20

(xs ’x)e+2πiω j s

—

s

10

T ’1

1

= (xt ’x)(xs ’x)

0

T

„ =’(T ’1) t’s=„

-10

— e’2πiω j „ -0.4 -0.2 0.0 0.2 0.4

∞ Frequency

c(„ ) e’2π iω j „ .

= (12.26)

„ =’∞

for nonzero j and where c(„ ) is zero for |„ | greater Figure 12.8: Window function HT (12.29) dis-

played for T = 10 (dashed curve) and T = 50

than T ’ 1.

(solid curve).

12.3.4 The Covariance Structure of the Fourier

Coef¬cients ZT j . We next derive the covariance

Finally the summation and integration operations

structure of the Fourier coef¬cients ZT j . The

are interchanged, and the summation is performed

main result, (12.28), is used in [12.3.5,6] to show

to obtain

that the periodogram ordinates are asymptotically

unbiased estimators of the spectral density and also 1

4 2

to show that they are asymptotically uncorrelated. (ω)HT (ω’ω j )e(T +1)πi(ω’ω j )

E jk = 2

T ’1

To simplify our derivation we will assume 2

that x1 , . . . , xT come from a zero mean, ergodic,

— HT (ω’ωk ) e’(T +1)πi(ω’ωk ) dω

weakly stationary process so that the auto-

(12.28)

covariance function can be estimated as

T

1

c(„ ) = xt xt’„ . (12.27) where

T t=„ +1

HT (ω) = sin(T π ω)/ sin(π ω)

The ergodicity assumption is particularly impor-

tant because it assures us that estimators such as T

’(T +1)πiω/2

=e e2πiωt .

(12.27) are consistent.

Let E jk = E ZT j Z— k . The ¬rst step towards t=1

T

understanding the structure of covariance E jk is (12.29)

to expand the random coef¬cients ZT j and then

exchange expectation and summation operators: Equation (12.28) links the covariance structure

of the Fourier coef¬cients ZT j to the spectral

4

Xt e’2πiω j t

E jk = E density function (ω) of the process through the

t

T2

window function HT (ω) given by equation (12.29).

— Xs e2πiωk s Figure 12.8 shows HT for T = 10 and T =

s

50. Note that, as T increases, HT develops into

4

E(Xt Xs )e’2πi(ω j t’ωk s)

= a function with a narrow central spike of height

t s

T2

T and width 1/ T and with side lobes that are

separated by zeros at ±1/ T, ±2/ T, . . ..

4 ’2πi(ω j t’ωk s) .

= s γ (t ’ s)e

t

2

T

The next step is to replace γ („ ) with its Fourier

transform: 12.3.5 The Periodogram Ordinates are Asymp-

1 totically Uncorrelated. For ¬xed j and k and in-

4 2

E jk = 2 (ω)e 2πiω(t’s)

creasing T , the windows HT (ω ’ ω j ) and HT (ω’

T t s ’1

ωk ) tend to narrow into adjacent spikes. Therefore,

2

’2πi(ω j t’ωk s)

—e since (ω) is continuous, we can approximate

dω.

12.3: Estimating the Spectrum 267

(12.28) for moderate to large T , as random variables. In particular, it can be shown

(see, e.g., Brockwell and Davis [68, p. 347]) that

ω j + ωk

4 ±

(T +1)πi(ωk ’ω j )

E jk = 2 e χ2

(0)χ (1) j =0

2

T

(ω j ) 2

1 ¤ j ¤ T ’1

IT j ∼

1

χ (2)

2

2

— HT (ω’ω j )HT (ω’ωk ) dω. (12.30) 2

χ

(1/2)χ 2 (1) j = T if T is even.

’1

2 2

(12.31)

Consequently,

Equation (12.31) clearly illustrates why the

E jk = E ZT j Z— k ≈ 0 for j = k.

T periodogram is such a poor spectral estimator.

Although it is asymptotically unbiased, it is not

That is, the Fourier coef¬cients, and therefore

consistent: its variance does not decrease with

the periodogram ordinates IT j , are approximately

increasing sample length.

uncorrelated.

This is illustrated in the upper two panels

of Figure 12.9, which shows two periodograms

computed from time series of length T = 120 and

12.3.6 What does the Periodogram Estimate?

T = 240 generated from a unit variance white

Continuing on from (12.30), we see that