"Frontmatter". In: Analysis of Financial Time Series

LONG-MEMORY MODELS 73=(k − d − 1)!k!(−d − 1)! .3. For −0.5 < d < 0.5, the ACF of x t isρ k =In particular, ρ 1 = d/(1 − d) andd(1 + d) ···(k − 1 + d), k = 1, 2,....(1 − d)(2 − d) ···(k − d)ρ k ≈(−d)!(d − 1)! k2d−1 , as k →∞.4. For −0.5 < d < 0.5, the PACF of x t is φ k,k = d/(k − d) for k = 1, 2,....5. For −0.5 < d < 0.5, the spectral density function f (ω) of x t , which is theFourier transform of the ACF of x t , satisfieswhere ω ∈[0, 2π] denotes the frequency.f (ω) ∼ ω −2d , as ω → 0, (2.45)Of particular interest here is the behavior of ACF of x t when d < 0.5. The propertysays that ρ k ∼ k 2d−1 , which decays at a polynomial, instead of exponential, rate. Forthis reason, such an x t process is called a long-memory time series. A special characteristicof the spectral density function in Eq. (2.45) is that the spectrum divergesto infinity as ω → 0. However, the spectral density function of a stationary ARMAprocess is bounded for all ω ∈[0, 2π].Earlier we used the binomial theorem for noninteger powers(1 − B) d =∞∑( )(−1) k dB k ,kk=0( d=k)d(d − 1) ···(d − k + 1).k!If the fractionally differenced series (1− B) d x t follows an ARMA(p, q) model, thenx t is called an ARFIMA(p, d, q) process, which is a generalized ARIMA model byallowing for noninteger d.In practice, if the sample ACF of a time series is not large in magnitude, but decaysslowly, then the series may have long memory. As an illustration, Figure 2.17 showsthe sample ACFs of the absolute series of daily simple returns for the CRSP valueandequal-weighted indexes from July 3, 1962 to December 31, 1997. The ACFs arerelatively small in magnitude, but decay very slowly; they appear to be significantat the 5% level even after 300 lags. For more information about the behavior ofsample ACF of absolute return series, see Ding, Granger, and Engle (1993). For thepure fractionally differenced model in Eq. (2.44), one can estimate d using either amaximum likelihood method or a regression method with logged periodogram at the