Stochastic Volatility Likelihood Inference and Comparison with ARCH Models

Stochastic Volatility Likelihood Inference and Comparison with ARCH Models

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Review of Economic Studies (1998) 65, 361-393 0 1998 The Review of Economic Studies Limited 0034-6527/98/00170361$02.00 Stochastic Volatility : Likelihood Inference and Comparison with ARCH Models SANGJOON KIM Salomon Brothers Asia Limited NEIL SHEPHARD Nufield College, Oxford and SIDDHARTHA CHIB Washington University, St. Louis First version received December 1994;Jinal version accepted August 1997 (Eds.) 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. INTRODUCTION The variance of returns on assets tends to change over time. One way of modelling this feature the data is to let the conditional variance be a function squares previous observations and past variances. This leads to the autoregressive conditional heteroscedasticity (ARCH) based models developed by Engle (1982) and surveyed in Bollerslev, Engle and Nelson ( 1994). An alternative to the ARCH framework is a model in which the variance is specified to follow some latent stochastic process. Such models, referred to as stochastic volatility (SV) models, appear in the theoretical finance literature on option pricing (see, for example, Hull and White (1987) their work generalizing Black-Scholes option formula to allow for volatility). Empirical versions SV model are typically formula- ted discrete time. The canonical model class for regularly spaced data is 361
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362 REVIEW OF ECONOMIC STUDIES where yr is the mean corrected return on holding the asset at time t, h, is the log volatility at time t which assumed to follow a stationary process (I 4 I < 1) with h, drawn from the stationary distribution, and qt are uncorrelated standard normal white noise shocks and JV ( - , ) is the normal distribution. The parameter p or exp (p/2) plays the role of the constant scaling factor and can be thought of as the modal instantaneous volatility, c# as the persistence in the volatility, and CT, the volatility of the log-volatility. For indenti- fiability reasons either p must be set to one or p to zero. We show later that the param- eterization with p equal to one in preferable and so we shall leave p unrestricted when we estimate the model but report results for p = exp (p/2) as this parameter has more economic interpretation.
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This note was uploaded on 02/28/2010 for the course ECO 211 taught by Professor Gilo during the Spring '10 term at Young Harris.

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Stochastic Volatility Likelihood Inference and Comparison with ARCH Models

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