readable discussion of this subject by Raftery

following table.

[328] and discussants. Hannan [158] shows that

The frequency of AIC selected order BIC is a consistent order determining criterion.10

of 1000 AR(2) time series of length T

The frequency of BIC selected order

(±1 , ±2 ) = (0.9, ’0.8)

of 1000 AR(2) time series of length T

Order

(±1 , ±2 ) = (0.9, ’0.8)

≥5

T 0 1 2 3 4

Order

60 0 0 736 143 46 75

≥5

T 0 1 2 34

120 0 0 744 120 57 79

240 0 0 742 116 62 80 60 0 0 950 40 8 2

120 0 0 969 28 3 0

(±1 , ±2 ) = (0.3, 0.3) 240 0 0 975 22 3 0

Order

(±1 , ±2 ) = (0.3, 0.3)

≥5

T 0 1 2 3 4

Order

60 60 220 505 101 55 59

≥5

T 0 1 2 34

120 1 41 715 105 58 80

240 0 1 717 111 61 111 60 196 371 403 26 6 0

120 22 198 754 18 4 4

AIC seldom underestimates the order of the pro- 240 0 15 967 14 4 0

cess for the larger sample sizes, at least for the

time series considered here. The tendency to over- The Monte Carlo experiment described in

estimate the order appears to strengthen slightly [12.2.10] was repeated using BIC. The results,

with increasing sample size. As anticipated, the which are given in the table above, illustrate that

variability of the estimated order does not decrease BIC tends to select more parsimonious models

with increasing sample size. (i.e., models with fewer parameters) than AIC. We

see that it generally identi¬es the correct order of

12.2.11 Objective Order Determination: BIC. the process more accurately than AIC, and that

The other order determining criterion that is often its skill improves with increasing sample size.

used is the Bayesian information criterion [360]. Overall, the BIC order estimates have much lower

It is also developed around a test statistic, but bias and variability than their AIC counterparts.

in a Bayesian rather than frequentist setting. The We therefore recommend the use of BIC over AIC.

statistic used in the development of the BIC is the

Bayes factor B( p+q), p that compares the evidence

12.3 Estimating the Spectrum

for the model of order p + q with that for the

model of order p. The Bayes factor is similar 12.3.0 Overview. We give a brief introduction to

to a likelihood ratio except that numerator and the estimation of power spectra in this section. As

denominator are average likelihoods integrated in many other parts of this book, our purpose is not

relative to a prior distribution on the parameters of to be exhaustive but rather to give a ¬‚avour of the

the process. When the sample is large, the prior reasoning and issues involved. Jenkins and Watts

distribution plays a relatively minor role in the [195] and Koopmans [229] give a much more

Bayes factor, and it can then be shown that detailed and competent exposition than we do

here. Bloom¬eld [49] provides a very accessible

2 ln B( p+q), p ≈ 2 δl ’ q ln(n).

introduction to the subject.

The fundamental tool that we will use is

The BIC is consequently de¬ned as

the periodogram.11 The connection between the

B I C p = ’2l(± p , σZ |xT ) + ( p+1) ln(n). periodogram and the auto-covariance function,

From our perspective, of course, the main the statistical properties of the periodogram, and

difference between the AIC and BIC is that the consequently the reasons for not using it as

penalty for using an extra parameter is much 10 That is, E ( p

B I C ’ p)

2 ’ 0 as T ’ ∞ where p

BIC

greater with the latter. This penalty re¬‚ects a is the order selected with BIC.

fundamental difference between the ways in which 11 We previously touched on the periodogram in [11.2.0] and

frequentists and Bayesians weigh evidence. These [12.2.8].

12: Estimating Covariance Functions and Spectra

264

for generating quasi-periodic behaviour. The only

an estimator of the autospectrum are discussed

time series models known before this time were

¬rst. However, we will see that, despite the

combinations of simple, almost periodic functions

periodogram™s poor properties as a raw spectral

(such as (11.25)) and white noise residuals.

estimator, spectral estimators with much more

The plan for the remainder of this section is

acceptable characteristics can be constructed from

as follows. We will explore the properties of

the periodogram.

the periodogram in subsections [12.3.1“7]. Data

A pitfall that many have encountered is to

tapers, which are used to counteract problems that

confuse harmonic analysis, the detection of

arise when a process has a periodic component

regular periodic signals, with spectral analysis,

or a spectrum with sharp peaks, are described

the description of how variance is distributed as

in [12.3.8]. Spectral estimators constructed from

a function of time scale in processes that do not

the periodogram are covered in subsections

vary periodically. The potential for confusion is

[12.3.9“19]. The ˜chunk™ estimator (which is also

clearly apparent from Shaw™s 1936 study [347]

sometimes referred to as the Bartlett estimator

that we cited in Chapter 10. The periodogram

[12.3.9,10]), is discussed ¬rst because it is

does not have better properties when applied to

easily adapted to climate problems in which,

harmonic analysis than when applied to spectral

for example, a daily time series of length 90

analysis, but it is truly useful as a tool for harmonic

days is observed at the same time every year.

analysis when the source of the periodicities is

We then go on to develop some ideas that will

clearly understood as in the analysis of tidal

help readers understand how spectral estimators

variations (e.g., Zwiers and Hamilton [447]) or the

are constructed and interpreted. This is done in

analysis of emissions from a rotating star (e.g.,

subsections [12.3.11“18] by describing smoothed

Bloom¬eld™s analysis of observations of a variable

periodogram estimators that are commonly used to

star [49]).

analyse time series that are contiguous in time (as

An important historical note is that Slutsky

opposed to time series composed of a number of

[351] was apparently suspicious of the way in

disjoint chunks). Subsection [12.3.19] contains a

which some economic data were being analysed.

summary of spectral estimators constructed from

He showed that variance can be con¬ned to a

the periodogram. An example intercomparing

narrow frequency band by passing white noise

spectral estimators is presented in [12.3.20], and

through a series of summing ¬lters

an alternative approach to spectral estimation

Yt = Zt + Zt’1 is brie¬‚y discussed in [12.3.21]. The effects of

aliasing are discussed in [12.3.22].

and differencing ¬lters

Yt = Zt ’ Zt’1 . 12.3.1 The Periodogram. Let {x1 , . . ., xT } be

a time series. Equation (C.1), which expands

In fact, if white noise is passed through m

{x1 , . . . , xT } in terms of complex exponentials,

summing ¬lters and n differencing ¬lters then

can be re-expressed in sine and cosine terms, as

the output process can be shown to have spectral

in equation (11.16), as

density function

q

xt = a 0 + a j cos(2πω j t) + b j sin(2π ω j t) ,

yy (ω) =2 (cos π ω) (sin π ω)2n σZ ,

m+n+1 2m 2