Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 425

and thephase Ph(ω)=arctan(−wI(ω)/wR(ω)). The former measures the amplitude
effect of the filter, so that if at some frequencies the gain is less than one, then those
frequency components will be attenuated in the filtered series; the latter measures
the displacement, or the phase shift, of the signal.
If fy(ω)denotes the spectrum of yt, the spectrum of w(L)yt is equal to

|w(e−iω)|^2 fy(ω), and therefore the square of the gain function (also known as the
power transfer function) provides the factor by which the spectrum of the input
series is multiplied to obtain that of the filtered series. In the important special
case when w(L)is symmetric, the phase displacement is zero, and the gain is simply
G(ω)=|w 0 + 2

∑m

j= 1 wjcosωj|.

9.6 Appendix B: The Wiener–Kolmogorov filter

The classical Wiener–Kolmogorov prediction theory, which is restricted to station-
ary processes, deals with optimal signal extraction of an unobserved component.
Lettingμtdenote some stationary signal andytan indeterministic linear process

with Wold representationyt=v(L)ξt,v(L)= 1 +v 1 L+v 2 L^2 +···,

∑

|vj|<∞,

ξt∼WN(0,σ^2 ), the minimum mean square linear estimator ofμt+lbased on a
semi-infinite sampleyt−j,j=0, 1,...,∞, is:

μ ̃t+l|t=

1

σ^2 v(L)

[

gμy(L)

v(L−^1 )

L−l

]

+

yt. (9.30)

Here,gμy(L)denotes the cross-covariance generating function ofμtandyt,gμy(L)=
∑
jγμy,jL

j, whereγ

μy,jis the cross-covariance at lagj,E[(μt−E(μt))(yt−j−E(yt))],

and forh(L)=

∑∞

j=−∞hjL

j,[h(L)]

+=

∑∞

j= 0 hjL

j, i.e., a polynomial containing only

non-negative powers ofL; see Whittle (1983, p. 42). The formula forl≤0 provides
the weights for signal extraction (contemporaneous filtering forl=0).
If an infinite realization of futureytwas also available, the minimum mean
square linear estimator is:

μ ̃t|∞=

gμy(L)

gy(L)

yt,

wheregy(L)is the autocovariance generating function ofyt,gy(L)=|v(L)|^2 σ^2 , and

|v(L)|^2 =v(L)v(L−^1 ). If the series is decomposed into two orthogonal components,
yt=μt+ψt,gμy(L)=gμ(L)(see Whittle, 1983, Ch. 5).
These formulae also hold whenytandμtare non-stationary (see Pierce, 1979).
As an example of their application, the expressions for the final and concurrent

estimators of the trend component for model (9.2), withση^2 =0 andσψ^2 /σζ^2 =λ
(Leser–HP filter), are:

μ ̃t|∞=

1

1 +λ| 1 −L|^4

yt, μ ̃t|t=

θ( 1 )

θ(L)

yt,