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336 Chapter 10 Further Topics
Figure 10-1
The sample correlation
functions ˆρij(h), of Example
10.1.1. Series 1 is { ˆZt }
and Series 2 is { ˆYt }.
is determined and filed using ITSM as described in step (5). It satisfies the equations
ˆNt Xt2 − 4.86B 3 (1 − .698B) −1 Xt1, t 5, 6,...,150.
Analysis of this univariate series with ITSM gives the MA(1) model
Nt (1 − .364B)Wt, {Wt} ∼WN(0,.0590).
Substituting these preliminary noise and transfer function models into equation
(10.1.1) then gives
Xt2 4.86B 3 (1 − .698B) −1 Xt1 + (1 − .364B)Wt, {Wt} ∼WN(0,.0590).
Now minimizing the sum of squares (10.1.7) with respect to the parameters
w0,v1,
as described in step (7), we obtain the least squares model
θ (N)
1
Xt2 4.717B 3 (1 − .724B) −1 Xt1 + (1 − .582B)Wt, (10.1.8)
where {Wt} ∼WN(0,.0486) and
Xt1 (1 − .474B)Zt, {Zt} ∼WN(0,.0779).
Notice the reduced white noise variance of {Wt} in the least squares model as compared
with the preliminary model.
The sample auto- and cross-correlation functions of the series ˆZt and ˆWt, t
5,...,150, are shown in Figure 10.2. All of the correlations lie between the bounds
±1.96/ √ 144, supporting the assumption underlying the fitted model that the residuals
are uncorrelated white noise sequences.