Inference for functions of partially identified parameters in moment inequality models

Bugni, Federico A.; Canay, Ivan A.; Shi, Xun

doi:10.1920/wp.cem.2015.5415

Cited by 10 publications

(13 citation statements)

References 31 publications

(15 reference statements)

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“…This implies that the setM n (θ n ) can also be treated as non-stochastic, whereM n is the set in equation (B.120) withK n replacing K Pn . The result then follows because by Lemma D.2.8 in Bugni, Canay, and Shi (2015), G Lemma B.7: Let (P n , θ n ) have the almost sure representations given in Lemma B.1, let J * be defined as in (B.37), and assume that J * = ∅. Let C collect all size d subsets C of {1, ..., J + 2d + 2} ordered lexicographically by their smallest, then second smallest, etc.…”

Section: Appendix B Main Lemmasmentioning

confidence: 74%

“…Under this assumption, the class of normalized moment functions is uniformly Donsker (Bugni, Canay, and Shi, 2015). This allows us to show that the first-order linear approximation to s(p, C n (ĉ n )) is valid and further establish the validity of our bootstrap procedure.…”

Section: Assumptionsmentioning

confidence: 75%

“…Computation of the extreme points of the confidence intervals is also computationally attractive thanks to an application, novel to this paper, of the response surface method for global optimization that can be of independent interest in the partial identification literature. The class of DGPs over which we can establish validity of our procedure is non-nested with the class over which the main alternative to our method (Romano and Shaikh, 2008;Bugni, Canay, and Shi, 2014) is asymptotically valid. For example, our method yields asymptotically uniformly valid confidence intervals for linear functions of best linear predictor parameters in models with interval valued outcomes and discrete covariates, while profiling based methods do not.…”

Section: Discussionmentioning

confidence: 99%

“…See Stoye (2009) and especially AS for further discussion and Bugni, Canay, and Shi (2015) for a paper that focuses on this interpretation and improves onĉ AS n for the purpose of specification testing. We leave a detailed analysis of our implied specification test to future research.…”

Section: Discussionmentioning

confidence: 99%

“…where the last equality follows from sup θ∈Θ |G n (θ)| = O P (1) due to asymptotic tightness of {G n } (uniformly in P ) by Lemma D.1 in Bugni, Canay, and Shi (2015), Theorem 3.6.1 and Lemma 1.3.8 in van der Vaart and Wellner (2000), and sup θ∈Θ |η n,j (θ)| = o P (κ n / √ n) by Assumption 3.3 (ii) respectively 3.3 (ii).…”

Section: P( Sup λ∈ρBmentioning

confidence: 99%

See 4 more Smart Citations

Confidence intervals for projections of partially identified parameters

Kaido¹,

Molinari²,

Stoye³

2016

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This paper proposes a bootstrap-based procedure to build confidence intervals for single components of a partially identified parameter vector, and for smooth functions of such components, in moment (in)equality models. The extreme points of our confidence 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 θ, instead of θ itself, is uniformly asymptotically covered with prespecified 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 confidence 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 finite sample) than the one used if projecting confidence regions designed to cover the entire parameter vector θ. Hence, our confidence interval is weakly shorter than the projection of established confidence 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 profiling 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 profiling based methods are not.

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Section: Appendix B Main Lemmasmentioning

confidence: 74%

Section: Assumptionsmentioning

confidence: 75%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: P( Sup λ∈ρBmentioning

confidence: 99%

See 3 more Smart Citations

Confidence intervals for projections of partially identified parameters

Kaido¹,

Molinari²,

Stoye³

2016

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Improving confidence set estimation when parameters are weakly identified

Battey

Feng

Smith

2016

Statistics & Probability Letters

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Constrained conditional moment restriction models

Chernozhukov¹,

Newey²,

Santos³

2015

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This paper examines a general class of inferential problems in semiparametric and nonparametric models defined by conditional moment restrictions. We construct tests for the hypothesis that at least one element of the identified set satisfies a conjectured (Banach space) "equality" and/or (a Banach lattice) "inequality" constraint. Our procedure is applicable to identified and partially identified models, and is shown to control the level, and under some conditions the size, asymptotically uniformly in an appropriate class of distributions. The critical values are obtained by building a strong approximation to the statistic and then bootstrapping a (conservatively) relaxed form of the statistic. Sufficient conditions are provided, including strong approximations using Koltchinskii's coupling.Leading important special cases encompassed by the framework we study include: (i) Tests of shape restrictions for infinite dimensional parameters; (ii) Confidence regions for functionals that impose shape restrictions on the underlying parameter; (iii) Inference for functionals in semiparametric and nonparametric models defined by conditional moment (in)equalities; and (iv) Uniform inference in possibly nonlinear and severely ill-posed problems.

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Inference for functions of partially identified parameters in moment inequality models

Cited by 10 publications

References 31 publications

Confidence intervals for projections of partially identified parameters

Confidence intervals for projections of partially identified parameters

Improving confidence set estimation when parameters are weakly identified

Constrained conditional moment restriction models

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