The EGARCH model of Nelson [29] is one of the most successfulARCH models which may exhibit characteristic asymmetries offinancial time series, as well as long memory. The paper studiesthe covariance structure and dependence properties of the EGARCHand some related stochastic volatility models. We show that thelarge time behavior of the covariance of powers of the (observed)ARCH process is determined by the behavior of the covariance ofthe (linear) log-volatility process; in particular, a hyperbolicdecay of the later covariance implies a similar hyperbolic decayof the former covariances. We show, in this case, that normalizedpartial sums of powers of the observed process tend to fractionalBrownian motion. The paper also obtains a (functional) CLT for thecorresponding partial sums' processes of the EGARCH model withshort and moderate memory. These results are applied to studyasymptotic behavior of tests for long memory using the R/Sstatistic.
