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104 Chapter 3 ARMA Models
and
2 E Xn+1 − ˆXn+1 σ 2 rn rn.
The results are shown in Table 3.1.
h-Step Prediction of an ARMA(p, q) Process
As in Section 2.5, we use PnY to denote the best linear predictor of Y in terms of
X1,...,Xn (which, as pointed out after (3.3.4), is the same as the best linear predictor
of Y in terms of W1,...,Wn). Then from (2.5.30) we have
PnWn+h
n+h−1
jh
θn+h−1,j
Wn+h−j − ˆWn+h−j
σ 2
n+h−1
jh
θn+h−1,j
Xn+h−j − ˆXn+h−j
Using this result and applying the operator Pn to each side of equations (3.3.1), we
conclude that the h-step predictors PnXn+h satisfy
⎧
n+h−1
θn+h−1,j Xn+h−j ⎪⎨
− ˆXn+h−j ,
jh
PnXn+h
p
n+h−1
⎪⎩ φiPnXn+h−i + θn+h−1,j Xn+h−j − ˆXn+h−j ,
1 ≤ h ≤ m − n,
h>m− n.
(3.3.11)
Table 3.1
i1
jh
If, as is almost always the case, n>m max(p, q), then for all h ≥ 1,
p
q
PnXn+h φiPnXn+h−i +
. (3.3.12)
i1
jh
θn+h−1,j
Xn+h−j − ˆXn+h−j
Once the predictors ˆX1,... ˆXn have been computed from (3.3.7), it is a straightforward
calculation, with n fixed, to determine the predictors PnXn+1,PnXn+2,PnXn+3,...
ˆXn+1 for the ARMA(2,3) Process of Example 3.3.4.
n Xn+1 rn θn1 θn2 θn3 ˆXn+1
0 1.704 7.1713 0
1 0.527 1.3856 0.8982 1.5305
2 1.041 1.0057 1.3685 0.7056 −0.1710
3 0.942 1.0019 0.4008 0.1806 0.0139 1.2428
4 0.555 1.0019 0.3998 0.2020 0.0732 0.7443
5 −1.002 1.0005 0.3992 0.1995 0.0994 0.3138
6 −0.585 1.0000 0.4000 0.1997 0.0998 −1.7293
7 0.010 1.0000 0.4000 0.2000 0.0998 −0.1688
8 −0.638 1.0000 0.4000 0.2000 0.0999 0.3193
9 0.525 1.0000 0.4000 0.2000 0.1000 −0.8731
10 1.0000 0.4000 0.2000 0.1000 1.0638
11 1.0000 0.4000 0.2000 0.1000
12 1.0000 0.4000 0.2000 0.1000
.