This paper develops a method for inference in dynamic discrete choice models with serially correlated unobserved state variables. Estimation of these models involves computing high-dimensional integrals that are present in the solution to the dynamic program and in the likelihood function. First, the paper proposes a Bayesian Markov chain Monte Carlo estimation procedure that can handle the problem of multidimensional integration in the likelihood function. Second, the paper presents an efficient algorithm for solving the dynamic program suitable for use in conjunction with the proposed estimation procedure.

We propose a framework for making Bayesian parametric models robust to local misspecification. Suppose in a baseline parametric model, a parameter of interest has an interpretation in an encompassing semiparametric model. Bayesian and maximum likelihood estimators are generally biased under local misspecification. We propose to augment the baseline likelihood by a multiplicative factor that involves scores for the baseline model, the efficient scores for the encompassing semiparametric model, and an auxiliary parameter that has the same dimension as the parameter of interest. We show that the marginal posterior for the parameter of interest in the augmented model is asymptotically normal with mean equal to the semiparametrically efficient estimator and variance equal to the semiparametric efficiency bound. The suggested augmentation robustifies the baseline parametric model to local misspecification, while preserving the appeal of Bayesian inference. We develop an MCMC algorithm for the augmented model and illustrate the approach in applications.

Confidence intervals are commonly used to describe parameter uncertainty. In nonstandard problems, however, their frequentist coverage property does not guarantee that they do so in a reasonable fashion. For instance, confidence intervals may be empty or extremely short with positive probability, even if they are based on inverting powerful tests. We apply a betting framework to formalize the “reasonableness ” of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the degree of unreasonableness of previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are reasonable by construction. We apply our framework to inference about a parameter near a boundary and a local-to-unity autoregressive root. We find that previously suggested confidence intervals are not reasonable, and numerically determine alternative confidence sets that satisfy our criteria.

We consider nonparametric estimation of a mixed discrete‐continuous distribution under anisotropic smoothness conditions and a possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for (up to a log factor) optimal adaptive estimation of mixed discrete‐continuous distributions. The proposed model demonstrates excellent performance in simulations mimicking the first stage in the estimation of structural discrete choice models.