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.

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

3
Fractionally integrated models
A class of models that exhibit this same behavior

• dgt.5 implies r is nonstationary
• 0ltdlt.5 implies r is stationary with long memory
  also dH-.5
• -.5ltdlt0 implies r is stationary with short memory

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.

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 Zivots manual for further details.

6
Testing for persistence

• R/S Statistic
• Range of deviations from the mean rescaled by the
  standard deviation.
• When rescaled and rs are iid normal this
  statistic onverges to the range of a Brownian
  bridge

7
GPH Test

• Base on the behavior of the spectral density as
  it approaches 0.
• See section 8.3 and 8.4 of Zivots manual.