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246 Chapter 7 Multivariate Time Series
Remark 5. The innovations algorithm also has a multivariate version that can be
used for prediction in much the same way as the univariate version described in
Section 2.5.2 (for details see TSTM, Proposition 11.4.2).
7.6 Modeling and Forecasting with Multivariate AR Processes
If {Xt} is any zero-mean second-order multivariate time series, it is easy to show from
the results of Section 7.5 (Problem 7.4) that the one-step prediction errors Xj − ˆXj,
j 1,...,n, have the property
′
E
0 for j k. (7.6.1)
Xj − ˆXj
Xk − ˆXk
Moreover, the matrix M such that
⎡ ⎤ ⎡ ⎤
X1 − ˆX1 X1
⎢ X2 − ˆX2
⎥ ⎢ ⎥
⎥ ⎢ X2 ⎥
⎢ ⎥ ⎢ ⎥
⎢ X3 − ˆX3 ⎥
⎢ ⎥ M⎢
X3 ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣ . ⎦ ⎣ . ⎦
Xn − ˆXn
Xn
(7.6.2)
is lower triangular with ones on the diagonal and therefore has determinant equal
to 1.
If the series {Xt} is also Gaussian, then (7.6.1) implies that the prediction errors
Uj Xj − ˆXj, j 1,...,n, are independent with covariance matrices V0,...,Vn−1,
respectively (as specified in (7.5.7)). Consequently, the joint density of the prediction
errors is the product
f(u1,...,un) (2π) −nm/2
n
j1
detVj−1
−1/2
exp − 1
2
n
j1
u ′ −1
jVj−1uj Since the determinant of the matrix M in (7.6.2) is equal to 1, the joint density of the
observations X1,...,Xn at x1,...,xn is obtained on replacing u1,...,un in the last
expression by the values of Xj − ˆXj corresponding to the observations x1,...,xn.
If we suppose that {Xt} is a zero-mean m-variate AR(p) process with coefficient
matrices {1,...,p} and white noise covariance matrix | , we can therefore
express the likelihood of the observations X1,...,Xn as
L(,| ) (2π) −nm/2
n
j1
detVj−1
−1/2
exp
− 1
2
n
j1
U ′ −1
jVj−1Uj where Uj Xj − ˆXj, j 1,...,n, and ˆXj and Vj are found from (7.5.4), (7.5.6),
and (7.5.7).
,
.