Confidence Intervals for Projections 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 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 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, and we establish its rate of convergence. The algorithm is of independent interest for optimization problems with simple objectives and complicated constraints. An empirical application estimating an entry game illustrates the usefulness of the method. Monte Carlo simulations confirm the accuracy of the solution algorithm, the good statistical as well as computational performance of calibrated projection (including in comparison to other methods), and the algorithm's potential to greatly accelerate computation of other confidence intervals.

Inference of Estimators Defined by Mathematical Programming

“…(2012), among others) to construct joint confidence sets for (θ, s, λ) and then obtain the marginal confidence set for θ as projection of the joint confidence sets. For methods that focuses on marginal confidence sets for θ (which usually yield tighter inference than the simple projection method above), one could use more elaborate methods like Kaido et al (2016) and Bugni, Canay, and Shi (2017) in case of moment inequality models, and Kim (2017) in other cases. , 3), (σ 2 1 , σ 2 2 ) = (1, 1), σ12 = 0 c µ = (5, 3), (σ 2 1 , σ 2 2 ) = (3, 1), σ12 = 0 d µ = (5, 3), (σ 2 1 , σ 2 2 ) = (3, 1), σ12 = 1.5 Note: The solution of portfolio selection (12) based on the estimated (R,Q) is located by two red lines.…”

A Random Attention Model

“…If the null hypothesis is rejected, then certain features shared by the collection of preferences is incompatible with the underlying decision theory (up to Type I error). See Bugni, Canay, and Shi (2015), Kaido, Molinari, and Stoye (2019) and references therein for further discussion of this idea and related methods.…”

Inference for Linear Conditional Moment Inequalities

“…In our setting one can test H 0 : µ n ∈ M 0 by first using one of the aforementioned approaches to test H 0 (δ) as defined in (9) for all δ and then applying the projection method. This yields a conservative test, but Kaido et al (2019a) show how to eliminate this conservativeness when considering projections based on D. Andrews & Soares (2010). Unfortunately, however, projection tests based on moment-selection procedures break the linear structure discussed in the last section.…”

“…Finally, there is a large literature on techniques which seek to reduce sensitivity to the inclusion of slack moments in settings without nuisance parameters, including D. Andrews & Soares (2010), D. Andrews & Barwick (2012), Romano et al (2014a), and Cox & Shi (2019). , Bugni et al (2017), Belloni et al (2018), and Kaido et al (2019a) build on related ideas to reduce sensitivity to slack moments in models with nuisance parameters. If applied in our setting, however, these techniques would eliminate the linear structure which simplifies computation.…”

“…The main text inRomano et al (2014a) uses bootstrap critical values, but the appendix,Romano et al (2014b), develops results for the normal model.8 Conditional variances were previously considered by e.g Chetverikov (2018). for inference with conditional moment inequalities, and byKaido et al (2019b) andBarseghyan et al (2019) for settings with a discrete instrument. We discuss estimation of Σ in Section 6 below.9 Though the diagonal terms in Σ are smaller than those in Ω(δ), and this will lead to larger values of the the test statistics introduced below, their off diagonal correlations also differ, which can generate larger critical values.…”