which has a peak at (12.18)

m’n where q = T , ω j = j/ T , j = 1, . . . , q, and the

ω0 = cos’1 . 2

m+n notation x indicates the largest integer contained

in x. The coef¬cients, given by equation (C.2), are

In the limit, if m and n are allowed to increase

1T

in¬nitely in such a way that (m ’ n)/(m + n)

a0 = xt (12.19)

tends to a constant, all the energy in the spectrum T t=1

is concentrated at a single frequency. Hence the

and

limiting process is a single sinusoid. We now

2T

call this the Slutsky effect. Slutsky con¬rmed

aj = xt cos(2πω j t) (12.20)

the effect by means of a simulation. Koopmans T t=1

[229] points out that Slutsky™s result was seminal T

in the development of ARMA models because b = 2 xt sin(2πω j t), (12.21)

j

T t=1

it illustrated a previously unknown mechanism

12.3: Estimating the Spectrum 265

for j = 1, . . . , q. Note that, for even T , where x is the sample mean. Now for notational

convenience, let c jt = cos(2π ω j t) and de¬ne

T

1

aq = (’1)q xt , (12.22) s jt similarly. Then, by applying the orthogonality

T properties of the discretized sine and cosine, we

t=1

bq = 0. (12.23) obtain

Thus the number of non-trivial coef¬cients a j T Var (Xt ) = t (xt ’ a0 )2

and b j is always T . This is as it should be

2

= a j c jt + b j s jt

since the Fourier transform is simply a coordinate t j

transformation in which information is neither lost

= a 2 (c jt )2 + b2 (s jt )2

t j

nor gained.12 j j

+ ai b j cit s jt

The time series can be recovered by substituting i= j

equations (12.19“12.22) into (12.18) and making

= t (c jt ) + t (s jt ) +0

a2 b2

2 2

j j

j j

use of the following orthogonality properties of the

= j (a j + b2 )

T 2

discretized sine and cosine functions: j

2

T

t=1 cos(2π ωk t) cos(2π ωl t) = 2 δkl

T

=2 j IT j .

a)

T

t=1 sin(2π ωk t) sin(2π ωl t) = 2 δkl

T

b)

Summations with respect to i and j are taken only

over i, j = 1, . . . , q where q = T .

T

t=1 cos(2π ωk t) sin(2π ωl t) = 0,

c) 2

Thus when T is odd, the periodogram partitions

where δkl = 1 if k = l and 0 otherwise.

the sample variance into q components as

The periodogram is de¬ned in terms of the

coef¬cients a j and b j as q

2

Var (Xt ) = IT j .

T2

IT j = (a + b2 ) T

(12.24)

4j j j=1

for j = ’ T ’1 , . . . , T . For negative j, a j = When T is even, the decomposition is

2 2

a’ j and b j = ’b’ j . We explicitly include the

q’1

subscript T to indicate that I is computed from a 2 1

Var (Xt ) = IT j + IT q .

time series of length T . The periodogram ordinates

T T

IT j correspond to the Fourier frequencies ω j and j=1

are sometimes referred to as intensities.

Note that the periodogram is symmetric in the 12.3.3 The Periodogram Carries the Same

Fourier frequencies ω j (except for ωq with even Information as the Sample Auto-covariance

T ) just as the spectral density function (ω) Function. The periodogram is the Fourier trans-

is symmetric. In fact, we show in [12.3.6] that form of the estimated auto-covariance function

the periodogram is an asymptotically unbiased evaluated at the Fourier frequencies ω j .

estimator of the spectral density. However, we ¬rst To show this, it is convenient to replace the

examine some other properties of the periodogram. sine and cosine transforms used above with the

complex exponential representation of the Fourier

12.3.2 The Periodogram Distributes the transform:

Sample Variance. The intensities are interesting

T

since they partition the sample variance into q IT j = |zT j |2

4

components.

The argument goes as follows. Assume, for

where

simplicity, that T is odd. Also assume that the time

series x1 , . . . , xT was obtained by observing an 2T

xt e’2πi ω j t

ergodic weakly stationary process. Then a natural, zT j =

T t=1

but slightly biased, estimator of the variance of XT

= aj ’ i bj. (12.25)

is

T

1

Var (Xt ) = (xt ’ x)2 With this representation, it is easily shown that

T the periodogram (12.24) is the Fourier transform

t=1

of the estimated auto-covariance function (12.2).

12 Note that equations (12.19)“(12.21) describe the Fourier

First replace xt in equations (12.25) with xt ’ x

transform of the in¬nite time series {zt : t ∈ Z} de¬ned by

zt = xt for t = 1, . . . , T and zt = 0 otherwise. and then substitute (12.25) into equation (12.24)

12: Estimating Covariance Functions and Spectra

266

to obtain

50

T

IT j = |zT j |2

40

4

1

(xt ’x) e’2πiω j t

=

30