MCMC Confidence Sets for Identified Sets
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.
Chen, Xiaohong; Christensen, Timothy M.; and Tamer, Elie, "MCMC Confidence Sets for Identified Sets" (2016). Cowles Foundation Discussion Papers. 2482.