The dissertation consists of three papers in Bayesian econometrics. In Chapter 1, "Asymptotic Properties of Bayesian Inference in Linear Regression with a Structural Break", we revisit the threshold regression models, an important class of models in economic analysis. For example, multiple equilibria can give rise to threshold effects. The challenge is to conduct a valid inference of the regression slope parameters when the threshold parameter needs to be estimated. The non-standard aspect of the estimation problem motivates the use of Bayesian methods, which can correctly reflect the finite-sample uncertainty of estimating the threshold upon inference of the regression coefficients. Our theoretical contribution is to establish a Bernstein-von Mises type theorem (Bayesian asymptotic normality) for the regression coefficients under a wide class of priors for the parameters, which essentially indicates an asymptotic equivalence between the conventional frequentist and the Bayesian inference. Our result is beneficial to both Bayesians and frequentists. A Bayesian user can invoke our theorem to convey his or her statistical result to the frequentist researchers. For a frequentist researcher, looking at the credible interval can serve as a robustness check for the finite sample uncertainty coming from the threshold estimation, and such sensitivity analysis is natural as our result guarantees the credible interval to converge to the frequentist confidence interval. The simulation studies show that the conventional confidence intervals tend to under-cover while credible intervals offer reasonable coverages in general. As sample size increases, both methods coincide, as predicted from our theoretical conclusion. Using the data from Durlauf and Johnson (1995) on economic growth, we illustrate that the traditional confidence intervals on might under-represent the true sampling uncertainty. Chapter 2, "Semiparametric Bayesian Estimation of Dynamic Discrete Choice Models" is a joint paper with Andriy Norets. We propose a tractable estimation method for dynamic discrete choice models where we relax distributional assumptions on the additive unobserved shocks in the per-period utility functions. The distribution of these shocks is modeled by mixtures of extreme value distributions. Our approach exploits the analytical tractability of extreme value distributions and the flexibility of location-scale mixtures. We consider the Bayesian approach to inference and implement it using Hamiltonian Monte Carlo for simulation from the posterior. In a binary choice model, our approach delivers estimation results that are consistent with the previous literature. As an application to multinomial choice problems, for which previous literature does not provide tractable estimation methods in general settings, we apply our method to Gilleskie (1998) model of medicare use and work absences. We also provide results on continuity of CCPs in the distribution of the unobserved shocks as well as existence of positive prior of any neighborhood of the true CCPs by location-scale mixtures. In addition, we obtain approximation of twice differentiable densities by location-scale mixtures of extreme value distributions. In future work, these results will be used for analysis of frequentist properties of the posterior distribution such as posterior concentration on the identified sets for model parameters. Chapter 3, "Estimating Dynamic Panel Data Models under Permanent Unobserved Heterogeneity when Number of Clusters is Unknown" applies the reversible jump MCMC type algorithm of Norets (2020) to a class of nonlinear dynamic panel model. In conventional approaches to estimate non-linear panel data models with permanent unobserved heterogeneity, the number of types or clusters M is often pre-specified or fixed a posteriori by the researcher. On the other hand, in Bayesian estimation based on a class of reversible jump MCMC methods, M is treated as a parameter to be estimated jointly with other structural parameters. The resulting inference and predictions can naturally incorporate the uncertainty involving the cluster structure. However, it is known that the proposal choice for the trans-dimensional move during MCMC can be difficult for complex econometric models. In a recent advance in the literature, Norets (2020) proposes an algorithm that does not require the researcher to choose a specific proposal and establish its theoretical optimality. In this paper, we apply the algorithm to a class of dynamic panel data models with heterogeneous coefficients. Through simulations, we show that this approach can flexibly estimate a wide types of unobserved heterogeneity and can deliver robust inference and counterfactual predictions.
