288 Investigating Economic Trends and Cycles
The periodic extension of the linearly trended sequence, which would be gen-
erated by traveling around the circle indefinitely, has a saw-tooth profile. The
corresponding spectrum or periodogram has aone-over-fprofile that descends, as
the frequency increases, in the manner of a rectangular hyperbola, from a high
point that is adjacent to the zero frequency to a low point at the limiting fre-
quency. Unless the data are adequately detrended, such a spectrum will serve to
conceal all but the most prominent of the harmonic characteristics of the data.
There are two simple ways in which the data may be detrended. The first, which
has been described already in section 6.7.3, is to apply the difference operator to the
data as many times as are necessary to reduce them to stationarity. The components
that are extracted by filtering the differenced data can be reinflated, in the manner
indicated by equations (6.133)–(6.137), to obtain the components of the original
data.
We denote the data byyand their differences byg=Q′y. The filtered sequence
that underlies the trend is denoted bydand the vector of initial conditions byd∗.
Then, if we set=I, the relevant equations for delivering the estimatexof the
trend component are:
x=S∗d∗+Sd and d∗=(S∗′S∗)−^1 S′∗(y−Sd). (6.154)The detrended sequence ish=y−x. Underlying the detrended sequence is the
filtered sequencek=g−d, from which the detrended data component may be
obtained directly via the equations:
h=S∗k∗+Sk and k∗=−(S′∗S∗)−^1 S′∗Sk. (6.155)Another way of reversing the effects of a differencing operation that has been
applied to the data to reduce them to stationarity is to re-inflate the Fourier ordi-
nates of the filtered sequence, using values from the frequency response function of
the anti-differencing summation operator. Once the ordinates had been reinflated
within the frequency domain, they can be transformed into the time domain to
produce the filtered sequence.
This method is applicable only to components that are bounded away from
the zero frequency, since the summation operator has infinite gain at zero. (See
Figure 6.8.) However, if one wishes to apply a lowpass filter to the data, then one
has the option of applying the complementary highpass filter and of subtracting
the filtered sequence from the original data to generate the lowpass component.
The second way of detrending the data is to extract a polynomial component
via an ordinary or a generalized least squares regression according to the formula
of (6.120). The formula will allow greater weight to be given to the points at both
ends of the sample, to ensure that the interpolated curve passes through their
midst. This can be achieved by allowing−^1 to be a diagonal matrix with large
values at the ends. In this way, a disjunction in the wrapped version of the residual
sequence, or in its periodic extension, can be avoided.
Example Figure 6.15 shows the logarithms of the data on aggregate household
expenditure in the UK for the years 1956–2005, through which a smooth trajectory