Confidence Intervals for Partially Identified Parameters

• Published in: Econometrica 2004-11, Vol.72 (6), p.1845-1857
• Main Authors: Imbens, Guido W., Manski, Charles F.
• Format: Article
• Language: English
• Subjects: Applications; Approximation; Bounds; Case studies; Confidence interval; Confidence intervals; Consistent estimators; Convergence; Econometrics; Economic theory; Estimation; Estimators; Exact sciences and technology; identification regions; Inference; Insurance, economics, finance; Interval estimators; Mathematics; Missing data; Nonparametric inference; Notes and Comments; Observational research; Parameter identification; Parametric inference; Probability; Probability and statistics; Random variables; Sciences and techniques of general use; Statistical variance; Statistics; Studies; uniform convergence
• ISSN: 0012-9682; 1468-0262
• DOI: 10.1111/j.1468-0262.2004.00555.x

Recently a growing body of research has studied inference in settings where parameters of interest are partially identified. In many cases the parameter is real-valued and the identification region is an interval whose lower and upper bounds may be estimated from sample data. For this case confidence intervals (CIs) have been proposed that cover the entire identification region with fixed probability. Here, we introduce a conceptually different type of confidence interval. Rather than cover the entire identification region with fixed probability, we propose CIs that asymptotically cover the true value of the parameter with this probability. However, the exact coverage probabilities of the simplest version of our new CIs do not converge to their nominal values uniformly across different values for the width of the identification region. To avoid the problems associated with this, we modify the proposed CI to ensure that its exact coverage probabilities do converge uniformly to their nominal values. We motivate this modified CI through exact results for the Gaussian case.