Essays on Multivariate Stochastic Volatility Models Using Wishart Processes: A General Discussion and Dimension Reduction by Latent Factor Structures

KU, YU-CHENG. Essays on Multivariate Stochastic Volatility Models Using Wishart Processes: A General Discussion and Dimension Reduction by Latent Factor Structures. (Under the direction of Peter Bloomfield.) 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. c © Copyright 2010 by Yu-Cheng Ku All Rights Reserved Essays on Multivariate Stochastic Volatility Models Using Wishart Processes: A General Discussion and Dimension Reduction by Latent Factor Structures by Yu-Cheng Ku A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Statistics Raleigh, North Carolina 2010 APPROVED BY: Ronald Gallant Sujit Ghosh Min Kang Brian Reich Peter Bloomfield Chair of Advisory Committee Biography Ku, Yu-Cheng was born in Taichung, Taiwan. In 2006, he came to North Carolina State University to pursue his doctorate degree in Statistics. During the doctoral program, he interned in Fannie Mae at Washington D.C. as a risk analyst in the Department of Credit Risk Oversight in 2008. He did his second internship as a data analyst in the Headquarter of Customer and Business Insights of GlaxoSmithKline (GSK) at Durham, NC in 2010. Those valuable experiences taught him how statistical methods can help solve real business problems. In his academic community, he has presented in international conferences for multiple times and was invited to referee papers for several journals, including Journal of Econometrics. Outside of school he loves to travel. For the past four years he has traveled seven countries across Asia, Europe, and North America. Also, he traveled more than thirty states including Hawaii and ten National Parks of the U.S. He will finish his Ph.D. with a Math minor in 2010. After which, he will serve as a postdoctoral fellow to continue his research career at the Australian School of Business, Sydney, Australia.