Confidence Intervals forProjections of Partially Identified Parameters

Abstract

We propose a bootstrap-based calibrated projection procedure to build confidence intervals for single components and for smooth functions of a partially identified parameter vector in moment (in)equality models. The method controls asymptotic coverage uniformly over a large class of data generating processes.The extreme points of the calibrated projection confidence interval are obtained by extremizing the value of the component (or function) of interest subject to a properrelaxation of studentized sample analogs of the moment (in)equality conditions. The degree of relaxation, or critical level, is calibrated so that the component (or function)of θ, not θ itself, is uniformly asymptotically covered with prespecified probability. This calibration is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive.Nonetheless, the program defining an extreme point of the confidence interval is generally nonlinear and potentially intricate. We provide an algorithm, based on the response surface method for global optimization, that approximates the solution rapidly and accurately. The algorithm is of independent interest for inference on optimal values of stochastic nonlinear programs. We establish its convergence under conditions satisfied by canonical examples in the moment (in) equalities literature.Our assumptions and those used in the leading alternative approach (a profiling based method) are not nested. An extensive Monte Carlo analysis confirms the accuracy ofthe solution algorithm and the good statistical as well as computational performance of calibrated projection, including in comparison to other methods.