D.S.G. Pollock 28910.010.511.011.512.01960 1970 1980 1990 2000Figure 6.15 The logarithms of quarterly household expenditure in the UK, for the years
1956–2005, together with an interpolated trend
has been interpolated. This has been obtained by selecting the Fourier coefficients
of the twice-differenced data that correspond to frequencies in the interval[0,π/ 8 ].
This frequency band has been chosen in the light of the periodogram of Figure 6.6,
which shows that it contains an isolated spectral structure.
The sequence that has been synthesized from these coefficients has been
reinflated in the manner indicated by (6.154) to produce the trajectory. The result
of this procedure is a composite of the trend and the business cycle. The same tra-
jectory of aggregate expenditure would have been obtained by adding the business
cycle that is depicted in Figure 6.5 to the linear trend of Figure 6.4.
6.9 Band-limited processes
The majority of the methods that we have described for extracting the components
of an econometric data sequence presuppose that the data can be described by a
univariate ARIMA model. The spectral density function of an ARIMA process is
supported on the entire frequency interval[0,π], where its ordinates are strictly
positive with the possible exception of a few zero-valued ordinates that constitute
a set of measure zero. Such zero values will be attributable to the presence of unit
roots within the moving-average operator.
It is commonly assumed that the component parts of an aggregate economet-
ric sequence can also be described by ARIMA models. It is on this basis that the
WK filters are derived. However, reference to the periodogram of Figure 6.6 and
to others like it suggests that the components often reside within strictly limited
frequency bands which are separated by dead spaces where the spectral ordinates
are virtually zeros.
In many circumstances, the disparity between the assumptions underlying the
WK filters and the nature of the data to which they are applied has no adverse
effects. A lowpass filter that achieves a gradual transition from a pass band to a stop
band within the region of a spectral dead space will be as effective in extracting a
low-frequency trend component as is a frequency-domain filter that achieves an
abrupt transition between two adjacent Fourier frequencies.