D.S.G. Pollock 27300.250.500.751.001.250 π/ 4 π/ 2 3 π/ 4 πB(^1600) H
Figure 6.11 The gain of the HP filterHand of the Butterworth filterBwith nominal cut-off
points atπ/4 radians, together with the gain of an HP filter with a smoothing parameter of
1600
it is possible to avoid the need for non-zero initial conditions. The estimate ofη(t)
can then be subtracted fromy(t)to obtain the estimate ofξ(t).
The HP filter has many antecedents. Its invention cannot reasonably be
attributed to Hodrick and Prescott (1980, 1997), who cited Whittaker (1923) as
one of their sources. Leser (1961) also provided a complete derivation of the filter
at an earlier date. The Butterworth filter is a commonplace of electrical engineer-
ing. The digital version of the filter has been described in an econometric context
by Pollock (2000) and by Gómez (2001). It has been applied to climatological data
by Harvey and Mills (2003).
Example Figure 6.11 shows the gain functions of the three filters overlaid on the
same diagram. The lowpass HP filter with a smoothing parameter ofλ=1600 is
commonly recommended for estimating the trend in quarterly economic data. The
corresponding gain function is marked in the diagram by the number 1600.
An alternative to specifying the smoothing parameter directly is to specify the
frequency valueωdfor which the gain isβ(ωd) = 0.5. For the HP filter, the
correspondence betweenωdandλis as follows:
λ=
1
4 { 1 −cos(ωd)}^2
and ωd=cos−^1 ( 1 − 1
√
4 λ). (6.98)
The frequencyωdcorresponds to the mid-point in the transition between the
pass band and the stop band of the filter. This might be described as the nominal
cut-off frequency, but, in the case of the HP filter, this is a misnomer, on account of
the very gradual transition of the gain. The Butterworth filter is capable of a much
more rapid transition. The curve labeledBcorresponds to the gain of a Butterworth
filter withn=6 andωd=π/4.
6.6.3 Structural ARIMA models
The HP filter and the Butterworth filter are appropriate to the task of extracting the
trend or the trend/cycle component from a data sequence without regard to the