Long memory or long range dependence

1. Long 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. K. Ensor, STAT 421

2. Behavior 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 K. Ensor, STAT 421

3. Fractionally integrated models A class of models that exhibit this same behavior • |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 memory K. Ensor, STAT 421

4. Prediction • 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 421

5. Long 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 421

6. Testing 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 K. Ensor, STAT 421

7. GPH Test • Base on the behavior of the spectral density as it approaches 0. • See section 8.3 and 8.4 of Zivot’s manual. K. Ensor, STAT 421