D.S.G. Pollock 245wherey(t)=lnY(t),τ(t)=lnT(t),c(t)=lnC(t),s(t)=lnS(t)andε(t)=lnE(t).
An additional assumption, which might be plausible, is that the componentsc(t),
s(t), andε(t)have amplitudes that remain roughly constant over time.
In the absence of extraneous information that correlates them with other vari-
ables, it is impossible to distinguish the components of (6.3) perfectly, one from
another, unless they occupy separate frequency bands. If their bands do overlap,
then any separation of the components will be tentative and doubtful. Thus, a
sequence that is deemed to represent one of the components will comprise, to
some extent, elements that rightfully belong to the other components.
However, as we shall see, the components of an econometric data sequence often
reside within bands of frequencies that are separated by wide dead spaces where
there are no spectral elements of any significance. The possibility of definitely
separating the components is greater than analysts are likely to perceive unless
they work in the frequency domain.
The exception concerns the separation of the business cycle from the trend.
These components are liable to be merged within a single spectral structure, and
there is no uniquely appropriate way of separating them. Their separation depends
upon adopting whatever convention best suits the purposes of the analysis. No
such difficulties will affect the simple schematic model of the business cycle that
we shall consider in the next section.
6.2 A schematic model of the business cycle
In order to extract the modulating components from the data, it is also neces-
sary to remove the trend component fromY(t). To understand what is at issue in
detrending the data, it is helpful to look at a simple schematic model comprising
an exponential growth trajectoryT(t)=βexp{rt}, withr>0, that is modulated
by a exponentiated cosine functionC(t)=exp{γcos(ωt)}to create a model for the
trajectory of aggregate income:
Y(t)=βexp{rt+γcos(ωt)}. (6.4)The resulting business cycles, which are depicted in Figure 6.1, have an asymmetric
appearance. Their contractions are of lesser duration than their expansions and
they become shorter as the growth raterincreases.
Eventually, when the rate exceeds a certain value, the periods of contraction will
disappear and, in place of the local minima, there will be only points of inflection.
In fact, the condition for the existence of local minima is thatωγ >r, which is
to say the product of the amplitude of the cycles and their angular velocity must
exceed the growth rate of the trend.
Next, we take logarithms of the data to obtain a model, represented in Figure 6.2,
that has additive trend and cyclical components. This gives:
ln{Y(t)}=y(t)=μ+rt+γcos(ωt), (6.5)