Statistical Finance Analyst - RetiredUBS Warburg, LLC, CT
Bayesian Time Series: Analysis Methods Using Simulation-Based Computation
This dissertation introduces new simulation-based analysis approaches, including both sequential and off-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 fact that an exponential power distribution is a scale mixture of normals. This model has application in signal processing, specifically image processing, with orthogonal wavelet transformations. We discuss our experience in applying the proposed algorithm to data from a grass image. As an alternative to MCMC methods, a generic sequential algorithm is proposed for a wide class of Bayesian dynamic models, including those with fixed parameters. This combines old ideas of kernel smoothing via shrinkage for modeling fixed parameter with newer ideas of auxiliary particle filtering for the time varying states. It is shown here that our specific smoothing approaches can interpret and suggest modifications to techniques that add ``artificial evolution noise'' to fixed model parameters at each time point to address problems of sample attrition and prior:data conflict. Our new approach permits smoothing and regeneration of sample points of model parameters without the ``loss of historical information" inherent in the ``artificial evolution noise'' method; the computational load required by our method is meaningfully reduced from the earlier kernel smoothing algorithms for Bayesian posterior simulation. Simple examples are used to demonstrate the efficacy of our algorithm. Following general discussions of the generic sequential algorithm, we turn our attention to the modeling of multivariate stochastic variance matrices in financial time series, an important application area of Bayesian dynamic models and their analysis approaches. The emphasis is on some research problems in applying Monte Carlo methods to multivariate, potentially singular, Dynamic Linear Models (DLMs) with variance discounting. Some theoretical results about singular Wishart/matrix-Beta distribution are derived. A simplified example is used to demonstrate the feasibility of the proposed sequential algorithm in such models using the theoretical results developed. A Forward Filtering Backward Sampling algorithm is simultaneously proposed for the DLM of interest. We also analyze with the proposed sequential algorithm a volatility model closely related to variance discounting models - a multivariate stochastic volatility model in dynamic factor representations studied with MCMC. With 30 fixed parameters, the factor model is used to demonstrate the performance of our algorithm in models of modest dimensions. We conclude with some discussion of general issues of practical relevance, and suggestions for further development.