Palgrave Handbook of Econometrics: Applied Econometrics

254 Investigating Economic Trends and Cycles

must hold for allj. For (6.25) and (6.26) to hold, it is sufficient that:

E(αjβk)=0 for all j,k, (6.27)

and that:

E(αjαk)=E(βjβk)=

{

0, ifj=k;

σj^2 ,ifj=k.

(6.28)

An implication of the equality of the variances ofαjandβjis that the phase angle
θjis uniformly distributed in the interval[−π,π].
Under these conditions, the autocovariance of the process at lagτ=t−swill be
given by:

γτ=

∑n

j= 0

σj^2 cosωjτ. (6.29)

The variance of the process is just:

γ 0 =

∑n

j= 0

σj^2 , (6.30)

which is the sum of the variances of thenindividual periodic components. This is
analogous to equation (6.13).
The stochastic model of equation (6.22) may be extended to encompass processes
defined over the entire set of positive and negative integers as well as processes that
are continuous in time. First, we may consider extending the lengthTof the sample
indefinitely. AsTandnincrease, the Fourier coefficients become more numerous
and more densely packed in the interval[0,π]. Also, given that the variance of the
process is bounded, the variance of the individual coefficients must decrease.
To accommodate these changes, we may writeαj =dA(ωj)andβj=dB(ωj),
whereA(ω),B(ω)are cumulative step functions with discontinuities at the points
{ωj;j=0,...,n}. In the limit, the summation in (6.22) is replaced by an integral,
and the expression becomes:

y(t)=

∫π

0

{cos(ωt)dA(ω)+sin(ωt)dB)ω)}

=

∫π

−π

eiωtdZ(ω),

(6.31)

where:

dZ(ω)=

1

2

{dA(ω)−idB(ω)} and

dZ(−ω)=dZ∗(ω)=

1

2

{dA(ω)+idB(ω)}.

(6.32)

Also,y(t)={yt;t=0,±1,±2,...}stands for a doubly-infinite data sequence.