D.S.G. Pollock 2930.000.050.100.15–0.05
–0.1
1960 1970 1980 1990 2000Figure 6.16 The residuals from fitting a polynomial of degree 4 to the logarithmic expendi-
ture data. The interpolated line, which represents the business cycle, has been synthesized
from the Fourier ordinates in the frequency interval[0,π/ 8 ]
00.050.100.150 π/ 4 π/ 2 3 π/ 4 πFigure 6.17 The periodogram of the data sequence of Figure 6.16
the data by a least squares regression. The effect is to remove some of the power
from the Fourier ordinates adjacent to the zero frequency.
The residual sequence from this polynomial interpolation is show in Figure 6.16,
together with an interpolated function that has been synthesized from the Fourier
ordinates that lie in the interval[0,π/ 8 ]. This, function, which purports to repre-
sent the business cycle, is devoid of any seasonal fluctuations. Figure 6.17 displays
the periodogram of the residual sequence.
After the removal of all elements of frequencies in excess ofπ/8 the data may be
resampled at 1/8th of the original rate of observation. This simple fractional rate
is a convenient one, since it implies taking one in every eight of the anti-aliased
data points. In that case, there is no need to synthesize a continuous function for
the purpose of resampling the data.
The periodogram of the sub-sampled anti-aliased data is show in Figure 6.18
with the parametric spectrum of an estimated AR(3) model superimposed. The
periodogram represents a rescaled version of the part of the periodogram of Figure
6.17 that occupies the frequency range[0,π/ 8 ]and it appears to be well represented
by the parametric spectrum.
The continuous band-limited function of Figure 6.16 can be recovered from the
sub-sample by associating to each of its elements an appropriately scaled Dirichlet