Stylized Facts of Financial Time Series and Three Popular Models of Volatility

Stylized Facts of Financial Time Series and Three Popular Models of Volatility

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Unformatted text preview: Stylized Facts of Financial Time Series and Three Popular Models of Volatility Hans Malmsten and Timo Tersvirta Department of Economic Statistics Stockholm School of Economics Box 6501, SE-113 83 Stockholm, Sweden SSE/EFI Working Paper Series in Economics and Finance No 563 August 2004 Abstract Properties of three well-known and frequently applied first-order models for modelling and forecasting volatility in financial series such as stock and exchange rate returns are considered. These are the standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH), the Exponential GARCH and the Autoregressive Stochastic Volatility model. The focus is on finding out how well these models are able to reproduce characteristic features of such series, also called stylized facts. These include high kurto- sis and a rather low-starting and slowly decaying autocorrelation function of the squared or absolute-valued observations. Another stylized fact is that the autocorrelations of absolute-valued returns raised to a positive power are maximized when this power equals unity. A number of results for moments of the three models are given as well as the autocorrelation function of squared observations or, when available, the autocorrelation function of the absolute-valued observations raised to a positive power. These results make it possible to consider kurtosis-autocorrelation combi- nations that can be reproduced with these models and compare them with ones that have been estimated from financial time series. The ability of the models to reproduce the stylized fact that the autocorrelations of pow- ers of absolute-valued observations are maximized when the power equals one is discussed as well. Finally, it is pointed out that none of these basic models can generate realizations with a skewed marginal distribution. Not unexpectedly, a conclusion that emerges from these considerations, largely based on results on the moment structure of these models, is that none of the models dominates the others when it comes to reproducing stylized facts in typical financial time series. 1 Keywords. Autoregressive conditional heteroskedasticity, evaluation of volatility models, exponential GARCH, GARCH, modelling return series, stochastic volatility JEL Classification Codes: C22, C52 2 Acknowledgements. This research has been supported by the Swedish Research Council for Humanities and Social Sciences, and by Jan Wallan- ders and Tom Hedeliuss Foundation, Grant No. J99/37. Material from this paper has been presented at 14th SINAPE, Caxambu, MG, July 2000, 4th erc/METU International Conference in Economics, Ankara, Septem- ber 2000, NoonToNoon Workshop Finance, Statistics and Stochastics, Helsinki, October 2000, 22nd International Symposium on Forecasting, Dublin, June 2002, and in seminars at Central European University, Bu- dapest, European University Institute, Florence, Humboldt-Universitt, Berlin, Ibmec Business School, So Paulo, Norges Bank (Central Bank of...
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Stylized Facts of Financial Time Series and Three Popular Models of Volatility

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