On a Dynamic Mixture GARCH Model 249

This paper proposes a new mixture GARCH model with a dynamic mixture proportion. The mixture Gaussian distribution of the error can vary from time to time. The Bayesian Information Criterion and the EM algorithm are used to estimate the number of parameters as well as the model parameters and their standard errors. The new model is applied to the S&P500 Index and Hang Seng Index and compared with GARCH models with Gaussian error and Student’s t error. The result shows that the IGARCH effect in these index returns could be the result of the mixture of one stationary volatility component with another non-stationary volatility component. The VaR based on the new model performs better than traditional GARCH-based VaRs, especially in unstable stock markets. Copyright © 2008 John Wiley & Sons, Ltd. key words mixture time series; GARCH; statistical arbitrage INTRODUCTION After its inception in Engle (1982), the GARCH model has been commonly used to model the conditional variance of many fi nancial data. One problem is raised, however, that the GARCH model seemed incapable of capturing a sudden and dramatic increase in the volatility of data in a very short period. Gray (1996) in his paper, tried to use GARCH to model some interest rate data, but found that a regime-switching process fi tted interest rate data much better than a single GARCH process. Chen et al. (2005) also found that regime-switching GARCH is more suitable than a single GARCH or ARCH process in modeling the huge volatility change driven by news arrival in fi nancial markets. Dumas (1992), Sercu et al. (1995), Michael et al. (1997) and Berka (2005) showed evidence of regime-switching dynamics existing in real exchange rate when there is trading cost. Further, Mikosch and Starica (2004) pointed out that fi tting a normal GARCH model to data with structural change would exhibit the integrated GARCH (IGARCH) effect, where the sum of the ARCH coeffi cient and the GARCH coeffi cient is close to 1. This IGARCH effect is widely shown in stock market data and other fi nancial market data. In order to explain this IGARCH effect and capture the characteristics of dramatic volatility change in a short time, two major approaches have been proposed: the threshold model and the mixture model. Journal of Forecasting J. Forecast. 28, 247–265 (2009) Published online 10 October 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/for.1093 * Correspondence to: W. K. Li, Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: hrntlwk@hku.hk 248 X. Cheng, P. L. H. Yu and W. K. Li Copyright © 2008 John Wiley & Sons, Ltd. J. Forecast. 28, 247–265 (2009) DOI: 10.1002/for In the approach of mixture modeling, Schwert (1989) proposes a model in which observations have either a higher volatility or a lower volatility, and this depends on a hidden variable Zt which follows a two-stage Markov process. Wong and Li (2001a) propose a similar model but the hidden volatility state variable Zt is linked to other variables (such as past observations) with a logistic linking function. These approaches, however, can be regarded as an extension of Tong and Lim’s (1980) threshold autoregressive model. Indeed, threshold time series with the threshold indicator variable satisfying a Markovian process was fi rst pointed out by Tong and Lim (1980) in their rejoinder to the discussion. As an attempt to extend the mixture model to cater for the conditional heteroscedasticity case, Wong and Li (2001b) propose a mixture ARCH model with a constant mixture probability. An early attempt to model changing structure in a time series is the Markov-switching model. Goldfeld and Quandt (1973) began the Markov-switching model. Hamilton and Susmel (1994) and Cai (1994) were the fi rst authors to propose the Markov-switching ARCH model. The mixture GARCH model with a classical GARCH specifi cation is a path-dependent model. Gray (1996) proposes a traceable mixture GARCH model, in which the s t−1 in the GARCH model is replaced by E(s t−1Ft−1). However, this modifi cation made the mixture model incapable of capturing the dramatic changes in volatility in a short period. This is because the expectation of volatility in the two regimes mingles the volatilities, so that the ‘mingled volatility’ cannot describe the two distinct volatility levels in the mixture data. Haas et al. (2004) propose another markov-switching mixture GARCH model, where a hidden state variable Zt determines which volatility regime is exhibited in the observation. In this paper, we propose a dynamic mixture GARCH model (DMGARCH), in which the mixture proportion is linked with other variables by means of a logistic link function. Generally, the link function can be any inverse cumulative distribution function. Because of its simplicity, we choose the logistic function as the link function in our model. The next section gives a detailed model specifi cation. We adopt the volatility selection idea in the work of Haas et al. (2004), so that there is an unobserved stage indicator variable Zt at each time point. Naturally, the introduction of the latent variable Zt makes the EM algorithm (Dempster et al., 1977; Ruud, 1991) the ideal estimation method. The Bayesian Information Criterion (Schwarz, 1978) is used for model selection. The third section gives the procedure of parameter estimation with the EM algorithm, and the fourth section discusses the BIC approach in the DMGARCH case. In the fi fth section, we apply this model to the S&P500 index and the Hang Seng Index on their daily log-return data. An interesting result from S&P500 and Hang Seng Index estimation is that the IGARCH effect in the normal GARCH model seems to be the result of a mixture of a stationary volatility component and a non-stationary volatility component. We also fi nd that the value-at-risk (VaR) based on the DMGARCH model performs better than the traditional GARCH-based VaR. THE ARX-DMGARCH MODEL The ARX-DMGARCH model with two components under consideration is defi ned as follows: F y p e t t k t t k t k F − = ( ) =     ∑ 1 1 2 , , Φ σ On a Dynamic Mixture GARCH Model 249 Copyright © 2008 John Wiley & Sons, Ltd. J. Forecast. 28, 247–265 (2009) DOI: 10.1002/for where e y y x t t i t i i p i t i i q = − − − − = − = ∑ ∑ γ γ δ 0 1 1 σ ω α β σ k t k k t k k t e k , , , 2 1 2 1 2 1 2 = + + = − −