D.S.G. Pollock 25110.010.511.011.512.01960 1970 1980 1990 2000Figure 6.4 The quarterly sequence of the logarithms of household expenditure in the UK for
the years 1956–2005, together with an interpolated linear trend
00.050.10–0.05
–0.10
–0.15
0 50 100 150Figure 6.5 The residual deviations of the logarithmic expenditure data from the linear trend
of Figure 6.4. The interpolated line, which represents the business cycle, has been synthesized
from the Fourier ordinates in the frequency interval[0,π/ 8 ]
circumference of a circle. A trend line fitted by ordinary least squares regression
would have a lesser gradient, which would raise the final years above the line. This
would be a reflection of the relative prosperity of the times.
The residual deviations of the expenditure data from the trend line of Figure 6.4
are represented in Figure 6.5, and their periodogram is shown in Figure 6.6. Within
this periodogram, the spectral structure extending from zero frequency up toπ/ 8
belongs to the business cycle. The prominent spikes located at the frequencyπ/ 2
and at the limiting Nyquist frequency ofπare the property of the seasonal fluc-
tuations. Elsewhere in the periodogram, there are wide dead spaces, which are
punctuated by the spectral traces of minor elements of noise.
The slowly varying continuous function interpolated through the deviations of
Figure 6.5 has been created by combining a set of sine and cosine functions of
increasing frequencies in the manner of (6.14), with the frequencies extending no
further thanωd=π/8, and by lettingtvary continuously in the interval[0,T). This
is a representation of the business cycle as it affects household expenditure. Observe
that, since it is analytic, the turning points of this function can be determined via
its first derivative.