This paper extends Imbens and Manski's (2004) analysis of confidence intervals for interval identified parameters. The extension is motivated by the discovery that for their final result, Imbens and Manski implicitly assumed locally superefficient estimation of a nuisance parameter.
I reanalyze the problem both with assumptions that merely weaken this superefficiency condition and with assumptions that remove it altogether. Imbens and Manski's confidence region is valid under weaker assumptions than theirs, yet superefficiency is required. I also provide a confidence interval that is valid under superefficiency, but can be adapted to the general case. A methodological contribution is to observe that the difficulty of inference comes from a preestimation problem regarding a nuisance parameter, clarifying the connection to other work on partial identification.