Confidence intervals for projections of partially identified parameters

This paper proposes a bootstrap-based procedure to build confi dence intervals for single components of a partially identifi ed parameter vector, and for smooth functions of such components, in moment (in)equality models. The extreme points of our confi dence interval are obtained by maximizing/minimizing the value of the component (or function) of interest subject to the sample analog of the moment (in)equality conditions properly relaxed. The novelty is that the amount of relaxation, or critical level, is computed so that the component (or function) of ?, instead of ? itself, is uniformly asymptotically covered with prespeci ed probability. Calibration of the critical level is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive. Computation of the extreme points of the con fidence interval is based on a novel application of the response surface method for global optimization, which may prove of independent interest also for applications of other methods of inference in the moment (in)equalities literature. The critical level is by construction smaller (in fi nite sample) than the one used if projecting con fience regions designed to cover the entire parameter vector. Hence, our con fidence interval is weakly shorter than the projection of established con fidence sets (Andrews and Soares, 2010), if one holds the choice of tuning parameters constant. We provide simple conditions under which the comparison is strict. Our inference method controls asymptotic coverage uniformly over a large class of data-generating processes. Our assumptions and those used in the leading alternative approach (a profi ling-based method) are not nested. We explain why we employ some restrictions that are not required by other methods and provide examples of models for which our method is uniformly valid but profi ling-based methods are not.