Stochastic Volatility and GARCH: a Comparison Based on UK Stock Data

Abstract This paper compares two types of volatility models for returns, ARCH-type and stochastic volatility (SV) models, both from a theoretical and an empirical point of view. In particular a GARCH(1,1) model, an EGARCH(1,1) model and a log-normal AR(1) stochastic volatility model are considered. The three models are estimated on UK stock data: a series of the British equity index FTSE100 is used to estimate the relevant parameters. Diagnostic tests are implemented to evaluate how well the models fit the data. The models are used to obtain daily volatility forecasts and these volatilities are used to estimate the "VaR" on a simple one-unit position on FTSE100. The VaR accuracy is tested by means of a backtest. While the results do not lead to a straightforward preference between GARCH(1,1) and SV, the EGARCH shows the best performance. Keywords: Volatility modelsstochastic volatilityGARCHvalue at risk Acknowledgements This paper is a revised version of one presented at the XXXII EWGFM in London (April 2003). I wish to thank the conference participants and the two anonymous referees for their comments and suggestions. I am also grateful to Michael Pitt and Ilias Tsiakas for helpful advice. Financial support from MIUR is gratefully acknowledged. The usual caveats apply. Notes 1. The process of computing the log-variance conditional on the observations up to the current time h t | R t , for , is called "filtering". In general, given a time series modelled as independent conditionally on an unobserved state, "filtering" means "to learn about the state given contemporaneously available information" (quote from Pitt and Shephard, 1999 Pitt, M. K. and Shephard, N. 1999. Filtering via simulation: auxiliary particle filter. Journal of the American Statistical Association, 94: 590–599. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). 2. The FTSE100 index prices have been downloaded from Datastream; the bank holidays have been taken out from the series. 3. Here and in the following most of the correlograms are omitted: these graphs are available on request. 4. The GARCH models (both GARCH(1,1) and EGARCH(1,1)) are estimated with the software E-Views. 5. The SV model is estimated through an Ox code related to CKS98. The estimation and filtering algorithms used for CKS98 are available at http://www.nuff.ox.ac.uk/users/shephard/ox/. The appropriate algorithms have been exploited by adapting the code to the context of this paper. 6. For technical details on the inefficiency factors and the standard errors see CKS98. 7. In CKS98 it is shown that more complex estimation algorithms are definitely more efficient (lower inefficiency factors) for all the parameters. In the present work the simplest and quickest algorithm has been chosen, considering that the resulting estimated parameters are anyway very close even using more efficient algorithm in CKS98. A more complex algorithm is also estimated for comparison. 8. If the highest absolute returns (e.g. higher than 4%) are considered as outliers and taken out of the series, the effects on the backtest is double: on one hand the misses corresponding to the outliers are clearly avoided. On the other hand, however, the volatility forecast, which is strongly affected by past returns, does not increase as a consequence of the high absolute returns, leading to new misses in the period following outliers, as in the case of 11 September. In the present case, the backtest after eliminating outliers present at least the same number of misses as before. 9. While the common VaR definition in Equation 18 is based on an approximation, the exact lower and upper limits for the interval forecast are respectively and .