Palgrave Handbook of Econometrics: Applied Econometrics

258 Investigating Economic Trends and Cycles

On writing

∑
ψje−iωj=ψ(ω), which is the frequency response function of the
filter, this becomes:

y(t)=

∫

ω

eiωtψ(ω)dZx(ω)

=

∫

ω

eiωtdZy(ω).

(6.41)

If the processx(t)has a spectral density functionfx(ω), which will allow one to
writedF(ω)=f(ω)dωin equation (6.36), then the spectral density functionfy(ω)
of the filtered processy(t)will be given by:

fy(ω)dω=E

{

dZy(ω)dZy∗(ω)

}

=ψ(ω)ψ∗(ω)E

{

dZx(ω)dZ∗x(ω)

}

=|ψ(ω)|^2 fx(ω)dω.

(6.42)

The complex-valued frequency-response functionψ(ω), which characterizes the
linear filter, can be written in polar form as:

ψ(ω)=|ψ(ω)|e−iθ(ω), (6.43)

The function|ψ(ω)|, which is described as the gain of the filter, indicates the extent
to which the amplitude of the cyclical components of whichx(t)is composed are
altered in the process of filtering.
Whenx(t)=ε(t)is a white-noise sequence of independently and identically
distributed random variables of varianceσ^2 , equation (6.42) gives rise to the expres-

sionfy(ω)=σ^2 |ψ(ω)|^2 =σ^2 ψ(ω)ψ∗(ω), which is the spectral density function of
y(t). Then, it is helpful to use the notation of thez-transform wherebyψ(ω)is

written asψ(z)=

∑

jψjz

j;z=e−iω. If we allowzto be an arbitrary complex num-

ber, then we can define the autocovariance generating functionγ(z)=

∑

τγτz

τ

whereinγτ=E(ytyt−τ). This takes the form of:

γ(z)=σ^2 ψ(z)ψ(z−^1 ). (6.44)

Example Figure 6.8 depicts the squared gain of the difference operator∇(z)=
1 −z, which is the curve labeledD. The squared gain of∇(z)is obtained by setting

z=exp{−iω}within|∇(z)|^2 =( 1 −z)( 1 −z−^1 )to giveD(ω)= 2 −2 cos(ω), whence

W(ω)=D−^1 (ω)can be obtained, which is the squared gain of the summation
operator. The product ofD(ω)andW(ω)is the constant functionN(ω)=1, which
also represents the spectral density function or power spectrum of a white-noise
process with a variance ofσ^2 = 2 π. Likewise,W(ω)represents the pseudo-spectrum
of a first-order random walk.
This is not a well-defined spectral density function, since the random walk does
not constitute a stationary process of a sort that can be defined over a doubly-
infinite set of time indices. The unbounded nature ofW(ω)asω→0 is testimony