Palgrave Handbook of Econometrics: Applied Econometrics

392 Structural Time Series Models

notice that this is 1 at the zero frequency and decreases monotonically to zero asω
approachesπ. This behavior enforces the interpretation of (9.3) as a lowpass filter,
and the corresponding detrending filter, 1−wμ(L), is the highpass filter derived
from it. We shall return to this issue in the next section.

9.2.3 Higher-order trends and lowpass filters

Alowpassfilter is a filter that passes low-frequency fluctuations and reduces the
amplitude of fluctuations with frequencies higher than a cut-off frequencyωc(see,
e.g., Percival and Walden, 1993). The frequency response function of an ideal
lowpass filter takes the following form: forω∈[0,π],

wlp(ω)=

{

1ifω≤ωc

0ifωc<ω≤π.

The notion of a highpass filter is complementary, its frequency response function
being whp(ω)= 1 −wlp(ω). The coefficients of the ideal lowpass filter are provided
by the inverse Fourier transform of wlp(ω):

wlp(L)=

ωc

π

+

∑∞

j= 1

sin(ωcj)

πj

(Lj+L−j).

A bandpass filter is a filter that passes fluctuations within a certain frequency
range and attenuates those outside that range. Given lower and upper cut-off fre-
quencies,ω 1 c<ω 2 cin(0,π), the ideal frequency response function is unity in
the interval[ω 1 c,ω 2 c]and zero outside. The notion of a bandpass filter is relevant
to business cycle measurement: the traditional definition, ascribed to Burns and
Mitchell (1946), considers all the fluctuations with a specified range of period-
icities, namely those ranging from one and a half to eight years. Thus, ifsis the
number of observations in a year, fluctuations with periodicity between 1.5sand
8 sare included. Baxter and King (1999; henceforth, BK) argue that the ideal fil-
ter for cycle measurement is a bandpass filter. Now, given the two business cycle
frequencies,ωc 1 = 2 π/( 8 s)andωc 2 = 2 π/(1.5s), the bandpass filter is:

wbp(L)=

ωc 2 −ωc 1

π

+

∑∞

j= 1

sin(ωc 2 j)−sin(ωc 1 j)

πj

(Lj+L−j). (9.4)

Notice thatwbp(L)is the contrast between the two lowpass filters with cut-off fre-
quenciesωc 2 andωc 1. The frequency response function of the ideal business cycle
bandpass filter for quarterly observations (s=4), which is equivalent to the gain
function (see Appendix A), is plotted in Figure 9.3 (p. 403).
The ideal bandpass filter exists and is unique, but as it entails an infinite number
of leads and lags, an approximation is required in practical applications. BK show
that theK-terms approximation to the ideal filter (9.4), which is optimal in the
sense of minimizing the integrated mean square approximation error, is obtained
from (9.4) by truncating the lag distribution at a finite integerK. They propose