Palgrave Handbook of Econometrics: Applied Econometrics

252 Investigating Economic Trends and Cycles

0

0.05

0.10

0.15

0.20

0 π/ 4 π/ 2 3 π/ 4 π

Figure 6.6 The periodogram of the residual sequence of Figure 6.5. A band, with a lower
bound ofπ/16 radians and an upper bound ofπ/3 radians, is masking the periodogram

6.3.2 Complex exponentials

In dealing with the mathematics of the Fourier transform, it is common to use com-
plex exponential functions in place of sines and cosines. This makes the expressions
more concise. According to Euler’s equations, these are:

cos(ωjt)=

1

2

(eiωjt+e−iωjt) and sin(ωjt)=

−i

2

(eiωjt−e−iωjt), (6.15)

where i=

√

−1. Therefore, equation (6.6) can be expressed as:

xt=α 0 +

[T∑/ 2 ]

j= 1

αj+iβj

2

e−iωjt+

[T∑/ 2 ]

j= 1

αj−iβj

2

eiωjt, (6.16)

which can be written concisely as:

xt=

T∑− 1

j= 0

ξjeiωjt, (6.17)

where:

ξ 0 =α 0 , ξj=

αj−iβj

2

and ξT−j=ξj∗=

αj+iβj

2

. (6.18)

Equation (6.17) may be described as the inverse Fourier transform. The direct
transform is the mapping from the data sequence within the time domain to the
sequence of Fourier ordinates in the frequency domain. The relationship between
the discrete periodic function and its Fourier transform can be summarized by
writing:

xt=

T∑− 1

j= 0

ξjeiωjt ←→ ξj=

1

T

T∑− 1

t= 0

xte−iωjtdt. (6.19)