Bayesian analysis in partially identified parametric and non-parametric models

This dissertation studies a general type of econometrc model characterized by moment conditions. Such a model, with different variations, has many important empirical applications in economics, biostatistics, and finance. The variations of the model have two dimensions: one is on the type of the moment conditions: either moment equality more inequality; the other is on the dimension of the structural parameter: either finite or infinite. As a result, the model contains most of the important econometric models. The key feature of the model that I am interested in is that the parameter is not completely identified. With limited knowledge of the underlying data distribution, it is only partially identified. I proceed with a Bayesian approach in this dissertation.;Chapter 1. This chapter introduces the model and corresponding Bayesian methods in the literature, followed by detailed examples of the models to be considered in this dissertation. I present in detail some closely related recent literature, from both frequentist and Bayesian perspectives.;Chapter 2. I study a type of moment condition that has been rapidly studied by econometricians in recent years: moment inequalities. Since the parameter of interest is allowed to be not point identified, the treatment is very flexible in dealing with incomplete data, such as missing data or censored data. I construct the posterior distribution of the structural parameter, and establish its large sample behaviors. Since in many applications, it is more straightforward to specify the moment inequalities than the distribution of the data generating process, hence instead of the true likelihood, the posterior density is derived based on the limited information likelihood, a moment condition based likelihood. It is shown that the posterior converges to zero exponentially fast outside any small neighborhood of the identified region. Inside the identified region, it is bounded below by a rate that is not exponentially small. The simulations provide evidence that the Bayesian approach has very attractive properties, in the sense that, with a proper choice of the prior, the posterior provides extra information about the true parameter inside the identified region.;Chapter 3. There exists a moment and model selection problem in the moment inequality model. Here only a subset of the moment inequalities are to be used and the true parameter vector is assumed to follow a submodel allowing only some selected components to be nonzero (which can be, e.g., the regression coefficients of some selected explanatory variables). The moment inequalities are called compatible if fixing the dimension of the parameter vector and the parameter space, the identified region defined by these moment inequalities is not empty. I derive the posterior distribution of the moment inequality/parameter subspace combination, and show that the incompatible combinations have exponentially small posteriors. While the posteriors of compatible combinations are positive, they are sensitive to the researchers' a priori information of the model, which is the choice of the priors.;Chapter 4. This chapter addresses the estimation of the semi-nonparametric conditional moment restricted model that involves a nonparametric structural function g0. The posterior distribution of the parameter of interest is derived based on the limited information likelihood. I focus on the frequentist properties of the posterior distribution, allowing the nonparametric structural function to be partially identified. It is shown that the posterior converges to any small neighborhood of the identified region. I then apply the results to the single index model and the nonparametric instrumental regression model. In particular, the compactness assumption on the parameter space for nonparametric instrumental regression is relaxed, and a regularized prior is used to overcome the ill-posedness.;Chapter 5. I consider a Bayesian approach to making joint probabilistic inference on the action and the associated risk in data mining. The posterior probability is based on an empirical likelihood, which imposes a moment restriction relating the action to the resulting risk, but does not otherwise require a probability model for the underlying data generating process. The moment restriction partially identifies the parameters of interest, which include both the theoretical risk of interest and the parameters describing the associated actions. I illustrate with examples how this framework can be used to describe the posterior probability of actions to take in order to achieve a low risk, or conversely, to describe the posterior distribution of the resulting risk for a given action. The posterior distribution will cluster around the true risk-action relation with high probability for large data size, and that the actions can be generated from this posterior to reliably control the true resulting risk.