D.S.G. Pollock 303An opinion to which we adhere in this chapter is that the trend is best regarded as
an analytic device, as opposed to an object that subsists within the data that might
be uncovered by an appropriate technique. If the trend is to be regarded as an
artificial benchmark, then its definition depends largely on what one is intending
to measure.
In some cases, when the economy has had an uninterrupted progress, it is
straightforward to define an appropriate benchmark. A case in point has been the
UK economy over the years 1956–2005, of which the aggregate consumption is
portrayed in Figures 6.4–6.6. For that period, a log-linear trend function provides
a datum about which to measure the cyclical variations in consumption.
In other eras and over longer periods, where there have been substantial disrup-
tions to the progress of the economy, the matter becomes more complicated. To
highlight the major disruptions, it is appropriate to fit a polynomial of a limited
degree over the entire span of the data. An example is provided by Figure 6.21.
There, a fourth-degree polynomial, which adheres quite well to the data in the
main, also reveals the uncommon circumstances in the periods surrounding the
ends of the two world wars.
If the purpose is also to illustrate the normal workings of the economy, then
it may be appropriate to fit similar polynomial trends of low degrees to the
sub-periods that did not experience any disruptions. The overall result will be a
segmented curve; and the issue arises of how to join the segments.
The answer that is favored in this chapter is illustrated in Figure 6.23, which
shows the effect of a filter with a variable smoothing parameter. The resulting
curve comprises segments that are virtually straight lines, interspersed by short
segments with rapidly changing slopes.
The disjunctions that occur within the data sequence as a consequence of disrup-
tions and breaks give rise to spectra that extend over the entire frequency range.
Unless the breaks are absorbed within the trend, the residual sequence will fail to
manifest the band-limited structure that we might expect to see in normal periods.
Therefore, one of the criteria of a successful elimination of a break is the restora-
tion of a band-limited spectral structure to the trend cycle component within the
residual sequence.
The recognition that, at least for limited periods, the trend cycle complex is liable
to be confined to a limited frequency band gives rise to further opportunities, but
it also poses additional problems. The opportunities arise from the possibility of
using a Fourier synthesis to create a continuous analytic function to represent the
business cycle in isolation or the trend and cycle in combination.
In Figure 6.5, the business cycle has been synthesized from a limited number
of the low-frequency Fourier ordinates of the linearly detrended logarithmic data.
The combination of the trend and the cycle can be formed by adding the business
cycle function to the linear trend of Figure 6.4. The result is shown in Figure 6.15.
The analytic nature of these functions means that they are amenable to dif-
ferentiation, and their turning points are identified as the points where the first
derivatives are zero-valued. This method of finding the turning points may be con-
trasted with the very different procedure of Bry and Boschan (1971), which had