CI_{\alpha }^{2}$ upper C upper I Subscript alpha Superscript 2

have identical power properties, and both compare favorably to

C I α 3 $CI_{\alpha }^{3}$ upper C upper I Subscript alpha Superscript 3

and

C I α 4 $CI_{\alpha }^{4}$ upper C upper I Subscript alpha Superscript 4

.