Confidence Intervals for 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 proper relaxation 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 of the solution algorithm and the good statistical as well as computational performance of calibrated projection, including in comparison to other methods. Keywords: Partial identification; Inference on projections; Moment inequalities; Uniform inference. * We are grateful to Elie Tamer and three anonymous reviewers for very useful suggestions that substantially improved the paper. We thank for their comments Ivan Canay and seminar and conference participants at Bonn, BC/BU joint workshop, Brown,

Commodity Storage and the Market Effects of Biofuel Policies

“…This interval covers the identified set with probability 0.90, because 90 percent of the estimated parameter sets lie entirely inside it. Imbens and Manski (2004) show that the confidence interval for the identified set is wider than the confidence interval for the true parameter within the set. Heuristically, this result follows from the fact that the true parameter (a single point within the identified set) necessarily covers a narrower range than the identified set (assuming that the set has positive measure).…”

The effect of degree attainment on arrests: Evidence from a randomized social experiment

“…For example, if JC administrators were to take the necessary steps to increase the effect of JC on degree attainment to 0.50, then, ceteris paribus, JC can be expected to reduce the probability of being arrested by at most (0.50)(0.118) = 0.059, or about 17%. 19 All of the point estimates of the effect of education on crime in Section 1 fall within our bounds for the full sample. Remarkably, even though we use a very different source of exogenous variation and methods, our results are very similar to those found using CSLs when comparing 95% confidence intervals (CI) covering the true effect of education on crime.…”

“…Job Corps is a comprehensive education and vocational training program that provides free services to economically disadvantaged youths (aged [16][17][18][19][20][21][22][23][24] in the U.S. Its goals are to educate, train, and prepare adolescents and young adults for employment in purposeful careers, allowing them to live independently. To achieve its goals, JC provides academic and vocational training leading to the attainment of degrees (high school or GED) or certificates in a myriad of trades.…”

“…It is important to keep in mind, however, that each of these estimates is for a different treatment and population. 19 Clearly, just as it is the case in studies estimating a LATE, this type of calculations rely on linear extrapolations. Comparing the results by gender, the estimated lower bound on the percentage-point effect for males is about four times that of females.…”

Set coverage and robust policy

“…When conducting inference on partially identified parameters, Imbens and Manski (2004) pointed out that confidence regions may cover the whole identified set with a prescribed probability, to which we shall refer as set coverage, or they may cover each of its points with a prescribed probability, to which we shall refer as point coverage. Since set coverage implies point coverage, confidence regions satisfying point coverage are generally preferred on the grounds that they may be more informative.…”