Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 249

Since the trigonometrical functions are mutually orthogonal, the Fourier co-
efficients can be obtained via a set ofTsimple inner-product formulae, which are
in the form of ordinary univariate least squares regressions, with the values of the
sine and cosine functions at the pointst=0, 1,...,T−1 as the regressors.
Letcj=[c0,j,...,cT−1,j]′andsj=[s0,j,...,sT−1,j]′represent vectors ofTvalues of

the generic functions cos(ωjt)and sin(ωjt)respectively, and letx=[x 0 ,...,xT− 1 ]′

be the vector of the sample data andι=[1,...,1]′a vector of units. The “regression”
formulae for the Fourier coefficients are:

α 0 =(ι′ι)−^1 ι′x=

1

T

∑

t

xt=x ̄, (6.9)

αj=(cj′cj)−^1 c′jx=

2

T

∑

t

xtcos(ωjt), (6.10)

βj=(s′jsj)−^1 s′jx=

2

T

∑

t

xtsin(ωjt), (6.11)

αn=(cn′cn)−^1 c′nx=

1

T

∑

t

(− 1 )txt. (6.12)

However, in calculating the coefficients, it is more efficient to use the family of
specialised algorithms known as fast Fourier transforms, which deliver complex-
valued spectral ordinates from which the Fourier coefficients are obtained directly.
(See, for example, Pollock 1999.)
The power of a sequence is the time average of its energy. It is synonymous with
the mean square deviation which, in statistical terms, is its variance. The power

of the sequencexj(t)=ρjcos(ωjt)isρj^2 /2. This result can be obtained in view

of the identity cos^2 (ωjt)={ 1 +cos( 2 ωjt)}/2, for the average of cos( 2 ωjt)over an

integral number of cycles is zero. The assemblage of valuesρj^2 /2;j=1, 2,...,[T/ 2 ]
constitutes the power spectrum ofx(t), which becomes the periodogram when
scaled by a factorT. Their sum equals the variance of the sequence. IfT= 2 nis
even, then:

1

T

T∑− 1

t= 0

(xt−x ̄)^2 =

1

2

n∑− 1

j= 1

ρj^2 +α^2 n. (6.13)

Otherwise, ifTis odd, then the summation runs up to(T− 1 )/2, and the term
α^2 nis missing.
The indefinite sequencex(t)={xt;t=0,±1,±2,...}, expressed in the manner of
(6.6), is periodic with a periodTequal to the length of the sample. It is described as
the periodic extension of the sample, and it may be obtained be replicating sample
elements over all preceding and succeeding intervals ofTpoints. An alternative
way of forming the periodic sequence is by wrapping the sample around a circle of
circumferenceT. Then, the periodic sequence is generated by traveling perpetually
around the circle.