Palgrave Handbook of Econometrics: Applied Econometrics

296 Investigating Economic Trends and Cycles

The coefficients of the corresponding time-domain filter are obtained by applying
an inverse Fourier transform to this response to give:

ψ(k)=

∫β

α

eikωdω=

1

πk

{sin(βk)−sin(αk)}

=

2

πk

cos{(α+β)k/ 2 }sin{(β−α)k/ 2 }

=

2

πk

cos(γk)sin(δk).

(6.164)

Here,γ=(α+β)/2 is the centre of the pass band andδ=(β−α)/2 is half its width.
The final equality, which follows from the identity sin(A+B)−sin(A−B)=
2 cosAsinB, suggests two interpretations. On the left-hand side is the difference
between the coefficients of two lowpass filters with cut-off frequencies ofβandα
respectively. On the right-hand side is the result of shifting a lowpass filter with a
cut-off frequency ofδso that its center is moved fromω=0toω=γ.
The process of frequency shifting is best understood by taking account of both
positive and negative frequencies when considering the lowpass filter. Then the
pass band covers the interval(−δ,δ). To convert to the bandpass filter, two copies
of the pass band are made that are shifted so that their new centers lie at−γ
andγ. In the limiting case, the copies are shifted to the centers−πandπ. There
they coincide, and we haveψ(k)=2 cos(πk)sin(δk)/πk, which constitutes an ideal
highpass filter. A bandpass filter can also be expressed as the difference of two such
highpass filters.
The coefficients of (6.164) constitute an infinite sequence, which needs to be
truncated to produce a practical filter. Alternatively, a wrapped or circular filter
may be obtained by sampling the frequency response at a set of equally spaced
points in the frequency range[−π,π), equal in number to the elements of the data
sequence. The wrapped filter is obtained by applying the discrete Fourier transform
to the sampled ordinates and it can be applied to the data sequence by circular
convolution.
Thez-transformof a setoffilter coefficients that are symmetric about the central

point and that sum to zero incorporates the factor( 1 −z)( 1 −z−^1 )=−z−^1 ( 1 −z)^2.
This operator is effective in nullifying a linear trend and in reducing a quadratic
trend to a constant. Therefore, such a filter can be applied by linear convolution to
a trended data sequence in the expectation that it will produce a stationary filtered
sequence.
This is one of the attractions of the truncated bandpass filter that has been pro-
posed to economists by Baxter and King (1999). To ensure that the coefficients
of the truncated filter do sum to zero, the filter can be expressed as the difference
between two truncated versions of the ideal lowpass filter, of which the coefficients
have been scaled so as to sum to unity.
The truncated filter has several disadvantages. In the first place, the truncation
leads to the phenomenon of leakage that has already been described in section 6.8.
This is illustrated by Figure 6.19. Also, a finite-order moving-average filter with