Particle filters (PFs) are powerful sampling-based inference/learning algorithms for dynamic Bayesian networks (DBNs). They allow us to treat, in a principled way, any type of probability distribution, nonlinearity and non-stationarity. They have appeared in several fields under such names as “condensation”, “sequential Monte Carlo” and “survival of the fittest”. In this paper, we show how we can exploit the structure of the DBN to increase the efficiency of particle filtering, using a technique known as Rao-Blackwellisation. Essentially, this samples some of the variables, and marginalizes out the rest exactly, using the Kalman filter, HMM filter, junction tree algorithm, or any other finite dimensional optimal filter. We show that Rao-Blackwellised particle filters (RBPFs) lead to more accurate estimates than standard PFs. We demonstrate RBPFs on two problems, namely non-stationary online regression with radial basis function networks and robot localization and map building. We also discuss other potential application areas and provide references to some Þnite dimensional optimal filters.

This article is available in electronic form on the ISDS web site, http://www.stat.duke.edu

We discuss the development of dynamic factor models for multivariate nancial time series, and the incorporation of stochastic volatility components for latent factor processes. Bayesian inference and computation is developed and explored in a study of the dynamic factor structure of daily spot exchange rates for a selection of international currencies. The models are direct generalisations of univariate stochastic volatility models, and represent speci c varieties of models recently discussed in the growing multivariate stochastic volatility literature. We also discuss connections and comparisons with the much simpler method of dynamic variance discounting that, for over a decade, has been a standard approach in applied nancial econometrics in the Bayesian forecasting world. We review empirical ndings in applying these models to the exchange rate series, including aspects of model performance in dynamic portfolio allocation. We conclude with comments on the potential practical utility of structured factor models and future potential developments and model extensions.

ABSTRACT. SVARs are widely used for policy analysis and to provide stylized facts for dynamic general equilibrium models. Yet there have been no workable rank conditions to ascertain whether an SVAR is globally identified and no efficient al-gorithms for small-sample statistical inference when identifying restrictions are di-rectly imposed on impulse responses. To fill these important gaps in the literature, this paper makes four contributions. First, we establish a general rank condition for both exactly and overidentified models. Second, we show that this condition can be easily checked analytically and applies to a wide class of identifying restrictions, including linear and certain nonlinear restrictions. Third, we establish a much sim-pler rank condition for exactly identified models that amounts to a straightforward counting exercise. Fourth, we develop a number of efficient algorithms for small-sample statistical inference. I.

() BAYESIAN TIME SERIES: FINANCIAL MODELS AND SPECTRAL ANALYSIS by Yang Chen Department of Statistics and Decision Sciences Duke University Date: Approved: Dr. Mike West, Supervisor Dr. Peter Muller Dr. Brani Vidakovic Dr. S. Viswanathan An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Statistics and Decision Sciences in the Graduate School of Duke University Abstract This dissertation studies two models in Bayesian time series analysis: the Stochastic Volatility Model in the time domain and the Harmonic Model in the frequency domain of time series. Volatility plays a central role in modern finance especially in the pricing of derivative securities. Research on changing volatility can be categorized into two groups: the time-varying volatility models represented by ARCH type models and the Stochastic Volatility Models. Research on ARCH type models offers straightforward implementation and...

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This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distributions. A diagonal matrix of discount factors is employed in order to discount the variances element by element and therefore allowing a flexible and pragmatic variance modelling approach. Diagnostic tests and sequential model monitoring are discussed in some detail. The proposed estimation theory is applied to a four-dimensional time series, comprising spot prices of aluminium, copper, lead and zinc of the London metal exchange. The empirical findings suggest that the proposed Bayesian procedure can be effectively applied to financial data, overcoming many of the disadvantages of existing volatility models.

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Abstract: Many economic applications call for simultaneous equations VAR modeling. We show that the existing importance sampler can be prohibitively inefficient for this type of models. We develop a Gibbs simulator that works for both simultaneous and recursive VAR models with a much broader range of linear restrictions than those in the existing literature. We show that the required computation is of an SUR type, and thus our method can be implemented cheaply even for large systems of multiple equations. JEL classification: C15, C32, E50 Key words: simultaneous equations, recursive systems, independence, importance sampling, Gibbs