D.S.G. Pollock 29700.250.500.751.001.250 π/ 4 π/ 2 3 π/ 4 πFigure 6.19 The frequency response of the truncated bandpass filter of 25 coefficients super-
imposed upon the ideal frequency response. The lower cut-off point is atπ/15 radians
(11.25◦), corresponding to a period of 6 quarters, and the upper cut-off point is atπ/3 radians
(60◦), corresponding to a period of 32 quarters
constant coefficients is incapable of reaching the ends of the sample. This problem
occasions a trade-off between the accuracy of the approximation to the ideal filter,
which increases with the number of coefficients, and the end-of-sample problem,
which is exacerbated by increasing the span of the filter.
There are numerous ways of overcoming the end-of sample problem, including
the obvious recourse of extrapolating the sample by forecasting and backcasting
it with the help of an ARIMA model that purports to describe the data. Another
recourse is to extend the sample by attaching its symmetric reflection to either end.
However, if the data are strongly trended this will tend to increase the values at
the beginning of the sample and to decrease the values at the end, relative to the
values obtained via a linear extrapolation of the sample.
A circular filter should not be applied directly to a trended data sequence. When
such a sequence is wrapped around a circle there is liable to be a radical disjunction
where the beginning and the end of the sample are joined. The effects of this
disjunction are liable to be carried into the filtered sequence in a manner that
does not affect the ordinary linear filter. One way of overcoming this difficulty
is to apply the circular filter to data that have been reduced to stationarity by
differencing. Thereafter, the filtered differenced sequence can be cumulated to
obtain an estimate of the business cycle component.
Example The filter of Baxter and King (1999) is a time-invariant moving average
comprising 2q+1 of the central coefficients of the ideal infinite-order bandpass
filter, which are symmetrically disposed around the central value. These coefficients
should rescaled so that they sum to zero.
The elements of the filtered sequence are given by:
xt=φqyt−q+φq− 1 yt−q+ 1 +···+φ 1 yt− 1 +φ 0 yt+φ 1 yt+ 1 +···+φq− 1 yt+q− 1 +φqyt+q.(6.165)