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Figure 1-25
The estimated seasonal
component of the
accidental deaths
data from ITSM.
1.5 Estimation and Elimination of Trend and Seasonal Components 33
(thousands)
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
1973 1974 1975 1976 1977 1978 1979
Method S2: Elimination of Trend and Seasonal Components by Differencing
The technique of differencing that we applied earlier to nonseasonal data can be
adapted to deal with seasonality of period d by introducing the lag-d differencing
operator ∇d defined by
∇dXt Xt − Xt−d (1 − B d )Xt. (1.5.15)
(This operator should not be confused with the operator ∇ d (1 − B) d defined
earlier.)
Applying the operator ∇d to the model
Xt mt + st + Yt,
where {st} has period d, we obtain
∇dXt mt − mt−d + Yt − Yt−d,
which gives a decomposition of the difference ∇dXt into a trend component (mt −
mt−d) and a noise term (Yt −Yt−d). The trend, mt −mt−d, can then be eliminated using
the methods already described, in particular by applying a power of the operator ∇.
Example 1.5.5 Figure 1.26 shows the result of applying the operator ∇12 to the accidental deaths
data. The graph is obtained from ITSM by opening DEATHS.TSM, selecting Transform>Difference,
entering lag 12, and clicking OK. The seasonal component evident
in Figure 1.3 is absent from the graph of ∇12xt, 13≤ t ≤ 72. However, there still
appears to be a nondecreasing trend. If we now apply the operator ∇ to {∇12xt} by
again selecting Transform>Difference, this time with lag one, we obtain the graph