298 Investigating Economic Trends and Cycles
Given a sampley 0 ,y 1 ,...,yT− 1 ofTdata points, onlyT− 2 qprocessed values
xq,xq+ 1 ,...,xT−q− 1 are available, since the filter cannot reach the ends of the
sample, unless some extrapolations are added to it.
To overcome this difficulty, Christiano and Fitzgerald (2003) have used a filter
that comprises selections of the coefficients of the ideal filter which vary as one
moves through the sample. At all times, the central coefficient of the ideal filter
is aligned with the current data value. The remainder of the selection consists of
the coefficients on either side that fall within the data window. Thus, the filtered
values are weighted combinations of all of the sample elements.
In the case of data that might have been generated by a random-walk process,
it is proposed to supplement the weighted sum by two additional terms based on
the first and the final sample elements, which are the appropriate predictors of the
elements of the process that fall outside the data window. In that case, the elements
of the filtered sequence will be given by:
xt=Ay 0 +φty 0 +···+φ 1 yt− 1 +φ 0 yt
+φ 1 yt+ 1 +···+φT− 1 −tyT− 1 +ByT− 1 ,(6.166)whereAandBare sums of the coefficients of the ideal filter that lie beyond either
end of the data window. Since the filter coefficients must sum to zero, it follows
that:
A=−(
1
2φ 0 +φ 1 +···+φt)
and B=−(
1
2φ 0 +φ 1 +···+φT−t− 1). (6.167)
For data that appear to have been generated by a first-order random walk with a
constant drift, it is appropriate to extract a linear trend before filtering the residual
sequence. In fact, this has proved to be the usual practice in most circumstances.
It has been proposed to subtract from the data a linear functionf(t)=α+βt
interpolated through the first and the final data points, such thatα=y 0 and
β =(yT− 1 −y 0 )/T. In that case, there should beA=B=0. This procedure
is appropriate to seasonally adjusted data. For data that manifest strong seasonal
fluctuations, such as the UK expenditure data, a line can be fitted by least squares
through the data points of the first and the final years. Figure 6.20 shows the effect
of the application of the filter to the UK data adjusted in this manner.
Figure 6.20 can be compared with Figure 6.5 and Figure 6.16, both of which also
purport to show the business cycles that affected the data in question. It is clear that
the bandpass filter fails to transmit the appropriate cyclical fluctuations. An expla-
nation for the failure can be found in Figure 6.6, which shows the periodogram of
the linearly detrended data.
The highlighted band in Figure 6.6 covers the frequency interval[π/16,π/ 3 ]
which, according to Baxter and King (1999), is the frequency range that defines
the business cycle. However, this figure indicates that only a small part of the
low-frequency component falls within the interval. Therefore, it appears that the
definition is at fault. In fact, the leakage that is associated with the filter does
allow some of the low-frequency power of the elements that reside in the interval
[0,π/ 16 ]to pass into the filtered sequence.