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186 Chapter 6 Nonstationary and Seasonal Time Series Models
ACF
-1.0 -0.5 0.0 0.5 1.0
Figure 6-8
The sample ACF of the
data in Figure 6.7. Lag
0 10 20 30 40
The autocorrelation function of the model (6.1.6) can be derived by noting that
1 − 2r −1 cos ω B + r −2 B 2 1 − r −1 e iω B 1 − r −1 e −iω B
(6.1.7)
and using (3.2.12). This gives
where
ρ(h) r
−h sin(hω + ψ)
, h ≥ 0, (6.1.8)
sin ψ
tan ψ r2 + 1
r2 tan ω.
− 1
It is clear from these equations that
(6.1.9)
ρ(h) → cos(hω) as r ↓ 1. (6.1.10)
With r 1.005 and ω π/3 as in the model generating Figure 6.7, the model
ACF (6.1.8) is a damped sine wave with damping ratio 1/1.005 and period 6. These
properties are reflected in the sample ACF shown in Figure 6.8. For values of r closer
to 1, the damping will be even slower as the model ACF approaches its limiting form
(6.1.10).
If we were simply given the data shown in Figure 6.7, with no indication of
the model from which it was generated, the slowly damped sinusoidal sample ACF
with period 6 would suggest trying to make the sample ACF decay more rapidly
by applying the operator (6.1.7) with r 1 and ω π/3, i.e., 1 − B + B 2 .Ifit
happens, as in this case, that the period 2π/ω is close to some integer s (in this case
6), then the operator 1 − B s can also be applied to produce a series with more rapidly