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7.6 Modeling and Forecasting with Multivariate AR Processes 253
Remark 6. In the special case where (z) I (i.e., in the purely autoregressive
case) the expression (7.6.18) for the h-step error covariance matrix is exact for all
n ≥ p (i.e., if there are at least p + r observed vectors) . The program ITSM allows
differencing transformations and subtraction of the mean before fitting a multivariate
autoregression. Predicted values for the original series and the standard deviations
of the prediction errors can be determined using the multivariate option Forecasting>AR
Model.
Remark 7. In the multivariate case, simple differencing of the type discussed in
this section where the same operator D(B) is applied to all components of the random
vectors is rather restrictive. It is useful to consider more general linear transformations
of the data for the purpose of generating a stationary series. Such considerations lead
to the class of cointegrated models discussed briefly in Section 7.7 below.
Example 7.6.4 Sales with a leading indicator
Assume that the model fitted to the bivariate series {Yt,t 0,...,149} in Example
7.6.2 is correct, i.e., that
(B)Xt Zt, {Zt} ∼WN 0, ˆ| ,
where
Xt (1 − B)Yt − (.0228,.420) ′ , t 1,...,149,
(B) I − ˆ1B −···− ˆ5B 5 , and ˆ1,..., ˆ5, ˆ| are the matrices found in Example
7.6.2. Then the one- and two-step predictors of X150 and X151 are obtained from
(7.6.11) as
and
P149X150 ˆ1X149 +···+ ˆ5X145
.163
−.217
P149X151 ˆ1P149X150 + ˆ2X149 +···+ ˆ5X146
with error covariance matrices, from (7.6.15),
.076 −.003
|
−.003 .095
and
respectively.
| +ˆ1| ˆ ′
1
.096 −.002
,
−.002 .095
−.027
.816