Bugs for a Bayesian Analysis of Stochastic Volatility Models

This paper reviews the general Bayesian approach to parameter estimation in stochastic volatility models with posterior computations performed by Gibbs sampling. The main purpose is to illustrate the ease with which the Bayesian stochastic volatility model can now be studied routinely via BUGS (Bayesian Inference Using Gibbs Sampling), a recently developed, user-friendly, and freely available software package. It is an ideal software tool for the exploratory phase of model building as any modifications of a model including changes of priors and sampling error distributions are readily realized with only minor changes of the code. BUGS automates the calculation of the full conditional posterior distributions using a model representation by directed acyclic graphs. It contains an expert system for choosing an efficient sampling method for each full conditional. Furthermore, software for convergence diagnostics and statistical summaries is available for the BUGS output. The BUGS implementation of a stochastic volatility model is illustrated using a time series of daily Pound/Dollar exchange rates.

[1]  S. Turnbull,et al.  Pricing foreign currency options with stochastic volatility , 1990 .

[2]  L. Fahrmeir,et al.  Penalized likelihood smoothing in robust state space models , 1999 .

[3]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[4]  N. Shephard,et al.  Estimation of an Asymmetric Stochastic Volatility Model for Asset Returns , 1996 .

[5]  Peter C. Schotman,et al.  An empirical application of stochastic volatility models , 1998 .

[6]  N. Shephard.  Statistical aspects of ARCH and stochastic volatility , 1996 .

[7]  Jón Dańıelsson.  Stochastic volatility in asset prices estimation with simulated maximum likelihood , 1994 .

[8]  G. Roberts,et al.  Adaptive Markov Chain Monte Carlo through Regeneration , 1998 .

[9]  Bent E. Sørensen,et al.  Efficient method of moments estimation of a stochastic volatility model: A Monte Carlo study , 1999 .

[10]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[11]  George Tauchen,et al.  THE PRICE VARIABILITY-VOLUME RELATIONSHIP ON SPECULATIVE MARKETS , 1983 .

[12]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

[13]  Peter E. Rossi,et al.  Stochastic Volatility: Univariate and Multivariate Extensions , 1999 .

[14]  Andrew Thomas,et al.  WinBUGS - A Bayesian modelling framework: Concepts, structure, and extensibility , 2000, Stat. Comput..

[15]  A. Sokal.  Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms , 1997 .

[16]  Walter R. Gilks,et al.  Strategies for improving MCMC , 1995 .

[17]  Cong Han,et al.  MCMC Methods for Computing Bayes Factors: A Comparative Review , 2000 .

[18]  S. Taylor.  Financial Returns Modelled by the Product of Two Stochastic Processes , 1961 .

[19]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[20]  N. Shephard,et al.  Multivariate stochastic variance models , 1994 .

[21]  Peter E. Rossi,et al.  Bayesian Analysis of Stochastic Volatility Models , 1994 .

[22]  A. Gallant,et al.  Estimation of Stochastic Volatility Models with Diagnostics , 1995 .

[23]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[24]  N. Wermuth,et al.  On Substantive Research Hypotheses, Conditional Independence Graphs and Graphical Chain Models , 1990 .

[25]  R. Kohn,et al.  On Gibbs sampling for state space models , 1994 .

[26]  Adrian E. Raftery,et al.  Hypothesis testing and model selection , 1996 .

[27]  John Geweke,et al.  Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments , 1991 .

[28]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[29]  L. Harris,et al.  A maximum likelihood approach for non-Gaussian stochastic volatility models , 1998 .

[30]  Siem Jan Koopman,et al.  Time Series Analysis of Non-Gaussian Observations Based on State Space Models from Both Classical and Bayesian Perspectives , 1999 .

[31]  N. Shephard,et al.  Stochastic Volatility: Likelihood Inference And Comparison With Arch Models , 1996 .

[32]  Howard E. Reinhardt.  Theory of Probability: A Critical Introductory Treatment, Vol. 2 (Bruno de Finetti) , 1978 .

[33]  Jurgen A. Doornik,et al.  Statistical algorithms for models in state space using SsfPack 2.2 , 1999 .

[34]  Peter E. Rossi,et al.  Models and Priors for Multivariate Stochastic Volatility , 1995 .

[35]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter. , 1991 .

[36]  M. Pitt,et al.  Likelihood analysis of non-Gaussian measurement time series , 1997 .

[37]  N. Shephard.  Partial non-Gaussian state space , 1994 .

[38]  L. Glosten,et al.  On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks , 1993 .

[39]  Radford M. Neal.  Markov Chain Monte Carlo Methods Based on `Slicing' the Density Function , 1997 .

[40]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[41]  Walter R. Gilks,et al.  A Language and Program for Complex Bayesian Modelling , 1994 .

[42]  M. Tanner,et al.  Facilitating the Gibbs Sampler: The Gibbs Stopper and the Griddy-Gibbs Sampler , 1992 .

[43]  Siem Jan Koopman,et al.  Estimation of stochastic volatility models via Monte Carlo maximum likelihood , 1998 .

[44]  Stephen L Taylor,et al.  Modelling Financial Time Series , 1987 .

[45]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[46]  Bent E. Sørensen,et al.  GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study , 1996 .