Springer - Read

250 Chapter 7 Multivariate Time Series
7.6.2 Forecasting Multivariate Autoregressive Processes
The technique developed in Section 7.5 allows us to compute the minimum mean
squared error one-step linear predictors ˆXn+1 for any multivariate stationary time
series from the mean µ and autocovariance matrices Ɣ(h) by recursively determining
the coefficients ni,i 1,...,n, and evaluating
ˆXn+1 µ + n1(Xn − µ) +···+nn(X1 − µ). (7.6.9)
The situation is simplified when {Xt} is the causal AR(p) process defined by
(7.6.4), since for n ≥ p (as is almost always the case in practice)
ˆXn+1 1Xn +···+pXn+1−p. (7.6.10)
To verify (7.6.10) it suffices to observe that the right-hand side has the required form
(7.5.2) and that the prediction error
Xn+1 − 1Xn −···−pXn+1−p Zn+1
is orthogonal to X1,...,Xn in the sense of (7.5.3). (In fact, the prediction error is
orthogonal to all Xj, −∞