Long memory in Volatility

A weakly stationary process has short memory when its autocorrelation function (ACF) ?(:) is geometrically bounded. The stationary stochastic processes frequently referred in financial time series, such as ARCH (Engle, 1982), GARCH (Bollerslev, 1986), IGARCH (Engle and Bollerslev, 1986), and EGARCH (Nelson, 1991) all have short memory for volatilities. A weakly stationary process have long memory if its ACF has a hyperbolic decay. The stationary stochastic processes such as LMSV (Bredit, Crato and de Lima, 1994), FIGARCH (Baillie, Bollerslev and Mikkelsen, 1996), FIEAGRCH (Nelson, 1991; Bollerslev and Mikkelsen, 1996) are long-memory models for volatilitis. FIGARCH, Fractionally Integrated ARCH model (e.g. Andersen and Bollslev 1997; Baillie et al 1996). Baillie et al. (1996) (hereafter denoted BBM) introduce the Fractionally Integrated GARCH (FIGARCH) model Absolute returns and squared returns are proxies for volatility.