Multivariate Time Series Analysis

Let {xt : t T} be a Multivariate time series.

Definition:

(t) = mean value function of {xt : t T}

= E[xt] for t T.

(t,s) = Lagged covariance matrix of {xt : t T} = E{[ xt - (t)][ xs - (s)]'} for t,s T

Definition:

The time series {xt : t T} is stationary if the joint distribution of

is the same as the joint distribution of

for all finite subsets t1, t2, ... , tk of T and all choices of h.

kttt xxx ,,,21

hththt k xxx ,,,21

In this case then for t T.

and(t,s) = E{[ xt - ][ xs - ]'}

= E{[ xt+h - ][ xs+h - ]'}

= E{[ xt-s - ][ x0 - ]'}

= (t - s) for t,s T.

μμ )()( itxEt

Definition:The time series {xt : t T} is weakly stationary if :

for t T.and

(t,s) = (t - s) for t, s T.

μμ )(t

In this case

(h) = E{[ xt+h - ][ xs - ]'}

= Cov(xt+h,xt )

is called the Lagged covariance matrix of the process {xt : t T}

The Cross Correlation Function and the Cross Spectrum

Note: ij(h) = (i,j)th element of (h),

and is called the cross covariance function of

jht

it xx ,cov

. and js

it xx

00 jjii

ijij

hh

is called the cross correlation function of . and j

si

t xx

Definitions:

. and js

it xxis called the cross spectrum of

ki

kijij ekf

21i)

Note: since ij(k) ≠ ij(-k) then fij() is complex.

If fij() = cij() - i qij() then cij() is called the Cospectrum (Coincident spectral density) and qij() is called the quadrature spectrum

ii)

If fij() = Aij() exp{iij()} then Aij() is called the Cross Amplitude Spectrum and ij() is called the Phase Spectrum.

iii)

Definition:

is called the Spectral Matrix

pppp

p

p

ijpp

fff

ffffff

f

21

22221

11211

F

The Multivariate Wiener-Khinchin Relations (p-variate)

and

h

hi

ppppeh

ΣF

21

dehpp

hi

ppFΣ

Lemma:

i) Positive semidefinite:a*F()a ≥ 0 if a*a ≥ 0, where a is any complex vector.

ii) Hermitian:F() = F*() = the Adjoint of F() = the complex conjugate transpose of F(). i.e.fij() = .

Assume that

||

hij h

Then F() is:

Corrollary:The fact that F() is positive semidefinite also means that all square submatrices along the diagonal have a positive determinant

0

jjji

ijii

ffffHence

jiijjjii ffff and

jjiiijijij fffff 2*or

Definition:

= Squared Coherency function

jjii

ijij ff

fK

2

2

12 ijKNote:

Definition:

ii

ijij f

f

. and with associated jt

it xxfunctionTransfer

Applications and Examples of Multivariate Spectral Analysis

Example I - Linear Filters

denote a bivariate time series with zero mean.

Let

t = ..., -2, -1, 0, 1, 2, ...

Tt

yx

t

t :

sstst xay

Suppose that the time series {yt : t T} is constructed as follows:

The time series {yt : t T} is said to be constructed from {xt : t T} by means of a Linear Filter.

httyy yyEh

'''

sshts

ssts xaxaE

ssht

sstss xxaaE '

''

s s

shtstss xxEaa'

''

continuing hyy

s sss sshaa

'' '

s s

xxsshi

ss dfeaa'

''

s s

xxsshi

ss dfeaa'

''

s s

xxsisi

sshi dfeeaae

'

''

dfeaeae xxs

sis

s

sis

hi

'

''

continuing hyy

dfeae xxs

sis

hi2

dfAe xxhi

2

Thus the spectral density of the time series {yt : t T} is:

xxxx

s

sisyy fAfeaf 2

2

Comment A:

is called the Transfer function of the linear filter.

is called the Gain of the filter while

is called the Phase Shift of the filter.

s

siseaA

A

Aarg

Also httxy yxEh

sshtst xaxE

s

shtts xxEa

sxxs sha

continuing

hxy

dfeas

xxshi

s

dfea xxs

shis

dfAe xxhi

Thus cross spectrum of the bivariate time series

Tt

yx

t

t :

is:

xx

sxx

sisxy fAfeaf

Comment B:

= Squared Coherency function.

yyxx

xyxy ff

fK

2

2

1 2

22

xxxx

xx

fAf

fA

Example II - Linear Filterswith additive noise at the output

denote a bivariate time series with zero mean.

Let

t = ..., -2, -1, 0, 1, 2, ...

Tt

yx

t

t :

Suppose that the time series {yt : t T} is constructed as follows:

ts

stst vxay

The noise {vt : t T} is independent of the series {xt : t T} (may be white)

httyy yyEh

shtshts

ststs vxavxaE

s

htstss s

shtstss vxEaxxEaa'

''

thts

tshts vvEvxEa

'''

hsshaa vvs s

xxss

'' '

dfedfeae vvhi

xxs

sis

hi

2

continuing

hyy

dffAe vvxxhi 2

s

siseaA where

Thus the spectral density of the time series {yt : t T} is:

vvxxyy ffAf 2

Also httxy yxEh

shtshtst vxaxE

htts

shtts vxExxEa

sxxs sha

continuing

hxy

dfeas

xxshi

s

dfea xxs

shis

dfAe xxhi

Thus cross spectrum of the bivariate time series

Tt

yx

t

t :

is:

xx

sxx

sisxy fAfeaf

Thus

= Squared Coherency function.

yyxx

xyxy ff

fK

2

2

vvxxxx

xx

ffAf

fA

2

22

11

1

1 2

xx

vv

fAf

Noise to Signal Ratio

Estimation of the Cross Spectrum

Let

T

T

yx

yx

yx

,,,2

2

1

1

denote T observations on a bivariate time series with zero mean.If the series has non-zero mean one uses

in place of

yyxx

t

t

t

t

yx

Again assume that T = 2m +1 is odd.

Then define:

and

with k = 2k/T and k = 0, 1, 2, ... , m.

T

tkt

xk

T

tkt

xk tx

Tbtx

Ta

11

)sin(2,)cos(2

T

tkt

yk

T

tkt

yk ty

Tbty

Ta

11

)sin(2,)cos(2

Also

and

for k = 0, 1, 2, ... , m.

T

tkt

xk

xkk tix

TibaX

1

exp2

T

tkt

yk

ykk tiy

TibaY

1

exp2

The Periodogram &

the Cross-Periodogram

Also

and

for k = 0, 1, 2, ... , m.

2

1

2

1

)cos()sin(2 T

tkt

T

tktk

xxT txtx

TI

222

222 kkkxk

xk XTXXTbaT

2

1

2

1

)cos()sin(2 T

tkt

T

tktk

yyT tyty

TI

222

222 kkkyk

yk YTYYTbaT

Finally

yk

yk

xk

xkkkk

xyT ibaibaTYXTI

22

yk

xk

yk

xk

yk

xk

yk

xk baabibbaaT

2

xk

xk

yk

ykkkk

yxT ibaibaTXYTI

22

yk

xk

yk

xk

yk

xk

yk

xk baabibbaaT

2

kxy

TI of conjugatecomplex

Note:

and

1

1 1

)exp(2 T

Thkht

hT

ttk

xxT hixx

TI

1

1

)exp(2T

Thkxx hihC

1

1 1

)exp(2 T

Thkht

hT

ttk

yyT hiyy

TI

1

1

)exp(2T

Thkyy hihC

Also

and

1

1 1

)exp(2 T

Thkht

hT

ttk

xyT hiyx

TI

1

1

)exp(2T

Thkxy hihC

1

1 1

)exp(2 T

Thkht

hT

ttk

yxT hixy

TI

1

1

)exp(2T

Thkyx hihC

The sample cross-spectrum, cospectrum

& quadrature spectrum

Recall that the periodogram

2

1

2

1

)cos()sin(2 T

tkt

T

tktk

xxT txtx

TI

has asymptotic expectation 4fxx().

kxy

TI Similarly the asymptotic expectation of

is 4fxy().

An asymptotic unbiased estimator of fxy() can be obtained by dividing by 4. k

xyTI

The sample cross spectrum

1

1

)exp(21

21ˆ

T

Thkxyk

xyTkxy hihCIf

The sample cospectrum

1

1

)cos(21ˆReˆ

T

Thkxykxykxy hihCfc

The sample quadrature spectrum

1

1

)sin(21ˆImˆ

T

Thkxykxykxy hihCfq

The sample Cross amplitude spectrum,

Phase spectrum &

Squared Coherency

Recall

22 xyxyxy qcSpectrumAmplitudeCrossA

xy

xyxy c

qSpectrumPhase 1tan

functionCoherencySquared

ffqc

ff

fK

yyxx

xyxy

yyxx

xyxy

222

2

Thus their sample counter parts can be defined in a similar manner. Namely

\ sample ˆ SpectrumAmplitudeCrossAxy

xy

xyxy c

qSpectrumPhase

ˆˆ

tan sample ˆ 1

functionCoherencySquared

ff

qc

ff

fK

yyxx

xyxy

yyxx

xy

xy

sample

ˆˆˆˆ

ˆˆ

ˆˆ

222

2

22 ˆˆ xyxy qc

Consistent Estimation of the Cross-spectrum fxy()

Daniell Estimator

= The Daniell Estimator of the Cospectrum

d

drrkk

xyTd c

dc ˆ

121ˆ ,

= The Daniell Estimator of the quadrature spectrum

d

drrkk

xyTd q

dq ˆ

121ˆ ,

Weighted Covariance Estimator

hhChwc kh

xymkxy

mTw

cos21ˆ ,,

hhChwq kh

xymkxy

mTw

sin21ˆ ,,

of sequence a are ,2,1,0: where hhwm

such that weights

100 i) mm whw

hwhw mm ii)

.for 0 iii) mhhwm

Again once the Cospectrum and Quadrature Spectrum have been estimated,The Cross spectrum, Amplitude Spectrum, Phase Spectrum and Coherency can be estimated generally as follows using either the

a) Daniell Estimator or b) the weighted covariance estimator

of cxy() and qxy():

Namely

22 ˆˆˆxyxyxy qcA

xy

xyxy c

qˆˆ

tanˆ 1

yyxx

xyxy

yyxx

xy

xy ff

qc

ff

fK ˆˆ

ˆˆˆˆ

ˆˆ

222

2

xyxyxy qicf ˆˆˆ