Springer - Read

Figure 10-2
The sample correlation
functions of the estimated
residuals from the
model fitted in Example
10.1.1. Series 1 is { ˆZt }
and Series 2 is { ˆWt }.
10.1 Transfer Function Models 337
10.1.1 Prediction Based on a Transfer Function Model
When predicting Xn+h,2 on the basis of the transfer function model defined by (10.1.1),
(10.1.4), and (10.1.7), with observations of Xt1 and Xt2, t 1,...,n, our aim is to
find the linear combination of 1,X11,...,Xn1,X12,...,Xn2 that predicts Xn+h,2 with
minimum mean squared error. The exact solution of this problem can be found with
the help of the Kalman recursions (see TSTM, Section 13.1 for details). The program
ITSM uses these recursions to compute the predictors and their mean squared errors.
In order to provide a little more insight, we give here the predictors ˜PnXn+h
and mean squared errors based on infinitely many past observations Xt1 and Xt2,
−∞