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

TL;DR

This paper analyzes the power properties of confidence intervals for partially-identified parameters with interval identified sets, comparing different estimators and their efficiency in producing powerful inference.

Contribution

It establishes the equivalence in power between CI1 and CI2, shows they outperform CI3, and demonstrates when more efficient bounds estimators lead to more powerful CIs.

Findings

CI1 and CI2 are equally powerful and dominate CI3.

More efficient bounds estimators improve the power of CI1 and CI2.

Efficiency gains do not necessarily translate to CI3.

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 CI1, CI2, and CI3. 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 CI1 and CI2 are equally powerful, and both dominate CI3. 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 CI1 and CI2, but not necessarily for CI3.