We have discussed that for a stationary time series the ACF decays exponentially to zero as lag increases. Yet for a unit-root nonstationary time series, it can be shown that the sample ACF converges to 1 for all fixed lags as the sample size increases; see Chan and Wei (1988) and Tiao and Tsay (1983). There exist some time series whose ACF decays slowly to zero at a polynomial rate as the lag increases. These processes are referred to as long-memory time series. One such example is the fractionally differenced process defined by
where {at} is a white noise series. Properties of model (2.52) have been widely studied in the literature (e.g., Hosking, 1981). We now summarize some of these properties:
1. If d < 0.5, then xt is a weakly stationary process and has the infinite MA representation
2. If d > − 0.5, then xt is invertible and has the infinite AR representation
3. For − 0.5 < d < 0.5, the ACF of xt is
In particular, ρ1 = d/(1 − d) and
4. For − 0.5 < d < 0.5, the PACF of xt is ϕk, k = d/(k − d) for k = 1, 2, … .
5. For − 0.5 < d < 0.5, the spectral density function f(ω) of xt, which is the Fourier transform of the ACF of xt, satisfies
where ω ∈ [0, 2π] denotes the frequency.
Of particular interest here is the behavior of ACF of xt when d < 0.5. The property says that ρk ∼ k2d−1, which decays at a polynomial, instead of exponential, rate. For this reason, such an xt process is called a long-memory time series. A special characteristic of the spectral density function in Eq. (2.53) is that the spectrum diverges to infinity as ω → 0. However, the spectral density function of a stationary ARMA process is bounded for all ω ∈ [0, 2π].
Earlier we used the binomial theorem for noninteger powers
If the fractionally differenced series (1 − B)dxt follows an ARMA(p, q) model, then xt is called an ARFIMA(p, d, q) process, which is a generalized ARIMA model by allowing for noninteger d.
In practice, if the sample ACF of a time series is not large in magnitude, but decays slowly, then the series may have long memory. As an illustration, Figure 2.22 shows the sample ACFs of the absolute series of daily simple returns for the CRSP value- and equal-weighted indexes from January 2, 1970, to December 31, 2008. The ACFs are relatively small in magnitude but decay very slowly; they appear to be significant at the 5% level even after 300 lags. For more information about the behavior of sample ACF of absolute return series, see Ding, Granger, and Engle (1993). For the pure fractionally differenced model in Eq. (2.52), one can estimate d using either a maximum-likelihood method or a regression method with logged periodogram at the lower frequencies. Finally, long-memory models have attracted some attention in the finance literature in part because of the work on fractional Brownian motion in the continuous-time models.
Figure 2.22 Sample autocorrelation function of absolute series of daily simple returns for CRSP value- and equal-weighted indexes: (a) value-weighted index return and (b) equal-weighted index return. Sample period is from January 2, 1970, to December 31, 2008.
