D.S.G. Pollock 295from a period that was characterized by uninterrupted economic growth at annual
rates that varied little. Therefore, the method of polynomial detrending works
well.
In other eras, where there have been marked disruptions, the polynomial method
is less appropriate. In order to serve as a benchmark for the ensuing periods of
stability, the trend must be made to absorb the disruptions, which implies that it
must have a segmented structure. In section 6.10.2 we will describe a method for
achieving this.
A prescription that is to be found in the pioneering work of Burns and Mitchell
(1946) is that the business cycle should be defined in terms of a limited band of
frequencies. A modern interpretation of this is that the band should comprise the
sinusoidal elements of the data that have cyclical durations of no more that eight
years and of no less than a year and a half. Such cycles can be extracted from the
data via a bandpass filter, as we will discuss below.
The definition seems arbitrary, but it might be justified by proposing that the
reactions of economic agents to cycles within the frequency band differ from their
reactions to cycles at other frequencies. Thus, it might be argued that their adapta-
tions to cycles of more than eight years’ duration occur mainly at a subconscious
level, whereas cycles of a lesser duration incite conscious reactions.
The growth of an economy may be likened to a process of biological growth,
which is affected by events that occur in the course of its evolution. Therefore,
a stochastic trend based on the accumulation of random increments has been
seen as an appropriate model for an economic trend. This idea has inspired
the Beveridge–Nelson decomposition of an ARIMA process, which depicts the
trend as an accumulation of disturbances that also give rise to accompanying
fluctuations.
In practice, the Beveridge–Nelson decomposition depends upon a linear filter
that is applied to the data sequence like any other filter. However, the filtered
sequence that represents the trend is liable to include a substantial proportion of
the high-frequency elements of the data and for that reason it may be regarded as
unacceptable.
6.10.1 Bandpass filters
In an attempt to separate a business cycle component from the trend, economists
have been resorting increasingly to the use of bandpass filters to implement the
definition of Burns and Mitchell (1946). This appears to be in response to the fact
that the structural time series methods, which use ARIMA models to represent the
unobserved components, fail to isolate the business cycle.
An ideal bandpass filter that transmits all elements within the frequency range
[α,β], and blocks all others, has the following frequency response:
ψ(ω)={
1, if|ω|∈(α,β);
0, otherwise.
(6.163)