Long memory or long range dependenceBehavior of the spectral densityFractionally integrated modelsPredictionLong memory extended to GARCH and EGARCH modelsTesting for persistenceGPH TestK. Ensor, STAT 4211Spring 2005Long memory or long range dependence•ARMA models are characterized by an exponential decay in the autocorrelation structure as the lag goes to infinity.•Long memory processes are stationary time series that exhibit a slower decay in the autocorrelation structure•The sum of the autocorrelation over all lags is infinite.10 where as )( kckkK. Ensor, STAT 4212Spring 2005Behavior of the spectral density•The spectral density has a special structure as it approaches 0.•The Hurst coefficient is defined as•H close to 1 implies longer memorykikcekf 0 as )(21)(115.0 where21  HHK. Ensor, STAT 4213Spring 2005Fractionally integrated modelsttdurL  )()1(A class of models that exhibit this same behaviorkkLkd)(0•|d|>.5 implies r is nonstationary•0<d<.5 implies r is stationary with long memory also d=H-.5•-.5<d<0 implies r is stationary with short memoryK. Ensor, STAT 4214Spring 2005Prediction•Write as AR()•Requires truncation at p lags•Or re-estimate the filter coefficients based on a lag of p for the given long memory covariance structure•This can be accomplished using the Durbin-Levinson recursive algorithm which takes you from the autocorrelation to the coefficients (and vice versa).–Note similar to the argument we had early on in the autoregessive setting using the Yule-Walker equations to come up with our optimal prediction coefficients.K. Ensor, STAT 4215Spring 2005Long memory extended to GARCH and EGARCH models•Persistence in the volatility can be modeled by extending these long memory constructs to the volatility models such as the GARCH and EGARCH.•See Zivot’s manual for further details.K. Ensor, STAT 4216Spring 2005Testing for persistence•R/S Statistic•Range of deviations from the mean rescaled by the standard deviation.•When rescaled and r’s are iid normal this statistic onverges to the range of a Brownian bridgekjjtkkjjtkttrrrrsQ1111)(min)(max1K. Ensor, STAT 4217Spring 2005GPH Test•Base on the behavior of the spectral density as it approaches 0.•See section 8.3 and 8.4 of Zivot’s