Palgrave Handbook of Econometrics: Applied Econometrics

250 Investigating Economic Trends and Cycles

6.3.1 Approximations, resampling and Fourier interpolation

By lettingt=0,...,T−1 in equation (6.6), the data sequence{xt;t=0,...,T− 1 }is
generated exactly. An approximation to the sequence may be generated by taking
a partial sum comprising the terms of (6.6) that are associated with the Fourier fre-
quenciesω 0 ,...,ωd, whered<[T/ 2 ]. It is straightforward to demonstrate that this
is the best approximation, in the least squares sense, amongst all of the so-called
trigonometrical polynomials of degreedthat comprise the sinusoidal functions in
question.
The result concerning the best approximation extends to the continuous func-
tions that are derived by allowingtto vary continuously in the interval[0,T).
That is to say, the continuous function derived from the partial Fourier sum com-
prising frequencies no higher thanωd = 2 πd/Tis the minimum mean square
approximation to the continuous function derived from (6.6) by lettingtvary
continuously.
We may exclude the sine function of frequencyωdfrom the Fourier sum. Then
the continuous approximation is given by:

z(t)=

∑d

j= 0

{

αjcos

(

2 πjt

T

)}

+

d∑− 1

j= 1

{

βjsin

(

2 πjt

T

)}

=

∑d

j= 0

{

αjcos

(

2 πjτ

N

)}

+

d∑− 1

j= 1

{

βjsin

(

2 πjτ

N

)}

,

(6.14)

whereτ=tN/TwithN= 2 d, which is the total number of the Fourier coefficients.
Here,τvaries continuously in[0,N), whereastvaries continuously in[0,T).On
the right-hand side, there is a new set of Fourier frequencies{ 2 πj/N;j=0, 1,...,d}.
TheNcoefficients{α 0 ,α 1 ,β 1 ,...,αd− 1 ,βd− 1 ,αd}bear a one-to-one correspon-
dence with the set ofNordinates{zτ =z(τT/N);τ =0,...,N− 1 }sampled at
intervals ofπ/ωd=T/Nfromz(t). The consequence is thatz(t)is fully represented
by the resampled datazτ;τ=0,...,N−1, from which it may be derived by Fourier
interpolation.
The result concerning the optimality of the approximation is a weak one; for it
is possible that the preponderance of the variance of the data will be explained by
sinusoids at frequencies that lie outside the range[ω 0 ,...,ωd]. The matter can be
judged with reference to the periodogram of the data sequence, which constitutes
a frequency-specific analysis of variance.

Example Figure 6.4 represents the logarithms of the data on quarterly real
household expenditure in the UK for the period 1956–2005, through which a linear
function had been interpolated so as to pass through the midst of the data points
of the first and the final years.
This interpolation is designed to minimize any disjunction that might other-
wise occur where the ends of the data sequence meet when it is mapped onto the