290 Investigating Economic Trends and Cycles
The ordinates at timest=0,...,T−1 of the business cycle that is represented
in Figure 6.5 have been obtained by a Fourier method; but they might have been
obtained by applying the Butterworth filter of ordern=6 and with a nominal
cut-off frequency ofωd=π/4 radians, of which the gain is depicted in Figure 6.11.
The principal advantage of the Fourier method, in this context, lies in the ease with
which a continuous function can be synthesised from the Fourier coefficients.
Difficulties do arise when an attempt is made to estimate the parameters of an
ARMA model from data such as those of Figure 6.5. A natural objective is to attempt
to characterize the business cycle via the parameters of a fitted ARMA model. Such
a model is liable to be applied to a seasonally adjusted version of the data, for which
the periodogram will lack the spectral spike at the seasonal frequency ofπ/2 and
at the harmonic frequency ofπ.
An AR(2) model with complex roots is the simplest of the models that might be
appropriate to the purpose. The modulus of its roots should reveal the damping
characteristics of the cycles, and their argument should indicate the angular veloc-
ity or, equivalently, the length, of the cycles. However, such a model will invariably
deliver estimates that imply real-valued roots, which fail adequately to represent
the dynamics of the business cycle. (See Pagan, 1997, for example.)
The problem of estimating the business cycle also affects the model-based
approaches to econometric signal extraction, which depend upon the prior estima-
tion of an aggregate ARIMA model or upon the estimation of ARIMA components.
A business cycle component is usually missing from such models, since the
estimation fails to deliver the appropriate complex roots. However, it is straight-
forward to include a business cycle component with a pre-specified frequency in a
disaggregated structural model. (See Harvey, 1985, for example.)
To obtain parametric estimates of the business cycle, it is necessary to remove
from the data all but the relevant low-frequency components. This is achieved
by selecting the relevant Fourier coefficients from which the business cycle can
be constituted via a Fourier synthesis in the manner of (6.14). Thereafter, it is
necessary to sample the continuous function at a rate that will ensure that the
Nyquist frequencyπcorresponds to the highest frequency that is present in the
component. A successful ARMA model which represents the complex dynamics of
the business cycle can be estimated from the resampled data sequence.
The Shannon–Whittaker sampling theorem indicates that the resampled data
contains sufficient information to reconstitute the continuous business cycle
function.
6.9.1 The Shannon–Whittaker sampling theorem
Letx(t)be a square integrable continuous signal of which the Fourier transform
ξ(ω)is band limited to the frequency interval[−ωd,ωd]. Then the signal can be
recovered from its sampled ordinates provided that these are separated by a time
interval of no more thanπ/ωd, which is to say that the sinusoidal element of the
highest frequency within the signal must take at least two sampling intervals to
complete a cycle.