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24 Chapter 1 Introduction
Figure 1-17
The natural logarithms
of the red wine data.
6.5 7.0 7.5 8.0
1980 1982 1984 1986 1988 1990 1992
model for the process Yt, to analyze its properties, and to use it in conjunction with
mt and st for purposes of prediction and simulation of {Xt}.
Another approach, developed extensively by Box and Jenkins (1976), is to apply
differencing operators repeatedly to the series {Xt} until the differenced observations
resemble a realization of some stationary time series {Wt}. We can then use the theory
of stationary processes for the modeling, analysis, and prediction of {Wt} and hence
of the original process. The various stages of this procedure will be discussed in detail
in Chapters 5 and 6.
The two approaches to trend and seasonality removal, (1) by estimation of mt
and st in (1.5.1) and (2) by differencing the series {Xt}, will now be illustrated with
reference to the data introduced in Section 1.1.
1.5.1 Estimation and Elimination of Trend in the Absence of Seasonality
In the absence of a seasonal component the model (1.5.1) becomes the following.
Nonseasonal Model with Trend:
where EYt 0.
Xt mt + Yt, t 1,...,n, (1.5.2)
(If EYt 0, then we can replace mt and Yt in (1.5.2) with mt + EYt and Yt − EYt,
respectively.)