Confidence intervals for partially identified parameters

In the last decade a growing body of research has studied inference on partially identified parameters (e.g., Manski, 1990, 2003). In many cases where the parameter of interest is realvalued, the identification region is an interval whose lower and upper bounds may be estimated from sample data. Confidence intervals may be constructed to take account of the sampling variation in estimates of these bounds. Horowitz and Manski (1998, 2000) proposed and applied interval estimates that asymptotically cover the entire identification region with fixed probability. Here we introduce conceptually different interval estimates that asymptotically cover each element in the identification region with fixed probability (but not necessarily every element simultaneously). We show that these two types of interval estimate are different in practice, the latter in general being shorter. The difference in length (in excess of the length of the identification set itself) can be substantial, and in large samples is comparable to the difference of one - and two - sided confidence intervals. A complication arises from the fact that the simplest version of the proposed interval is discontinuous in the limit case of point identification, leading to coverage rates that are not uniform in important subsets of the parameter space. We develop a modification depending on the width of the identification region that restores uniformity. We show that under some conditions, using the estimated width of the identification region instead of the true width maintains uniformity.