(untitled) (European Journal of Pure and Applied Mathematics, 2010)
Properties of three well-known and frequently applied first-order models for modelling and forecasting volatility in daily or weekly 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 kurtosis 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. 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.