ABSTRACT
In the previous chapters, we have considered ARMA, ARUMA, seasonal, signal-plus-noise, and ARCH/GARCH processes. For the current discussion, we restrict ourselves to nonseasonal models. Stationary ARMA(p,q) processes have autocorrelations that decay (go to zero) exponentially as k gets large. That is, ρk ≃ α|k|, |α| < 1 as k → ∞ where k is the lag. We use the notation fk ≃ gk as k → ∞ to indicate that lim ( )k k kf g c→∞ =/ , where c is a finite nonzero constant. For example, for an AR(1) model with ϕ1 = 0.9, ρk = 0.9|k|. In Example 3.10, it was shown that the autocorrelation function for the AR(3) model (1 − 0.95B)(1 − 1.6B + 0.9B2)Xt = at is given by
ρ πk k k k= + +0 66 0 95 0 3518 0 95 2 0 09 1 31. . . . sin . .( ) ( ) [ ( ) ],
so that in both cases ρk ≃ α|k| as k → ∞. In Chapter 5 we viewed ARUMA(p,d,q) processes as limits of ARMA processes as one or more roots of the characteristic equation approach the unit circle uniformly. Since the autocorrelation of an ARMA process decays exponentially, such processes are referred to as “short-memory” processes. It was noted in Section 5.2 that the correlation function decays more slowly as one or more roots of the characteristic equation approach the unit circle, and in fact the extended autocorrelations of the limiting ARUMA(p,d,q) process with integer d > 0 do not decay as k goes to infinity. Signal-plus-noise processes also have autocorrelations that do not decay, but these models involve functional restrictions that often prove to be too limiting in practice. In Chapter 6, we showed that the correlation-driven (ARMA and ARUMA) models are more adaptive than the signal-plus-noise models in that the ARMA/ARUMA models “react” strongly to the most current data and very little to the data of the distant past.
