Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 287

0.00

0.25

0.50

0.75

1.00

1.25

–0.25

−π −π/ 2 0 π/ 2 π

Figure 6.14 The frequency response of the 17-point wrapped filter defined over the interval
[−π,π). The values at the Fourier frequencies are marked by circles

By applying an inverse discrete Fourier transform to these weights, the co-
efficients of a circular filter are obtained, of which the values are given by:

β◦(k)=

⎧

⎪⎪⎨

⎪⎪⎩

2 d+ 1

T

, ifk=0,

sin([d+ 1 / 2 ]ω 1 k)

Tsin(ω 1 k/ 2 )

, for k = 1, ... , [T/2],

(6.152)

whereω 1 = 2 π/T. These coefficients would be obtained by wrapping coefficients
of (6.150) around a circle of circumferenceTand adding the overlying values:

βk◦=

∑∞

j=−∞

βjT+k. (6.153)

Applying the wrapped filter to the finite data sequence via a circular convolution
is equivalent to applying the original filter to an infinite periodic extension of the
data sequence.
The function of (6.152) is just an instance of the Dirichlet kernel – see Pollock
(1999), for example. Figure 6.14 depicts the frequency response for this filter at
the Fourier frequencies, whereλj=0, 1 in the case whereωd=π/2. It also depicts
the continuous frequency response that would be the consequence of applying an
ordinary filter with these coefficients to a doubly-infinite data sequence.

6.8.1 Applying the Fourier method to trended data

In an ideal application of the Fourier method, it should be possible to wrap the
data sequenceyt;t=0,...,T−1 seamlessly around the circle, such that there is no
disjunction at the point where the head of the sequence joins the tail. To achieve
such an effect, it is common to taper the data so as reduce both ends to zero. To
avoid corrupting the sample data, the taper can be applied to some extrapolations
of the ends of the sample. However, a data sequence that follows a linear trend
is not amenable to tapering, since there is liable to be a radical disjunction at the
point where the head joins the tail.