random variable. Thus

a sample of length T = 384 from the process with

r = 0.9 and u = 0.95. It has an e-folding time of

H0 : κx y (ω j ) = 0 versus Ha : κx y (ω j ) > 0

approximately 9.5 time units, a rotation frequency

can be tested at the (1 ’ p) — 100% signi¬cance

˜ · = 0.050 (approximately 20 time units) and there

level by comparing κx y (ω j ) with is a peak in the coherency spectrum at ω0 = 0.053

(approximately 18 time units).

2Fp˜ Three Daniell cross-spectral estimates with

, (12.59)

r ’ 2 + 2Fp

˜ different amounts of smoothing are shown in

Figure 12.23. The left hand column, with n =

where Fp is the p critical value of the F(2, r ’ 2)

˜

˜

2, uses almost no smoothing. This is the cross-

distribution (see Appendix G). Con¬dence in-

spectral estimator that is obtained when adjacent

tervals should only be computed when the null

bivariate periodogram ordinates are averaged. The

hypothesis that κx y (ω j ) is zero is rejected.

upper panel shows the estimate of the spectrum

of the X component of the process on the

12.5.6 A Con¬dence Interval for the Phase decibel scale. The true spectrum is indicated

Spectrum. Hannan [157, p. 257] shows that by the long-dashed curve. The spectral estimate

approximate p — 100% con¬dence limits for the

˜ is noisy, but otherwise satisfactory. Despite the

phase spectrum x y are given by noise, the estimate conveys useful information

and gives us an indication of the shape of the

t(1+˜ )/2

’1

(κx y (ω j ))’1 ’ 1

p

x y (ω j ) ± sin spectrum and the location of the spectral peak.

r ’2 The middle panel shows the derived coherency

estimate (solid curve) and the true coherency

29 Koopmans [229, p. 283] gives a slightly re¬ned version

(long-dashed curve). Note that this estimate is

of this interval. He also points out that the quality of the

approximation depends upon the equivalent degrees of freedom very noisy with many large spikes that grossly

r and κx y (ω j ), and that it is best when r > 40 and 0.4 <

overestimate the true coherency. It does not give

κx y (ω j ) < 0.95. However, in our experience, interval (12.58)

any useful information about the true coherency

gives useful, although perhaps not precise, information when

spectrum, except to suggest that it is probably

there are substantially fewer equivalent degrees of freedom.

12: Estimating Covariance Functions and Spectra

286

Spectral Density Spectral Density Spectral Density

20

20

20

10

10

10

0

0

0

-10

-10

-10

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

Frequency Frequency Frequency

Coherence Coherence Coherence

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

Frequency Frequency Frequency

Phase Phase Phase

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

Frequency Frequency Frequency

Figure 12.23: Cross-spectral estimates computed from a time series of length T = 384 generated from

a bivariate AR(1) process with a rotational parameter matrix (11.51) (cf. [11.3.8] and [11.4.8,9]). The

columns contain Daniell estimates for n = 2, 16, and 64, from the left.

Top row: The estimated spectrum of the X component of the process, in decibels. The dashed curve

indicates the true spectrum. The cross indicates the bandwidth (horizontal) and width of the 95%

con¬dence interval (vertical).

Middle row: The estimated coherency. The long-dashed curve indicates the true coherency spectrum.

The short-dashes indicate the critical value for the 5% signi¬cance level test of zero coherency.

Bottom row: The estimated phase. The dashed line indicates the true phase.

nonzero for frequencies in the interval (0.02, 0.1). The centre column in Figure 12.23 shows the

The horizontal short-dashed line in this diagram Daniell cross-spectral estimate that is obtained

with a moderate amount of smoothing (n =