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6.1 ARIMA Models for Nonstationary Time Series 181
useful for representing data with trend (see Sections 1.5 and 6.2). It should be noted,
however, that ARIMA processes can also be appropriate for modeling series with no
trend. Except when d 0, the mean of {Xt} is not determined by equation (6.1.1),
and it can in particular be zero (as in Example 1.3.3). Since for d ≥ 1, equation
(6.1.1) determines the second-order properties of {(1 − B) d Xt} but not those of {Xt}
(Problem 6.1), estimation of φ, θ, and σ 2 will be based on the observed differences
(1 − B) d Xt. Additional assumptions are needed for prediction (see Section 6.4).
Example 6.1.1 {Xt} is an ARIMA(1,1,0) process if for some φ ∈ (−1, 1),
Figure 6-1
200 observations of the
ARIMA(1,1,0) series
Xt of Example 6.1.1.
(1 − φB)(1 − B)Xt Zt, {Zt} ∼WN 0,σ 2 .
We can then write
where
Xt X0 +
t
Yj, t ≥ 1,
j1
Yt (1 − B)Xt
∞
j0
φ j Zt−j.
A realization of {X1,...,X200} with X0 0, φ 0.8, and σ 2 1 is shown in
Figure 6.1, with the corresponding sample autocorrelation and partial autocorrelation
functions in Figures 6.2 and 6.3, respectively.
A distinctive feature of the data that suggests the appropriateness of an ARIMA
model is the slowly decaying positive sample autocorrelation function in Figure 6.2.
0 20 40 60 80
0 50 100 150 200