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In complicated/nonlinear parametric models, it is generally hard to determine whether the model parameters are (globally) 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 via Monte Carlo simulations directly from 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 Monte Carlo 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 Monte Carlo 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. Finally, our proposed procedures are applied in two empirical examples.
Chen, Xiaohong; Christensen, Timothy M.; and Tamer, Elie, "MCMC Confidence Sets for Identified Sets" (2016). Cowles Foundation Discussion Papers. 2483.