294 Investigating Economic Trends and Cycles
00.00020.00040.00060 π/ 4 π/ 2 3 π/ 4 πFigure 6.18 The periodogram of the sub-sampled anti-aliased data with the parametric
spectrum of an estimated AR(3) model superimposed
kernel and, thereafter, by adding these kernels. This demonstrates the one-to-one
correspondence that exists between the continuous function and the sub-sampled
sequence. This is precisely the one-to-one correspondence that exists between the
periodic functionz(t), synthesized by equation (6.14), and its sampled ordinates
{zτ=z(τT/N);τ=0, 1,...,N− 1 }.
The AR(3) model that underlies the spectral density function of Figure 6.18 pro-
vides a statistical description both of the continuous band-limited function of
Figure 6.16 and of the ordinates sampled from it at the rate of one observation
in eight sample periods.
6.10 Separating the trend and the cycles
The remaining issue to be discussed in this chapter is the matter of separating
the trend of an economic data sequence from the cycles that surround it. This
is a difficult problem. The trend and the cycles are combined within the same
spectral structure and there is rarely any indication, within the periodogram, of
where the trend ends and the cycles begin. In the absence of objective criteria
for achieving a separation, the definition of the trend is liable to reflect the pur-
poses of the study as well as the circumstances of the economy over the period in
question.
A simple prescription that was offered by the pioneering econometrician Tint-
ner (1940, 1953) is that the trend should contain no cyclical motions. This can
be interpreted to mean that, if the trend is a differentiable function, then its first
derivative should have no more than one local maximum or one local minimum.
Such a function can be described as a pure trend. A polynomial function of low
degree fitted to the data by least squares regression is liable to fulfill the require-
ment and it can provide an appropriate benchmark for measuring the cyclical
variations.
An example of such a trend is the linear function of Figure 6.5, which has been
applied to logarithmic data. When a quadratic function was fitted to the data
by least squares regression, the result was virtually a straight line. The data are