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In complicated/nonlinear parametric models, it is generally hard to know whether the model parameters are point identiﬁed. We provide computationally attractive procedures to construct conﬁdence sets (CSs) for identiﬁed sets of full parameters and of subvectors in models deﬁned through a likelihood or a vector of moment equalities or inequalities. These CSs are based on level sets of optimal sample criterion functions (such as likelihood or optimally-weighted or continuously-updated GMM criterions). The level sets are constructed using cutoﬀs that are computed via Monte Carlo (MC) simulations directly from the quasi-posterior distributions of the criterions. 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 regular models and some non-regular 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 with reduced-form parameters on the boundary. Our MC CSs for identiﬁed sets of subvectors are shown to have exact asymptotic coverage in models with singularities. We also provide results on uniform validity of our CSs over classes of DGPs that include point and partially identiﬁed models. We demonstrate good ﬁnite-sample coverage properties of our procedures in two simulation experiments. Finally, our procedures are applied to two non-trivial empirical examples: an airline entry game and a model of trade flows.
Chen, Xiaohong; Christensen, Timothy M.; and Tamer, Elie, "Monte Carlo Confidence Sets for Identified Sets" (2016). Cowles Foundation Discussion Papers. 2484.