generally only be two-dimensional and, unless

ideas developed in Section 12.3 naturally carry

stated otherwise, matrices will be 2 — 2.

over to the multivariate setting.

The ˜— ™ operator will denote the conjugate

The procedure used for univariate spectral

transpose when applied to a matrix or vector

analysis is also used here. We ¬rst describe the

quantity.

multivariate extension of the periodogram, then

brie¬‚y describe the extension of the periodogram- We will use the notation Xt or, more precisely,

{Xt : t ∈ Z} to represent a bivariate stochastic

based spectral estimators. The basic approach is

to use periodogram averaging (i.e., chunks) or process, and we will identify the components of

smoothing to construct good estimates of the Xt as Xt and Yt .

cross-spectral density function or, equivalently, The same assumptions made about Xt in

the co- and quad-spectra. These estimates are Section 12.4 also apply here.

then used in the obvious way to estimate derived

quantities, such as the coherency and phase

spectra. Approximate con¬dence intervals are 12.5.2 The Bivariate Periodogram. Let

presented for both. x1 , . . . , xT be a time series of length T that is

The tradeoff between bias and variance (see observed from process {Xt : t ∈ Z}.

again Figure 5.3) is delicate in cross-spectral

analysis. As with univariate spectral analysis,

The bivariate periodogram I T j is given by

large equivalent bandwidth is associated with

low variance and (potentially) large bias. The

—

I T j = (T /4)ZT j ZT j

bias induced by excessive periodogram smoothing (12.56)

can be quite misleading since large bandwidth

spectral estimators have the potential to shift

for j = ’q, . . . , q, q = T

where

2

the location of peaks in the coherency spectrum.

26 Note that this procedure may reject the null hypothesis

T

2

γx y („ ) = 0 when it is true more or less frequently than

Xt e’2πiω j t .

=

ZT j (12.57)

the nominal 5% signi¬cance level. The procedure uses an T t=1

approximation in which the true cross-correlation function has

been replaced with an estimate. Also, in most applications, the

estimated cross-correlation function is screened at many lags „ . We will use notation such as Z X T j and ZY T j

Thus the effect of multiplicity (conducting many related tests at

to identify the elements of ZT j , and use I x x T j ,

a given signi¬cance level, cf. Section 6.8) must be accounted

I x yT j and I yyT j to identify the elements of the

for when interpreting the test results.

12.5: Estimating the Cross-spectrum 283

2 — 2 bivariate periodogram matrix IT j . As in the Var Im(I x y T j ) ≈

univariate case, it is easily demonstrated that: x x (ω j ) yy (ω j )

1

2

’ x y (ω j )2 + x y (ω j ) .

2

1 the bivariate periodogram distributes the total

lag zero sample covariance matrix:

5 The co-variance between the real and

cx x (0) cx y (0)

ΣX,X = imaginary parts of the cross-periodogram can

c yx (0) c yy (0) be approximated by

q

2

= IT j ; Cov Re(I x y T j ), Im(I x y T j ) ≈

T j=1

x y (ω j ) x y (ω j ).

2 the estimated bivariate auto-covariance func-

tion 6 The real and imaginary parts of the cross-

periodogram are correlated with the peri-

cx x („ ) cx y („ )

Σ(„ ) = odograms of the X and Y components of X

c yx (’„ ) c yy („ )

(see Bloom¬eld [49] for details).

and the bivariate periodogram are a Fourier

7 The periodograms of the X and Y compo-

transform pair.

nents of X are also correlated (see Bloom¬eld

[49] for details).

12.5.3 Properties of the Bi-variate Periodo-

gram. The following are some of the properties 8 The bivariate periodogram ordinates have

of the bivariate periodogram (12.56). a complex Wishart distribution that has

properties analogous to those of the χ 2

1 The bivariate periodogram is Hermitian, that

distribution. The theory was developed by

is,

Goodman [144]. See Brillinger [66] or

IT j = I— j . Brockwell and Davis [68] for details.

T

2 The bivariate periodogram ordinates are One reason the bivariate periodogram is not

asymptotically uncorrelated. This is proven a good spectral estimator is that, just as with

using an argument exactly analogous to that its univariate counterpart, its variability can not

in [12.3.5]. be reduced by taking larger and larger samples.

Instead we end up with increasing numbers of

3 The bivariate periodogram ordinates I T j are

periodogram ordinates, all with approximately the

asymptotically unbiased estimators of the

same information content. This is demonstrated by

bivariate spectral density function

items 4“8 above.

x x (ω j ) x y (ω j ) Another dif¬culty with the bivariate peri-

— (ω )

yy (ω j ) odogram is that it produces degenerate coherency

j

xy

estimates. To see this, let us represent the X and Y

evaluated at the Fourier frequencies ω j . The components of Fourier transform (12.57) as

argument is also analogous to that in [12.3.6].

Zx T j = Ax j + iBx j

In particular, the cross-periodogram

Zy T j = Ay j + iBy j .

T

I x y T j = Zx T j Z— T j y

4 Then

T2

is an unbiased estimator of the cross-spectral I

xx T j = (Ax j + B2 j )x

density x y (ω j ). 4

T

Iyy T j = (A2 j + B2 j )

We will see below, and in [12.5.4], that I x y T j

4y y

is not a very good estimator of x y (ω j ).

T

Ixy T j = Ax j Ay j + Bx j By j

4 The variance of the real and imaginary parts 4

of the cross-periodogram can be approxi- + i (Bx j Ay j ’ Ax j By j ) .

mated by (see Bloom¬eld [49, Section 9.4])

The resulting coherency estimate

Var Re(I x y T j ) ≈

|Ixy T j |2

(ω j ) yy (ω j )

1

κx y (ω j ) =

xx

2

+ x y (ω j )2 ’ x y (ω j )2 , Ixx T j Iyy T j

12: Estimating Covariance Functions and Spectra

284

is easily shown to be unity at all Fourier can be thought of as a squared correlation

frequencies ω j . coef¬cient that depends upon frequency.

This is most easily appreciated by considering

Thus the bivariate periodogram can not be