Stochastic Volatility: Likelihood Inference And Comparison With Arch Models

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.

[1]  M. Rosenblatt.  Remarks on a Multivariate Transformation , 1952 .

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  D. Cox.  Tests of Separate Families of Hypotheses , 1961 .

[4]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[5]  L. Shenton,et al.  Omnibus test contours for departures from normality based on √b1 and b2 , 1975 .

[6]  B. Ripley.  Modelling Spatial Patterns , 1977 .

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

[8]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[9]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  F. Diebold,et al.  The dynamics of exchange rate volatility: a multivariate latent factor ARCH model , 1986 .

[11]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

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

[13]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[14]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[15]  T. Bollerslev,et al.  A CONDITIONALLY HETEROSKEDASTIC TIME SERIES MODEL FOR SPECULATIVE PRICES AND RATES OF RETURN , 1987 .

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

[17]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[18]  Franco Peracchi,et al.  Testing non-nested hypotheses , 1988 .

[19]  Stuart German,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1988 .

[20]  S. K. Park,et al.  Random number generators: good ones are hard to find , 1988, CACM.

[21]  M. Chesney,et al.  Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model , 1989, Journal of Financial and Quantitative Analysis.

[22]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[23]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[24]  Peter C. B. Phillips,et al.  To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends , 1991 .

[25]  D. Cox.  LONG‐RANGE DEPENDENCE, NON‐LINEARITY AND TIME IRREVERSIBILITY , 1991 .

[26]  H. V. Dijk,et al.  A Bayesian analysis of the unit root in real exchange rates , 1991 .

[27]  A. F. M. Smith,et al.  REPARAMETRIZATION ASPECTS OF NUMERICAL BAYESIAN METHODOLOGY FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS , 1992 .

[28]  John Hinde,et al.  Choosing Between Non-nested Models: a Simulation Approach , 1992 .

[29]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[30]  Neil Shephard,et al.  Fitting Nonlinear Time-Series Models with Applications to Stochastic Variance Models , 1993 .

[31]  M. Pesaran,et al.  A simulation approach to the problem of computing Cox's statistic for testing nonnested models , 1993 .

[32]  M. West.  Approximating posterior distributions by mixtures , 1993 .

[33]  John Geweke,et al.  Bayesian Analysis of Stochastic Volatility Models: Comment , 1994 .

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

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

[36]  S. Chib,et al.  Bayes inference in regression models with ARMA (p, q) errors , 1994 .

[37]  Enrique Sentana,et al.  Volatiltiy and Links between National Stock Markets , 1990 .

[38]  Peter E. Rossi,et al.  Bayesian Analysis of Stochastic Volatility Models: Comments: Reply , 1994 .

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

[40]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[41]  R. Mahieu,et al.  Stochastic volatility and the distribution of exchange rate news , 1994 .

[42]  Daniel B. Nelson,et al.  ARCH MODELS a , 1994 .

[43]  Tim Bollerslev,et al.  Chapter 49 Arch models , 1994 .

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

[45]  N. Shephard,et al.  The simulation smoother for time series models , 1995 .

[46]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[47]  W. Gilks,et al.  Adaptive Rejection Metropolis Sampling Within Gibbs Sampling , 1995 .

[48]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .

[49]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[50]  S. Chib.  Marginal Likelihood from the Gibbs Output , 1995 .

[51]  G. Kitagawa.  Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models , 1996 .

[52]  M. Pitt,et al.  Analytic Convergence Rates and Parameterization Issues for the Gibbs Sampler Applied to State Space Models , 1999 .

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

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

[55]  Michael Isard,et al.  Contour Tracking by Stochastic Propagation of Conditional Density , 1996, ECCV.

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

[57]  Siddhartha Chib,et al.  Markov Chain Monte Carlo Simulation Methods in Econometrics , 1996, Econometric Theory.