In the present paper we study switching state space models from a Bayesian point of view. For estimation, the model is reformulated as a hierarchical model. We discuss various MCMC methods for Bayesian estimation, among them unconstrained Gibbs sampling, constrained sampling and permutation sampling. We address in detail the problem of unidentifiability, and discuss potential information available from an unidentified model. Furthermore the paper discusses issues in model selection such as selecting the number of states or testing for the presence of Markov switching heterogeneity. The model likelihoods of all possible hypotheses are estimated by using the method of bridge sampling. We conclude the paper with applications to simulated data as well as to modelling the U.S./U.K. real exchange rate. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

This dissertation explores the estimation of endogenous treatment effects in the presence of heterogeneous responses. A Bayesian Nonparametric approach is taken to model the heterogeneity in treatment effects. Specifically, I adopt the Dirichlet Process Mixture (DPM) model to capture the heterogeneity and show that DPM often outperforms Finite Mixture Model (FMM) in providing more flexible function forms and thus better model fit. Rather than fixing the number of components in a mixture model, DPM allows the data and prior knowledge to determine the number of components in the data, thus providing an automatic mechanism for model selection. Two DPM models are presented in this dissertation. The first DPM model is based on a two-equation selection model. A Dirichlet Process (DP) prior is specified on some or all the parameters of the structural equation, and marginal likelihoods are calculated to select the best DPM model. This model is used to study the incentive and selection effects of having prescription drug coverage on total drug expenditures among Medicare beneficiaries. The second DPM model utilizes a three-equation Roy-type framework to model the observed heterogeneity that arises due to the treatment status, while the unobserved heterogeneity is handled by separate DPM models for the treated and untreated outcomes. This Roy-type DPM model is applied to a data set consisting of 33,081 independent individuals from the Medical Expenditure Panel Survey (MEPS), and the treatment effects of having private medical insurance on the outpatient expenditures are estimated. Key Words: Treatment Effects, Endogeneity, Heterogeneity, Finite Mixture Model, Dirichlet Process Prior, Dirichlet Process Mixture, Roy-type Modeling, Importance Sampling, Bridge Sampling

In the present paper we explore various approaches of computing model likelihoods from the MCMC output for mixture and switching models, among them the candidate's formula, importance sampling, reciprocal importance sampling and bridge sampling. We demonstrate that the candidate's formula is sensitive to label switching. It turns out that the best method to estimate the model likelihood is the bridge sampling technique, where the MCMC sample is combined with an iid sample from an importance density. The importance density is constructed in an unsupervised manner from the MCMC output using a mixture of complete data posteriors. Whereas the importance sampling estimator as well as the reciprocal importance sampling estimator are sensitive to the tail behaviour of the importance density, we demonstrate that the bridge sampling estimator is far more robust in this concern. Our case studies range from from selecting the number of classes in a mixture of multivariate normal distributions, testing for the inhomogeneity of a discrete time Poisson process, to testing for the presence of Markov switching and order selection in the MSAR model. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

In the present paper we discuss the problem of estimating model likelihoods from the MCMC output for a general mixture and switching model. Estimation is based on the method of bridge sampling (Meng and Wong, 1996), where the MCMC sample is combined with an iid sample from an importance density. The importance density is constructed in an unsupervised manner from the MCMC output using a mixture of complete data posteriors. Whereas the importance sampling estimator as well as the reciprocal importance sampling estimator are sensitive to the tail behaviour of the importance density, we demonstrate that the bridge sampling estimator is far more robust in this concern. Our case studies range from computing marginal likelihoods for a mixture of multivariate normal distributions, testing for the inhomogeneity of a discrete time Poisson process, to testing for the presence of Markov switching and order selection in the MSAR model. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"