used to estimate the coherency spectrum directly,

equally to the other spectral estimators summa-

even though the bivariate periodogram is itself an

rized in [12.3.19].

asymptotically unbiased estimator of the cross-

spectral density function.27 This tells us that The Daniell cross-spectral estimator [12.3.11] is

given by

if we want to estimate the coherency well,

we must construct cross-spectral estimators that

j+(n’1)/2

1

average across a number of nearly independent

x y (ω j ) = Ix y T k .

realizations of the bivariate periodogram. The n k= j’(n’1)/2

chunk and smoothed periodogram estimators

discussed in Section 12.3 do exactly this. Also, it is We can view x y (ω j ) as an estimate of the

intuitive that a relatively large number of bivariate (complex) covariance between processes X and Y

at time scales between ω’1 ’1

periodogram ordinates need to be averaged to

j+(n’1)/2 and ω j’(n’1)/2 .

overcome the bias induced by the degeneracy To appreciate this, we substitute equation (12.56)

of the individual bivariate periodogram ordinates. for the cross-periodogram to obtain

Thus the art of cross-spectral estimation involves

trade offs between variance and at least two types j+(n’1)/2

T

Zx T k Z— T k .

x y (ω j ) =

of bias. y

4n k= j’(n’1)/2

Except for the factor T , this expression looks

12.5.4 Smoothed Periodogram Estimators.

just like an estimate of the (complex) covari-

Since the bivariate periodogram ordinates are

ance between a pair of zero mean random var-

asymptotically independent, cross-periodogram-

iables Zx T and Z y T that is computed from

based cross-spectral estimators are constructed

a sample {(Zx T k , Z y T k ): k = j ’ (n ’

using chunk, or smoothing, techniques in the

1)/2, . . . , j + (n ’ 1)/2}. This interpretation be-

same way that univariate spectral estimators are

comes even stronger when we assume that the

constructed. Equivalent bandwidths and degrees

cross-spectral density function is constant in the

of freedom are also computed in exactly the

interval (ω j’(n’1)/2 , ω j+(n’1)/2 ) because then the

same way, and similar considerations are made for

random pairs (Zx T k , Z y T k ) are approximately

the choice of spectral or lag-window. Goodman

independent and identically distributed.

[144] derived the asymptotic distribution of

We can estimate the correlation between the

periodogram-based bivariate spectral estimators.28

X and Y processes in the frequency range

Goodman™s approximation is used to derive

(ω j’(n’1)/2 , ω j+(n’1)/2 ) by normalizing x y (ω j )

con¬dence intervals for cross-spectral parameters

with estimates of the standard deviations of the X

such as the coherency and phase (see, e.g.,

and Y in this frequency range. The latter are just

Koopmans [229], Hannan [157] or Brillinger [66]

the square roots of the estimated auto-spectra of X

and also [12.5.5,6]).

and Y. Thus we have

x y (ω j )

ρx y (ω j ) = .

12.5.5 A Con¬dence Interval for the Coherency

1/2

x x (ω j ) yy (ω j )

Spectrum. The smoothed coherency spectrum

Consequently the estimated coherency

27 This is not logically inconsistent. For example, suppose

that Z1 and Z2 are independent and identically distributed

κx y (ω j ) = |ρx y (ω j )|2

complex random variables such that Zi ∼ ei2π Ui where Ui

is distributed uniformly on the interval [0, 1). Then Z1 Z— is 2

an unbiased estimator of the centre of the unit circle (i.e., can be viewed as a measure of the squared

E(Z1 Z— ) = 0) even though |Z1 Z— | = 1. If we averaged across

correlation, or proportion of common variance

2 2

a large sample, say {(z1,i , z2,i ): i = 1, . . . , n} we would ¬nd

that is shared by X and Y in the ω’1

j+(n’1)/2 to

| n i=1 z1,i z— | ≈ 0, even though |z1,i z— | = 1 for all i.

n

1

2,i 2,i

ω’1

j’(n’1)/2 time scale range.

28 This distribution, known as the complex Wishart distri-

bution, describes the behaviour of random 2 — 2 Hermitian This interpretation of the coherency carries over

matrices (see Brillinger [66] or Hannan [157]). It has a

to other periodogram-based spectral estimators as

property similar to that of the χ 2 distribution: the sum of two

well and can be used to construct con¬dence

independent complex Wishart random matrices again has a

intervals.

complex Wishart distribution.

12.5: Estimating the Cross-spectrum 285

where x y (ω j ) is the phase estimate obtained

Fisher™s z-transform was used in [8.2.3]

to construct con¬dence intervals for ordinary by substituting a periodogram-based estimator

x y (ω j ) of the cross-spectral density into equa-

correlation coef¬cients. The same method can

be used here for nonzero κx y (ω j ). Fisher™s z- tions (11.64)“(11.66), r is the equivalent degrees

transform (8.5) of the square root of the coherency, of freedom of the spectral estimator, and t(1+˜ )/2

p

is the (1 + p)/2 critical value of the t(r ’ 2)

˜

1 + κx y (ω j )1/2

1 distribution (see Appendix F).

= tanh’1 (κx y (ω j ) 2 ),

1

ln

1 ’ κx y (ω j )1/2

2

12.5.7 Bias in the Coherency and Phase

is approximately normally distributed with mean Spectra: An Example. We return to the prob-

tanh’1 (κx y (ω j )1/2 ) and variance 1/r , where r lem of bias in the estimated coherency spectrum

is the equivalent degrees of freedom of the because of the con¬‚icting demands that good co-

spectral estimator. Therefore approximate p — ˜ herency estimates place on the spectral estimator.

100% con¬dence limits for the squared coherency In univariate spectral estimation, small numbers

are of equivalent degrees of freedom are associated

with high variability and low bias. In cross-spectral

Z(1+˜ )/2 2

tanh tanh’1 κx y (ω j )1/2 ±

p

,

√ estimation, small numbers of degrees of freedom

r

are also associated with large positive bias in

(12.58)

coherency estimates, which arises from the de-

where Z(1+˜ )/2 is the (1 + p)/2 critical value of the

˜ generacy of the coherence of the periodogram. In

p

standard normal distribution (Appendix D).29 addition, we will see that large equivalent band-

width leads to bias not only in the magnitude of

The approximation that leads to interval (12.58)

breaks down when κx y (ω j ) is zero. Then the coherency but also in the location of coherency

peaks.

(r/2 ’ 1)κx y (ω j ) We will use a time series generated from

1 ’ κx y (ω j ) a bivariate AR(1) process with a rotational

parameter matrix (11.51) to illustrate these

is approximately distributed as an F(2, r ’ 2)