Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 253

For matrix representations of these transforms, one may define:

U=T−^1 /^2 [exp{−i2πtj/T};t,j=0,...,T− 1 ],

U ̄=T−^1 /^2 [exp{i2πtj/T};t,j=0,...,T− 1 ],

(6.20)

which are unitary complex matrices such thatUU ̄=UU ̄ =IT. Then:

x=T^1 /^2 U ̄ξ ←→ ξ=T−^1 /^2 Ux, (6.21)

wherex=[x 0 ,x 1 ,...xT− 1 ]′andξ=[ξ 0 ,ξ 1 ,...ξT− 1 ]′are the vectors of the data
and of their Fourier ordinates respectively.

6.4 Spectral representations of a stationary process

The various equations of the Fourier analysis of a finite data sequence can also be
used to describe the processes that generate the data. Thus, within the equation:

yt=

∑n

j= 0

{

αjcos(ωjt)+βjsin(ωjt)

}

=ζ 0 +

∑n

j= 1

{

ζjeiωjt+ζj∗e−iωjt

}

,

(6.22)

the quantitiesαj,βjcan be taken to represent independent real-valued random
variables, and the quantities:

ζj=

αj−iβj

2

and ζj∗=

αj+iβj

2

(6.23)

can be regarded as complex-valued random variables.
The autocovariance of the elementsytandysis given by:

E(ytys)=

∑n

j= 0

∑n

k= 0

E

[

ζjζkei(ωjt+ωks)+ζjζk∗ei(ωjt−ωks)

+ζj∗ζkei(ωks−ωjt)+ζj∗ζk∗e−i(ωjt+ωks)

]

.

(6.24)

The condition of stationarity requires that the covariance should be a function
only of the temporal separation|t−s|ofytandys. For this, it is necessary that:

E(ζjζk)=E(ζj∗ζk∗)=E(ζj∗ζk)=E(ζjζk∗)=0, (6.25)

wheneverj=k. Also, the conditions:

E(ζj^2 )=0 and E(ζj∗^2 )=0, (6.26)