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In this paper, Markov chain Monte Carlo sampling methods are exploited to provide a unified, practical likelihood-based framework for the analysis of stochastic volatility models. A highly effective method is developed that samples all the unobserved volatilities at once using an approximating offset mixture model, followed by an importance reweighting procedure. This approach is compared with several alternative methods using real data. The paper also develops simulation-based methods for filtering, likelihood evaluation and model failure diagnostics. The issue of model choice using non-nested likelihood ratios and Bayes factors is also investigated. These methods are used to compare the fit of stochastic volatility and GARCH models. All the procedures are illustrated in detail.

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This article provides a framework for estimating the marginal likelihood for the purpose of Bayesian model comparisons . The approach extends and completes the method presented in Chib (1995) by overcoming the problems associated with the presence of intractable full conditional densities. The proposed method is developed in the context of MCMC chains produced by the Metropolis-Hastings algorithm, whose building blocks are used both for sampling and marginal likelihood estimation, thus economizing on prerun tuning effort and programming. Experiments involving the logit model for binary data, hierarchical random effects model for clustered Gaussian data, Poisson regression model for clustered count data, and the multivariate probit model for correlated binary data, are used to illustrate the performance and implementation of the method. These examples demonstrate that the method is practical and widely applicable.

1995) ML estimation of the t distribution using EM and its extensions, ECM and ECME. Statist. Sin., 5, 19±39. Maronna, R. A. (1976) Robust M-estimators of multivariate location and scatter. Ann. Statist., 4, 51±67. Mendoza-Blanco, J. R. (1995) Bayesian analysis for seemingly unrelated regression models with tdistributed errors. PhD Dissertation. Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh. Tanner, M. A. (1993) Tools for Statistical Inference, 2nd edn. New York: Springer. Zellner, A. (1976) Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms. J. Am. Statist. Ass., 71, 400±405. Algorithm AS 316Language Fortran 77 Description and PurposeMoser and Fei (1991) developed a sequential regression design based on the Robbins± Monro process. The design provides estimates of a regression function and its roots which are strongly convergent and asymptotically normal with minimum variance. The algorithm presented here implements the Moser±Fei procedure when EYjx Mx, L is an increasing regression function of x and K unknown parameters L L p1 , . . ., L pK H .

This paper provides a practical simulation-based Bayesian and non-Bayesian analysis of correlated binary data using the multivariate probit model. The posterior distribution is simulated by Markov chain Monte Carlo methods and maximum likelihood estimates are obtained by a Monte Carlo version of the EM algorithm. A practical approach for the computation of Bayes factors from the simulation output is also developed. The methods are applied to a dataset with a bivariate binary response, to a four-year longitudinal dataset from the Six Cities study of the health effects of air pollution and to a sevenvariate binary response dataset on the labour supply of married women from the Panel Survey of Income Dynamics.