shift in the filter output. In time domain, the impact of the filter on an input
series yt is given by the finite moving average. In the frequency domain, the
filter is characterised by its Fourier transform α(ω).17 To find the weights aj,
one solves the minimisation problem
Z7Γ
∣∕3(ω) — α(ω)∣2 dω, s.t. α(0) = 0; (A3)
-π
where ∣∕3(ω) ∣ is the “ideal” filter gain with cut-off frequencies ω1 and ω2.18 The
constraint ensures that the resulting filter has trend reducing properties.19
Solving the minimisation problem leads to the following results:20
a j■ = bj ⅛ θ ; j' = 0, ± 1,. . . , ± K ;
bj =
ω2 — ω1
π
⅛ (sinω2j - sinωιj)
if J = O
if j = ±1,±2,.√
(A4)
κ h
'J = -K 0J
2K + 1
The original Baxter-King filter has an Undesireable property, which is
known as Gibb ,s phenomenon, due to the fact that the ideal filter, which is a
discontinuous function of ω, is approximated by a finite Fourier series. This
approximation leads to side lobes in the gain function of the biter. (Priestley
1981, p. 561-3, Koopmans 1974, p. 187-9). While the relative contribution of
some components for the overall variance of the series is exaggerated (i.e. they
are multiplied by a gain greater than 1), other components are suppressed (i.e.
they are multiplied by a gain less than 1).
An obvious solution to this problem is to increase the Flter length. But
since we are restricted by the limited availability of economic data, there is not
much to be gained from changing the length of the Flter. A more appropriate
solution is to apply spectral windows. As an example, consider the so called
17See e.g. Koopmans (1974), p. 165 ff.
18The gain of a filter measures the change in the amplitude of the input components if
transformed by the hlter. The ideal bandpass filter gain ∣∕3(ω)∣ takes the value 1 in the
frequency interval [ωι,ω2] and O outside this interval.
19In order to remove the component with the frequency ω = O from the series, the filter
weights must sum to zero.
20The filter is symmetric (i.e. αj∙ = a_j), and therefore does not impose a phase shift on
the output.
25