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242 Chapter 7 Multivariate Time Series
for the multivariate AR(1) series {Xt}. By exactly the same argument as used in
Example 2.2.1, we can express Xt as
∞
Xt j Zt−j, (7.4.4)
j0
provided that all the eigenvalues of are less than 1 in absolute value, i.e.„ provided
that
det(I − z) 0 for all z ∈ C such that |z| ≤1. (7.4.5)
If this condition is satisfied, then the coefficients j are absolutely summable, and
hence the series in (7.4.4) converges; i.e., each component of the matrix n j0 jZt−j converges (see Remark 1 of Section 2.2). The same argument as in Example 2.2.1 also
shows that (7.4.4) is the unique stationary solution of (7.4.3). The condition that all
the eigenvalues of should be less than 1 in absolute value (or equivalently (7.4.5))
is just the multivariate analogue of the condition |φ| < 1 required for the existence
of a causal stationary solution of the univariate AR(1) equations (2.2.8).
Causality and invertibility of a multivariate ARMA(p, q) process are defined
precisely as in Section 3.1, except that the coefficients ψj, πj in the representations
Xt ∞ j0 ψjZt−j and Zt ∞ j0 πjXt−j are replaced by m × m matrices j
and j whose components are required to be absolutely summable. The following
two theorems (proofs of which can be found in TSTM) provide us with criteria for
causality and invertibility analogous to those of Section 3.1.
Causality:
An ARMA(p, q) process {Xt} is causal, oracausal function of {Zt}, if there
exist matrices {j} with absolutely summable components such that
∞
Xt jZt−j for all t. (7.4.6)
j0
Causality is equivalent to the condition
det (z) 0 for all z ∈ C such that |z| ≤1. (7.4.7)
The matrices j are found recursively from the equations
j j +
∞
kj−k, j 0, 1,..., (7.4.8)
k1
where we define 0 I, j 0 for j>q, j 0 for j>p, and j 0 for
j