is then assumed that the remaining deviations are

approximately 1/M. Using (12.31), it is easily

roughly stationary within a given season (e.g.,

shown that

DJF).17 This yields one natural, disjoint chunk per

± (0) 2

m χ (m) j =0

year, of about 90 days in length.

A pleasing property of the chunk estimator is

(ω j ) 2

(ω j ) ∼ 2m χ (2m) 1 ¤ j ¤

M’1

that its variance goes to zero as 1/m where m is

2

the number of years in the data set. (1/2) 2

m χ (m) j = M (M even).

A dif¬culty with the chunk estimator is that its 2

bias is determined by the chunk length. In fact, (12.35)

the expectation of the chunk estimator is given

This estimator can be made consistent and

by equation (12.32) when T is set to the chunk

asymptotically unbiased when the time series is

16 We use the expression ˜chunk™ estimator to avoid contiguous by ensuring that both the number of

confusion with another estimator (described in [12.3.16]) that chunks m and the chunk length M increase with

statisticians and statistical packages frequently refer to as the

increasing sample length.

Bartlett estimator.

We can construct an asymptotic p — 100%

˜

17 While this assumption is never strictly correct, it is often

con¬dence interval (see Section 5.4) for (ω j )

accurate enough to allow use of the chunk estimator.

12.3: Estimating the Spectrum 271

from the chunk estimator as follows. Equation The asymptotic properties of the periodogram

(12.35) says that asymptotically can be extended to the Daniell estimator if n is

small relative to T and if the spectral density

2m (ω j ) function is smooth enough so that it is roughly

∼ χ 2 (2m).

(ω j ) constant in every frequency interval of length

n/ T . Under these conditions it can be shown that

Therefore

the Daniell estimator has the following properties

2m (ω j ) for frequencies (n + 1)/2 T ¤ ω j ¤ (2q ’

p ≈ P a¤

˜ ¤b (12.36)

n)/2 T :19

(ω j )

2m (ω j ) 2m (ω j ) 1 The Daniell estimator is asymptotically

=P ¤ (ω j ) ¤

b a unbiased. That is,

where a and b are the (1 ’ p)/2 and (1 +

˜

E (ω j ) ≈ (ω j ).

p)/2 critical values of the χ

˜ 2 (2m) distribution

(see Appendix E). The width of this interval can

2

be made independent of the spectral estimate (ω j ) ≈ (1/2n) (ω j ) .

2 Var

by taking logs. Re-expressed in this way, the

Therefore, the Daniell estimator can be made

approximate p — 100% con¬dence interval is

˜

consistent by letting n tend to in¬nity as T

tends to in¬nity in such a way that n/ T ’ 0.

2m

+log (ω j )

log (12.37)

b

(ω j ), (ωk ) ≈

3 Cov

2m

¤ log (ω j ) ¤ log +log (ω j ) .

a n’| j’k|

(ω j ) (ωk ) | j ’ k| ¤ n

n2

Remember that it is the end points of this interval

0 otherwise.

that are random. For every 100 independent

interval estimates that are made, the interval is

expected to cover the true parameter p —100 times,

˜ That is, (ω j ) and (ωk ) are approximately

uncorrelated if frequencies ω j and ωk are

on average.

separated by a bandwidth n/ T or more.

12.3.11 The Daniell Spectral Estimator. We

(ω j ) 2

(ω j ) ∼ χ (2n).

develop a number of smoothed spectral estimators 4

2n

in the following subsections and show how the user

can determine their properties by controlling either This last property allows us to construct

a spectral window or a lag window. The estimators asymptotic con¬dence intervals for the spectral

are typically applied to contiguous time series, but density. Proceeding in the same way as we did with

the chunk estimator, the approximate p — 100%

˜

can also be applied to individual chunks and then

con¬dence interval for (ω j ) is given by

averaged, as with the chunk estimator. A summary

is available in [12.3.19].

2n

The results of [12.3.7] suggest a natural way to +log( (ω j ))

log

b

reduce the variance of the periodogram, namely

2n

to smooth it, an idea that was ¬rst proposed ¤ log( (ω j )) ¤ log +log( (ω j ))

a

by Daniell [99]. The simplest of all smoothed

periodogram spectral estimators, which carries where a and b are the (1 ’ p)/2 and (1 +

˜

Daniell™s name, is just a moving average of the p/2) critical values of the χ

˜ 2 (2n) distribution (see

periodogram ordinates IT j . Given an odd integer Appendix E).

n such that 1 ¤ n ¤ q, the Daniell estimator is18

j+(n’1)/2 12.3.12 Bias Versus Variance. Although the

1

(ω j ) = IT k . (12.38) Daniell estimator has nice asymptotic properties

n k= j’(n’1)/2

(cf. [12.3.11]), tradeoffs must be made between

18 The Daniell estimator is de¬ned here as the average of an bias and variance (see [5.3.7] and Figure 5.3) when

odd number of periodogram ordinates. It can also be de¬ned samples are ¬nite.

as the average of an even number of periodogram ordinates,

19 Similar results can be obtained for frequencies j/ T , j =

in which case the estimates should be thought of as being

1, . . . , (n ’ 1)/2 and j = ((2q ’ n + 1)/2 T ), . . . , q/ T , where

representative of the frequencies midway between adjacent

q = T /2.

Fourier frequencies.

12: Estimating Covariance Functions and Spectra

272

substantial bias at frequencies near the spectral

Smoothed Periodogram

n = 11 peak by spreading and ¬‚attening the peak.

20

However, when the true spectrum has no large

peaks, a bandwidth this large may induce very