Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models

Abstract

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.

Modelling Extreme-Value Dependence in International Stock Markets

“…We downloaded Ox from the website of Jurgen Doornik (http://www.nuff.ox.ac.uk/users/ doonik) and obtained the code used in the Kim, Shephard and Chib (1998) from the web-site http://www.nuff.ox.ac.uk/users/sheppard/ox). After experimenting with the various estimation methods, we settled for the Gibbs sampling algorithm presented in Kim, Shephard and Chib (1998). We refer to the stochastic volatility model as the SV filter.…”

“…Table 2 presents descriptive statistics for returns that are filtered for autocorrelation and those that are filtered for heteroskedasticity in addition. Three volatility filters were used to remove heteroskedasticity: AGARCH is an asymmetric version of GARCH; ADC is the Asymmetric Dynamic Covariance model in Kroner and Ng (1998), which is a multivariate GARCH model; SV is the stochastic volatility model in Kim, Shephard and Chib (1998). Detail specifications of the volatility filters are presented in the Appendix.…”

“…A second, more general form of volatility filter, is the stochastic volatility model presented by Kim, Shephard and Chib (1998): R t = βe ht/2 Z t , h t = µ +…”

Estimating a Semiparametric Asymmetric Stochastic Volatility Model with a Dirichlet Process Mixture

“…To reduce the influence of the starting values, we first perform 1000 sweeps over the log-volatilities using the step-by-step volatility sampler of Kim et al (1998) while holding the other parameters constant. We then let the entire sampler of Section 3 iterate 40,000 times, keeping only the last 30,000 draws from the two models for inference purposes.…”

“…1 In the model, log-volatility belongs to the parametric, first-order autoregressive, AR(1), class of stochastic volatility which can accomodate stationarity as imposed in the literature (Jacquier et al (1994) and Kim et al (1998)) as well as nonstationary deviations from this assumption. The rest of the model is nonparametric in the sense that no assumptions are made about the underlying joint distribution of returns and volatility.…”