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Monte Carlo Confidence Sets for Identified Sets

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  • Xiaohong Chen
  • Timothy M. Christensen
  • Elie Tamer

Abstract

It is generally difficult to know whether the parameters in nonlinear econometric models are point‐identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of the full parameter vector and of subvectors in models defined through a likelihood or a vector of moment equalities or inequalities. The CSs are based on level sets of “optimal” criterion functions (such as likelihoods, optimally‐weighted or continuously‐updated GMM criterions). The level sets are constructed using cutoffs that are computed via Monte Carlo (MC) simulations from the quasi‐posterior distribution of the criterion. We establish new Bernstein–von Mises (or Bayesian Wilks) type theorems for the quasi‐posterior distributions of the quasi‐likelihood ratio (QLR) and profile QLR in partially‐identified models. These results imply that our MC CSs have exact asymptotic frequentist coverage for identified sets of full parameters and of subvectors in partially‐identified regular models, and have valid but potentially conservative coverage in models whose local tangent spaces are convex cones. Further, our MC CSs for identified sets of subvectors are shown to have exact asymptotic coverage in models with singularities. We provide local power properties and uniform validity of our CSs over classes of DGPs that include point‐ and partially‐identified models. Finally, we present two simulation experiments and two empirical examples: an airline entry game and a model of trade flows.

Suggested Citation

  • Xiaohong Chen & Timothy M. Christensen & Elie Tamer, 2018. "Monte Carlo Confidence Sets for Identified Sets," Econometrica, Econometric Society, vol. 86(6), pages 1965-2018, November.
  • Handle: RePEc:wly:emetrp:v:86:y:2018:i:6:p:1965-2018
    DOI: 10.3982/ECTA14525
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    3. Xiaohong Chen & Matthew Gentry & Tong Li & Jingfeng Lu, 2020. "Identification and Inference in First-Price Auctions with Risk Averse Bidders and Selective Entry," Cowles Foundation Discussion Papers 2257, Cowles Foundation for Research in Economics, Yale University.
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    8. Liao, Yuan & Simoni, Anna, 2019. "Bayesian inference for partially identified smooth convex models," Journal of Econometrics, Elsevier, vol. 211(2), pages 338-360.
    9. Andrew Chesher & Adam Rosen, 2018. "Generalized instrumental variable models, methods, and applications," CeMMAP working papers CWP43/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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    11. Jean-Jacques Forneron, 2019. "Detecting Identification Failure in Moment Condition Models," Papers 1907.13093, arXiv.org, revised May 2021.
    12. Gregory Cox, 2020. "Weak Identification with Bounds in a Class of Minimum Distance Models," Papers 2012.11222, arXiv.org.
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    18. Panos Toulis, 2020. "Estimation of COVID-19 Prevalence from Serology Tests: A Partial Identification Approach," Working Papers 2020-54_Revised, Becker Friedman Institute for Research In Economics.

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