Bayesian Time Series: Analysis Methods Using Simulation-Based Computation

Statisti s) BAYESIAN TIME SERIES: ANALYSIS METHODS USING SIMULATION-BASED COMPUTATION by Fang Liu Institute of Statisti s and De ision S ien es Duke University Date: Approved: Professor Mike West, Supervisor Brani Vidakovi Merlise Clyde Gary Rosner An abstra t of a dissertation submitted in partial ful llment of the requirements for the degree of Do tor of Philosophy in the Institute of Statisti s and De ision S ien es in the Graduate S hool of Duke University 2000 Abstra t This dissertation introdu es new simulation-based analysis approa hes, in luding both sequential and o -line learning algorithms, for various Bayesian time series models. We provide a Markov Chain Monte Carlo (MCMC) method for an autoregressive (AR) model with innovations following exponential power distributions using the fa t that an exponential power distribution is a s ale mixture of normals. This model has appli ation in signal pro essing, spe i ally image pro essing, with orthogonal wavelet transformations put forth in [58℄. We dis uss our experien e in applying the proposed algorithm to data from a grass image. As an alternative to MCMC methods, a generi sequential algorithm is proposed for a wide lass of Bayesian dynami models, in luding those with xed parameters. This ombines old ideas of kernel smoothing via shrinkage ([92℄, [93℄) for modeling xed parameter with newer ideas of auxiliary parti le ltering ([67℄) for the time varying states. It is shown here that our spe i smoothing approa hes an interpret and suggest modi ations to te hniques that add \arti ial evolution noise" to xed model parameters at ea h time point to address problems of sample attrition and prior:data on i t. Our new approa h permits smoothing and regeneration of sample points of model parameters without the \loss of histori al information" inherent in the \arti ial evolution noise" method; the omputational load required by our method is meaningfully redu ed from the earlier kernel smoothing algorithms for Bayesian posterior simulation. Simple examples are used to demonstrate the eÆ a y of our algorithm. Following general dis ussions of the generi sequential algorithm, we turn our attention to the modeling of multivariate sto hasti varian e matri es in nan ial time series, an important appli ation area of Bayesian dynami models and their analysis approa hes. The emphasis is on some resear h problems in applying Monte Carlo iv methods to multivariate, potentially singular, Dynami Linear Models (DLMs) with varian e dis ounting. Some theoreti al results about singular Wishart/matrix-Beta distribution are derived based on the work of [86℄. A simpli ed example is used to demonstrate the feasibility of the proposed sequential algorithm in su h models using the theoreti al results developed. A Forward Filtering Ba kward Sampling algorithm is simultaneously proposed for the DLM of interest. We also analyze with the proposed sequential algorithm a volatility model losely related to varian e dis ounting models a multivariate sto hasti volatility model in dynami fa tor representations studied with MCMC in [1℄ and by other resear hers. With 30 xed parameters, the fa tor model is used to demonstrate the performan e of our algorithm in models of modest dimensions. We on lude with some dis ussion of general issues of pra ti al relevan e, and suggestions for further development. v A knowledgements I would like to express my deep gratitude to my advisor, Professor Mike West. I am grateful for the pre ious experien e of working with him. His onstant input of insights has made this thesis possible; his ontinual en ouragement has inspired me to progress; and his devotion to resear h has set me a life time example. I would like to thank Professor Brani Vidakovi , Merlise Clyde and Gary Rosner for taking interest in my work and making suggestions for improvement. Dr. Jose Quintana at CDC North Ameri a rst exposed me to the open questions in nan ial time series modeling and provided patient resear h guidan e. The nal revision of the thesis was made while I was a member of the Global Dynami Asset Allo ation Group at CDC North Ameri a. I appre iate my supervisors and olleagues, espe ially Dr. Jose Quintana, for their support. I thank the fa ulty and my fellow graduate students at ISDS for providing su h a ooperative environment of resear h and study. I am also indebted to Dr. Mi hael Pitt and Omar Aguilar who generously shared with me their data and MCMC simulation programs whi h are used in the applied examples for the sequential algorithm. My interest in s ienti resear h rst developed in my early hildhood under the in uen e of my parents, who have never failed their support in my pursuit of knowledge. My sister has remained a powerful \ hallenger" as well as the most intimate friend throughout my student life. To them, and my numerous friends, I owe spe ial thanks. vi