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MCMC Confidence sets for Identified Sets

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In complicated/nonlinear parametric models, it is hard to determine whether a parameter of interest is formally point identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of parameters in econometric models defined through a likelihood or a vector of moments. The CSs for the identified set or for a function of the identified set (such as a subvector) are based on inverting an optimal sample criterion (such as likelihood or continuously updated GMM), where the cutoff values are computed directly from Markov Chain Monte Carlo (MCMC) simulations of a quasi posterior distribution of the criterion. We establish new Bernstein-von Mises type theorems for the posterior distributions of the quasi-likelihood ratio (QLR) and profile QLR statistics in partially identified models, allowing for singularities. These results imply that the MCMC criterion-based CSs have correct frequentist coverage for the identified set as the sample size increases, and that they coincide with Bayesian credible sets based on inverting a LR statistic for point-identified likelihood models. We also show that our MCMC optimal criterion-based CSs are uniformly valid over a class of data generating processes that include both partially- and point- identified models. We demonstrate good finite sample coverage properties of our proposed methods in four non-trivial simulation experiments: missing data, entry game with correlated payoff shocks, Euler equation and finite mixture models.

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  • Xiaohong Chen & Timothy Christensen & Elie Tamer, 2016. "MCMC Confidence sets for Identified Sets," Cowles Foundation Discussion Papers 2037, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:2037
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    Cited by:

    1. Yu-Wei Hsieh & Xiaoxia Shi & Matthew Shum, 2017. "Inference on Estimators defined by Mathematical Programming," Papers 1709.09115, arXiv.org.
    2. Chen, Xiaohong & Gao, Fuchang, 2017. "A reverse Gaussian correlation inequality by adding cones," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 84-87.
    3. Francis J. DiTraglia & Camilo Garcia-Jimeno, 2020. "A Framework for Eliciting, Incorporating, and Disciplining Identification Beliefs in Linear Models," Papers 2011.07276, arXiv.org.
    4. Jiang, Wenxin, 2017. "On limiting distribution of quasi-posteriors under partial identification," Econometrics and Statistics, Elsevier, vol. 3(C), pages 60-72.
    5. JoonHwan Cho & Thomas M. Russell, 2018. "Simple Inference on Functionals of Set-Identified Parameters Defined by Linear Moments," Papers 1810.03180, arXiv.org, revised Dec 2020.

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