H16
The discussion of
bivariate
processes is readily extended to the general
multivariate
case.
If we have
m
processes,
, each having zero mean, we define
,1
,2
,
{
},{
}, .
..,{
}
tt
t
XX
X
m
m
the
covariance matrix at lag
h
by
( )
[
( )],
1,.
..,
,
1,.
..,
ij
hh
i
m
j
γ
Γ=
=
=
,
where
.
For
,,
()
ij
t h i
t j
hE
X
X
+
⎡⎤
=
⎣⎦
,(
ii
ij h
)
=
denotes the ACVF of
,
,
ti
X
while for
ij
)
≠
denotes the cross-covariance function between
and
.
,
X
,
tj
X
We assume that the
m
processes are
jointly stationary
,
i.e. that for all
ij
ij
h
is a function of
h
only and does not depend on
t
.
Each of the three main types of univariate models has its corresponding multivariate extension, which is
obtained, essentially, by replacing the
scalar
parameters in the univariate model by
matrix
parameters.
Let
and
be
m
-dimensional random vectors.
,
[
,...,
]'
t
m
=
X
,
[
,...,
t
m
ZZ
=
Z
A zero-mean,
m
-variate
model
is defined by
ARMA( , )
pq
(*)
11
...
...
p
t
p
t
t
q
−−
−
=
Φ
++
Φ
+ +
Θ
Θ
X
Z
Z
Z
t
q
−
q
where AR and MA coefficients,
{
and
{
} (
1,.
.., )
i
ip
Φ=
} (
1,.
.., )
i
i
Θ
=
, are real
matrices, and
mm
×
is the
multivariate white noise,
i.e., with zero mean vector and uncorrelated values at different times