-0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.01 0.02 0.03 0.04

-0.01 0.0

-1.0 -0.5 0.0 0.5 1.0 -0.4 -0.2 0.0 0.2 0.4

Lag/T Frequency

Figure 12.15: The spectral window W R (ω; M, T )

Figure 12.14: The lag window for the Daniell

spectral estimator with n = 5. that corresponds to the rectangular lag window

(12.42) with cutoff M = 5 for time series of length

T = 240.

where

covariance function is to use a rectangular lag

j+(n’1)/2

window

1

e’2πi(ωk ’ω j )„

wD („ ; n, T ) =

1 if |„ | ¤ M

n k= j’(n’1)/2

wR („ ; M, T ) = (12.42)

T n/ 2T ’2πiω„ 0 otherwise

≈ e dω

n ’n/ 2T that explicitly leaves out all estimated auto-

sin(πn„/ T ) covariances beyond some predetermined lag M.

= .

π n„/ T The corresponding spectral window, which is

shown in Figure 12.15 for M = 5, is

2M sin(2π ωM)

WR (ω; M, T ) ≈

.

Thus the Daniell estimator has equivalent spec- 2π ωM

T

tral window (12.39) and lag window (12.41) rep-

The resulting spectral estimator has equivalent

resentations. Since the periodogram and estimated

representations

auto-covariance function are a Fourier transform

T ’1

pair, smoothing in the frequency domain is equiv-

wR („ ; n, T )c(„ )e’2πiω j „ (12.43)

(ω j ) =

alent to smoothing in the time domain. We will

„ =’(T ’1)

see that the same is true for other smoothed

periodogram estimators as well. and

The lag window representation (12.41) gives q

(ω j ) = WR (ωk ’ ω j )IT k . (12.44)

us a somewhat different and useful perspective

k=’q

on why the Daniell estimator has lower variance

than the periodogram. The lag window, shown in Unfortunately, this particular estimator has

Figure 12.14 for n = 5, decays to zero with some undesirable properties.

increasing lag so that contributions to Fourier

• First, the spectral window (see Figure 12.15)

transform (12.41) from the large lag part of the

has large side lobes that permit variance

estimated auto-covariance are damped. Since we

leakage from frequencies far from ω j . Note

expect the true auto-covariance function to decay

that this source of variance leakage is

to zero at some lag, the window can be adjusted,

different from that discussed in [12.3.8]; it

either in the spectral or time domains, to exclude

will occur whether or not the data have been

lags for which the true auto-covariance function is

tapered. This problem exists to some extent

expected to be zero. We can therefore avoid the

with all spectral estimators that are designed

noise that is contributed by these lags.

with a truncated lag window.

• Second, the spectral window has negative

values at some frequencies. Consequently,

12.3.14 The Rectangular Spectral Estimator.

equation (12.43) or (12.44) can produce

This discussion motivates another simple, but

negative spectral density estimates with some

poor, spectral estimator. One simple way to

realizations of the periodogram.

exclude the large lag part of the estimated auto-

12: Estimating Covariance Functions and Spectra

274

Thus, the rectangular spectral estimator is best

0.005 0.010 0.015 0.020

avoided.

12.3.15 The Bartlett Spectral Estimator.

The chunk estimator is ˜almost™ a smoothed

periodogram estimator or, equivalently, a weighted

covariance estimator. When the time series is

0.0

contiguous, the chunk estimator can be modi¬ed -0.4 -0.2 0.0 0.2 0.4

Frequency

slightly to improve its properties and also

permit a smoothed periodogram or weighted

Figure 12.16: The spectral window WB (ω; M, T )

covariance representation. The resulting estimator

is commonly known as the Bartlett spectral that corresponds to the Bartlett lag window

(12.48) with cutoff M = 5 for time series of length

estimator.

Let c („ ) be the auto-covariance function esti- T = 240.

mate that is computed from the th chunk. Then,

using equation (12.26), we can write estimator

where c(„ ) is the auto-covariance function

(12.34) as the average of the Fourier transforms of

estimate computed from the full time series. When

the estimated auto-covariance functions,

we do this, and then substitute back into equations

m M’1 (12.46) and (12.45) we see that the ˜chunk™

1

c („ )e’2πiω j „ .

(ω j ) = estimator can be closely approximated as

m =1 „ =’(M’1)

|„ |

1’

M’1

c(„ )e’2πiω j „ .

(ω j ) = M

By rearranging the order of summation, we ¬nd

|„ |

1’

that estimator (12.34) is the Fourier transform of „ =’(M’1) T

the average estimated auto-covariance function:

This weighted covariance spectral estimator,

M’1 which has lag window

c(„ )e’2πiω j „ .

(ω j ) = ±

(12.45)

1 ’ |„ |

„ =’(M’1)