Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 393

using a three-year window, i.e.,K= 3 s, as a valid rule of thumb for macroeconomic
time series. They also constrain the weights to sum to zero, so that the resulting
approximation is a detrending filter: denoting the truncated filter wbp,K(L)=w 0 +
∑K
1 wj(L

j+L−j), the weights of the adjusted filter will be w
j−wbp,K(^1 )/(^2 K+^1 ). The
gain of the resulting filter is displayed in Figure 9.3 (henceforth we shall refer to it
as the BK filter). The ripples result from the truncation of the ideal filter and are
referred to as the Gibbs phenomenon (see Percival and Walden, 1993, p. 177). BK
do not entertain the problem of estimating the cycle at the extremes of the available
sample; as a result the estimates for the first and last three years are unavailable.
Christiano and Fitzgerald (2003) provide the optimal finite-sample approximations
for the bandpass filter, including the real-time filter, using a model-based approach.
Within the class of parametric structural models, an important category of
lowpass filter emerges from the application of Wiener–Kolmogorov optimal signal
extraction theory to the following model:

yt = μt+ψt, t=1, 2,...,n,

mμt = ( 1 +L)rζt, ζt∼NID(0,σζ^2 ),

ψt ∼ NID(0,λσζ^2 ),E(ζt,ψt−j)=0,∀j,

(9.5)

whereμtis the signal or trend component, andψtis the noise.
Assuming a doubly-infinite sample, the minimum mean square estimators of the
components (see Appendix B) are, respectively,μ ̃t=wμ(L)ytandψ ̃t=yt− ̃μt=
[ 1 −wμ(L)]yt, where:

wμ(L)=

| 1 +L|^2 r

| 1 +L|^2 r+λ| 1 −L|^2 m

. (9.6)

The expression (9.6) defines a class of filters which depends on the order of integra-
tion of the trend (m, which regulates its flexibility), on the number of unit poles at
the Nyquist frequencyr, whichceteris paribusregulates the smoothness ofmμt,
andλ, which measures the relative variance of the noise component.
The Leser–HP filter arises form =2,r = 0,λ= 1600 (quarterly data). The
two-sided EWMA filter arises form=1,r=0. The filters arising form=rare
Butterworth filters of the tangent version (see, e.g., Gómez, 2001). The analytical
expression of the gain is:

wμ(ω)=

{

1 +

[

tan(ω/ 2 )

tan(ωc/ 2 )

] 2 m}− 1

,

and depends solely onmandωc.Asm→∞the gain converges to the frequency
response function of the ideal lowpass filter.
The previous discussion enforces the interpretation of the trend filter wμ(L)as
a lowpass filter. Its cut-off frequency depends on the triple(m,r,λ). Frequency
domain arguments can be advocated for designing these parameters so as to select
the fluctuations that lie in a predetermined periodicity range. In particular, let us
consider the Fourier transform of the trend filter (9.6), wμ(ω)=wμ(e−ıω),ω∈[0,π],