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230 Chapter 7 Multivariate Time Series
The second-order properties of the multivariate time series {Xt} are then specified by
the mean vectors
⎡ ⎤
µt1
⎢ ⎥
µt : EXt ⎢ ⎥
⎣ . ⎦
(7.2.4)
µtm
and covariance matrices
⎡
⎤
γ11(t + h, t)
⎢
Ɣ(t + h, t) : ⎢
⎣ .
···
. ..
γ1m(t + h, t)
⎥
. ⎦ , (7.2.5)
γm1(t + h, t) ··· γmm(t + h, t)
where
γij (t + h, t) : Cov(Xt+h,i,Xt,j).
Remark 1. The matrix Ɣ(t + h, t) can also be expressed as
Ɣ(t + h, t) : E[(Xt+h − µt+h)(Xt − µt) ′ ],
where as usual, the expected value of a random matrix A is the matrix whose components
are the expected values of the components of A.
As in the univariate case, a particularly important role is played by the class of
multivariate stationary time series, defined as follows.
Definition 7.2.1 The m-variate series {Xt} is (weakly) stationary if
and
and
(i) µX(t) is independent of t
(ii) ƔX(t + h, t) is independent of t for each h.
For a stationary time series we shall use the notation
⎡ ⎤
µ1
⎢ ⎥
µ : EXt ⎢ ⎥
⎣ . ⎦
µm
(7.2.6)
Ɣ(h) : E[(Xt+h − µ)(Xt − µ) ′ ⎡
⎤
γ11(h)
⎢
] ⎢
⎣ .
···
. ..
γ1m(h)
⎥
. ⎦ . (7.2.7)
γm1(h) ··· γmm(h)
We shall refer to µ as the mean of the series and to Ɣ(h) as the covariance matrix at
lag h. Notice that if {Xt} is stationary with covariance matrix function Ɣ(·), then for