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188 Chapter 6 Nonstationary and Seasonal Time Series Models
ACF
-0.5 0.0 0.5 1.0
Figure 6-10
The sample ACF
of (1 − B6 )Xt with
0 10 20 30 40
{Xt } as in Figure 6.7. Lag
the Australian monthly red wine sales from January 1980 through October 1991,
and Figure 1.17 shows how the increasing variability with sales level is reduced
by taking natural logarithms of the original series. The logarithmic transformation
Vt ln Ut used here is in fact appropriate whenever {Ut} is a series whose standard
deviation increases linearly with the mean. For a systematic account of a general class
of variance-stabilizing transformations, we refer the reader to Box and Cox (1964).
The defining equation for the general Box–Cox transformation fλ is
fλ(Ut)
−1 λ
λ (Ut − 1), Ut ≥ 0,λ>0,
ln Ut, Ut > 0,λ 0,
and the program ITSM provides the option (Transform>Box-Cox) of applying fλ
(with 0 ≤ λ ≤ 1.5) prior to the elimination of trend and/or seasonality from the data.
In practice, if a Box–Cox transformation is necessary, it is often the case that either
f0 or f0.5 is adequate.
Trend and seasonality are usually detected by inspecting the graph of the (possibly
transformed) series. However, they are also characterized by autocorrelation functions
that are slowly decaying and nearly periodic, respectively. The elimination of trend
and seasonality was discussed in Section 1.5, where we described two methods:
i. “classical decomposition” of the series into a trend component, a seasonal component,
and a random residual component, and
ii. differencing.