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30 Chapter 1 Introduction
If the operator ∇ is applied to a linear trend function mt c0 +c1t, then we obtain the
constant function ∇mt mt − mt−1 c0 + c1t − (c0 + c1(t − 1)) c1. In the same
way any polynomial trend of degree k can be reduced to a constant by application of
the operator ∇ k (Problem 1.10). For example, if Xt mt +Yt, where mt k
j0
and Yt is stationary with mean zero, application of ∇ k gives
∇ k Xt k!ck +∇ k Yt,
a stationary process with mean k!ck. These considerations suggest the possibility,
given any sequence {xt} of data, of applying the operator ∇ repeatedly until we find
a sequence ∇k
xt that can plausibly be modeled as a realization of a stationary
process. It is often found in practice that the order k of differencing required is quite
small, frequently one or two. (This relies on the fact that many functions can be
well approximated, on an interval of finite length, by a polynomial of reasonably low
degree.)
Example 1.5.3 Applying the operator ∇ to the population values {xt,t 1,...,20} of Figure 1.5, we
find that two differencing operations are sufficient to produce a series with no apparent
trend. (To carry out the differencing using ITSM, select Transform>Difference,
enter the value 1 for the differencing lag, and click OK.) This replaces the original
series {xt} by the once-differenced series {xt − xt−1}. Repetition of these steps gives
the twice-differenced series ∇ 2 xt xt − 2xt−1 + xt−2, plotted in Figure 1.23. Notice
that the magnitude of the fluctuations in ∇ 2 xt increases with the value of xt. This effect
can be suppressed by first taking natural logarithms, yt ln xt, and then applying the
operator ∇ 2 to the series {yt}. (See also Figures 1.1 and 1.17.)
Figure 1-23
The twice-differenced series
derived from the population
data of Figure 1.5.
(millions)
-80 -60 -40 -20 0 20 40 60 80 100
1820 1840 1860 1880 1900 1920 1940 1960 1980
cjt j