Volatility Components and Long Memory-Effects Revisited

The goal of this paper is to illuminate the capability of the component GARCH model of Ding and Granger (1996) and Engle and Lee (1999) to reproduce the long memory-type behavior of financial volatility. The potential of this model to capture the long memory dynamics observed in measures of financial volatility has been documented recently by Maheu (2005) and Deo et al. (2006), who base their conclusions on simulation techniques and a forecasting exercise, respectively. In this paper, a simple explanation for these observations is provided, which is based on the theoretical autocorrelation function (ACF) of the component GARCH model. We also elucidate why even higher-order GARCH models with Bollerslev's (1986) nonnegativity constraints enforced cannot mimic the long memory effects. The reasoning is supported with several empirical examples, for which we explicitly calculate the theoretical ACF implied by a couple of different fitted models, and find that their structure is just as predicted by our argument. To conveniently conduct these computations, a general simple method for computing the theoretical ACF of GARCH models is suggested, which is easier to use than the formulas developed so far, and particularly so for higher lag-orders. The ability of the component model to approximate long memory is also validated on the basis of a visual comparison between the empirical and the implied theoretical ACFs.