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7.4 Multivariate ARMA Processes 241
7.4 Multivariate ARMA Processes
and
ˆWt1 . A study of the sample values of ˆWt2 − 4.74 ˆWt−3,1 suggests the model
(1 + .345B)Nt Ut, {Ut} ∼WN(0,.0782) (7.3.3)
for {Nt}. Finally, replacing ˆWt2 and ˆWt−3,1 in (7.3.2) by Zt2 and Zt−3,1, respectively,
and then using (7.1.1) and (7.1.2) to express Zt2 and Zt−3,1 in terms of {Dt2} and
{Dt1}, we obtain a model relating {Dt1}, {Dt2}, and {Ut1}, namely,
Dt2 + .0773 (1 − .610B)(1 − .838B) −1 [4.74(1 − .474B) −1 Dt−3,1
+ (1 + .345B) −1 Ut].
This model should be compared with the one derived later in Section 10.1 by the
more systematic technique of transfer function modeling.
As in the univariate case, we can define an extremely useful class of multivariate stationary
processes {Xt} by requiring that {Xt} should satisfy a set of linear difference
equations with constant coefficients. Multivariate white noise {Zt} (see Definition
7.2.2) is a fundamental building block from which these ARMA processes are constructed.
Definition 7.4.1 {Xt} is an ARMA(p, q) process if {Xt} is stationary and if for every t,
Xt − 1Xt−1 −···−pXt−p Zt + 1Zt−1 +···+qZt−q, (7.4.1)
where {Zt} ∼WN(0,| ).({Xt} is an ARMA(p, q) process with mean µ if {Xt −µ}
is an ARMA(p, q) process.)
Equations (7.4.1) can be written in the more compact form
(B)Xt (B)Zt, {Zt} ∼WN(0,| ), (7.4.2)
where (z) : I − 1z − ··· − pz p and (z) : I + 1z + ··· + qz q are
matrix-valued polynomials, I is the m×m identity matrix, and B as usual denotes the
backward shift operator. (Each component of the matrices (z), (z) is a polynomial
with real coefficients and degree less than or equal to p, q, respectively.)
Example 7.4.1 The multivariate AR(1) process
Setting p 1 and q 0 in (7.4.1) gives the defining equations
Xt Xt−1 + Zt, {Zt} ∼WN(0,| ), (7.4.3)