Long Memory in Volatility

How persistent is volatility? In other words, how quickly do financial markets forget large volatility shocks? Figure 1.1, Shephard (attached) shows that daily squared returns on exchange rates and stock indices can have autocorrelations which are significant for many lags. In any stationary ARCH or GARCH model, memory decays exponentially fast. For example, if {εt } are ARCH (1), the {εt} have autocorrelations ρk = α . Specifically, if α = .8 and k = 20, we get ρ20 = .012. This seems an unrealistically fast decay. On the other hand, for any integrated ARCH or GARCH , ρk = 1 for all k , so there is no decay at all. This seems unrealistically slow. The progression from ARCH (1) to ARCH (q ) to GARCH represents an attempt to allow for the strong volatility persistence observed in actual data. Typically, the exponential decay inherent in any stationary ARCH or GARCH model is too rapid to adequately describe the data (especially high frequency data), forcing the estimated models to be integrated. In reality, however, volatility may not be integrated, and the behavior of the estimated ARCH and GARCH models may simply be a signal that the memory is decaying relatively slowly compared to the exponential rate. What is needed, then, is a richer class of models allowing for intermediate degrees of volatility persistence.