| This dissertation consists of three essays. The first (Chapter 1) gives a general discussion of modeling dynamic correlations in multivariate stochastic volatility (MSV) models using Wishart processes. We explore the nonlinear relationship between the intertemporal sensitivity parameter and the covariance/correlation structure of the series of interest. Moreover, we prove concavity of the univariate log posteriors of the persistence parameter and of the degrees of freedom. Consequently, instead of using a grid sampler or the adaptive rejection Metropolis sampling, we can directly apply adaptive rejection sampling (ARS) to draw samples from these complicated densities, which is more efficient provided that log-concavity is assured. Moreover, we suggest using the Sherman-Morrison-Woodbury (SMW) formula in the update of the correlation matrices. Our empirical study shows that ARS together with SMW formula can considerably improve MCMC efficiency. Other issues about the assessment of hyperparameters and model parameterizations for this type of models are also discussed. Since the Wishart process plays the key role in this dissertation, it is essential to correctly generate random Wishart matrices for model estimation. Unfortunately, however, most (if not all) statistical software packages do not treat the generation of random Wishart matrices in a correct manner. For this reason, in the second essay (Chapter 2), based on Gyndikin's theorem and Bartlett's decomposition, the OX package "WishPack" is developed for generating random Wishart/inverse-Wishart matrices. To make the package more complete, the density functions for the Wishart and inverse Wishart distributions are also provided. The most important feature of this package is that it takes into account the singular Wishart matrices and distributions, since they have been well defined and are useful in practical problems. In the final essay (Chapter 3), to provide a parsimonious model for high-dimensional data, a dynamic correlation factor MSV (DCFMSV) model is proposed in which the evolution of the factor correlations is characterized by Wishart processes. The most advantageous feature of this model compared to existing models is that it retains the latent factor structure and therefore has more model flexibility. The estimation procedure is developed using Markov chain Monte Carlo (MCMC) methods. The real data example shows that the DCFMSV model is informative, useful, and close to the real world. |
