10

little bias. Since we generally know little about

0

the features of the true spectrum, balancing bias

and variance in spectral estimation is a matter of

-10

subjective judgement.

-20

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

12.3.13 An Alternative Representation of

Smoothed Periodogram

the Daniell Spectral Estimator. The Daniell

n = 21

estimator (cf. [12.3.9] and (12.38)) can be

20

re-expressed as the convolution between the

10

periodogram and a box car shaped spectral

window

0

q

-10

(ω j ) = WD (ωk ’ ω j ; n, T )IT k (12.39)

-20

k=’q

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

where the spectral window is given by

Smoothed Periodogram

n = 41

if |ω| ¤ (n/2 T )

1

n

WD (ω; n, T ) = (12.40)

20

0 otherwise.

10

We will see in the following subsections that other

0

smoothed periodogram spectral estimators can be

represented similarly.

-10

The Daniell estimator can also be expressed

-20

as the Fourier transform of the product of the

0.0 0.1 0.2 0.3 0.4 0.5

estimated auto-covariance function c(„ ) and a lag

Frequency

window, wD („ ; n, T ):

T ’1

Figure 12.13: Daniell estimates computed from the

wD („ ; n, T )c(„ )e’2πiω j „ . (12.41)

(ω j ) =

periodogram in the lower panel of Figure 12.9

„ =’(T ’1)

and plotted on the decibel scale. The cross in

the upper right corner indicates the width of the The lag window is derived as follows. Recall

95% con¬dence interval (vertical bar) and the from (12.26) that the periodogram is the Fourier

bandwidth (horizontal bar). transform of the auto-covariance function. There-

fore, expanding (12.38) we obtain

j+(n’1)/2

Figure 12.13 shows spectral estimates computed 1

(ω j ) = IT k

from the periodogram displayed in the lower n k= j’(n’1)/2

panel of Figure 12.9 using the Daniell estimator

with n = 11, 21, and 41. The dashed curve j+(n’1)/2 T’1

1

c(„ )e’2πiωk „ .

=

shows the spectral density that these estimators

nk= j’(n’1)/2 „ =’(T’1)

are trying to approximate. The Daniell estimator

with n = 11, which has a bandwidth of about Then, rearranging the order of summation, we ¬nd

0.046, is quite smooth in comparison with the

T’1

periodogram (Figure 12.9) and yet is nearly

c(„ )e’2πiω j „

(ω j ) =

unbiased. The true spectral density generally lies

„ =’(T’1)

within the approximate 95% con¬dence interval.

j+(n’1)/2

The estimators with n = 21 (bandwidth 0.088) 1

e’2πi(ωk ’ω j )„

—

and n = 41 (bandwidth 0.17) do not capture n k= j’(n’1)/2

the spectral peak well because they smooth the

T’1

periodogram excessively.

c(„ )e’2πiω j „ wD („ ; n, T )

=

There is not a correct choice of bandwidth.

„ =’(T’1)

In this example a bandwidth of 0.17 induces

12.3: Estimating the Spectrum 273