M

w(„ ; M, T ) = 1 ’ |„ |

Note that T

0 otherwise,

m

1

c(„ ) = c („ ) is generally called the Bartlett estimator.

m =1

The Bartlett estimator, as it is usually computed

m

M ’ |„ | 1 however (see, e.g., Jenkins and Watts [195]), uses

= Mc („ ).

m(M ’ |„ |) the slightly modi¬ed lag window

M =1

(12.46) |„ |

1’ if |„ | < M

M

wB („ ; M, T ) = (12.48)

Now assume, for convenience, that Xt is a 0 otherwise

zero mean process so that the auto-covariance

function estimate c („ ) can be computed without since it is then possible to derive a closed

subtracting the chunk mean. Then it is easily form representation for the corresponding spectral

window:

shown that

M sin(π ωM) 2

m

WB (ω; M, T ) ≈ .

1 (12.49)

Mc („ ) π ωM

(12.47) T

m(M ’ |„ |) =1

This window is shown in Figure 12.16 for M = 5.

is an unbiased estimate of γ („ ). But, since Note that this spectral window is wider than that

estimator (12.47) does not include all possible of the rectangular spectral estimator (Figure 12.15)

products X t X t’|„ | that can be computed from the for the same lag-window cutoff M, but that the

full time series, it is not the most ef¬cient unbiased side lobes are substantially reduced. The degrees

estimator of γ („ ). It therefore makes sense to of freedom and bandwidth of this estimator are

given in [12.3.19].20

replace estimator (12.47) with

T 20 Degrees of freedom and bandwidth are discussed in

c(„ )

T ’ |„ | [12.3.17].

12.3: Estimating the Spectrum 275

It should be noted that the Bartlett estimator

0.015

computed with (12.48) or (12.49) is not the

˜chunk™ estimator described in [12.3.9,10]. The

0.010

present estimator has lower variance and greater

bandwidth (see [12.3.19]).

0.005

The main problem with the Bartlett estimator

is that the side lobes of its spectral window

0.0

(12.49) are quite substantial when compared -0.4 -0.2 0.0 0.2 0.4

Frequency

with those of an estimator such as the Parzen

estimator (discussed in [12.3.16]). The Bartlett

spectral window (see Figure 12.16) has peaks Figure 12.17: The spectral window WP (ω; M, T )

at frequencies ±3/2 M that are about 4% of that corresponds to the Parzen lag window (12.50)

the height of the central peak. Therefore, since with cutoff M = 5 for time series of length T =

the unsmoothed periodogram can vary randomly 240.

across a couple of orders of magnitude, the Bartlett

estimator has the potential for signi¬cant unwanted

were easily identi¬ed because this estimator places

variance leakage.

equal weight on a ¬xed number of periodogram

ordinates that are asymptotically independent and

12.3.16 The Parzen Spectral Estimator. An-

identically distributed as χ 2 (2) random variables.

other popular smoothed periodogram spectral esti-

However, other smoothed periodogram estimators,

mator is the Parzen [305] spectral estimator. It has

such as the Bartlett estimator [12.3.15] and the

lag window

Parzen estimator [12.3.16] do not weight the

± |„ | 2 |„ | 3

1 ’ 6 M + 6 M periodogram ordinates equally. Thus it is not quite

so easy to determine their bandwidth and degrees

if |„ | < M

of freedom.

2

wP („ ; M, T ) = 2 1 ’ |„ | 3 Inferences about spectra estimated with a

M

general smoothed periodogram estimator are made

if M ¤ |„ | ¤ M

with the help of approximating χ 2 distributions.

2

That is, the equivalent degrees of freedom r

0 otherwise.

is found by matching the asymptotic mean and

(12.50)

variance of the spectral estimator with the mean

and variance of a χ 2 (r ) random variable.21

and corresponding spectral window

Standard texts, such as Koopmans [229] or

3M sin(π ωM/2) 4

WP (ω; M, T ) ≈ . Priestley [323], give the equivalent degrees of

π ωM/2

4T freedom of many smoothed periodogram spectral

estimators. The equivalent degrees of freedom for

The primary advantage of this estimator over

the estimators we have described are given in

the Bartlett estimator is that its spectral window

[12.3.19].

has virtually no side lobes (see Figure 12.17).

Once the equivalent degrees of freedom

The Parzen estimator also has somewhat lower

have been determined, con¬dence intervals can

variance than the Bartlett estimator for the same

be computed using the method outlined in

lag cutoff, since its spectral window has a wider

[12.3.10,11].

central peak and thus more bandwidth. However,

The moment matching exercise described

for this same reason, its estimates also have

above essentially identi¬es the Daniell spectral

somewhat more bias when the spectrum varies

estimator that is ˜equivalent™ to the smoothed

quickly relative to the bandwidth. This estimator

periodogram estimator. Thus, in addition to

has a wider spectral peak than the Bartlett

identifying equivalent degrees of freedom, this