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168 Chapter 5 Modeling and Forecasting with ARMA Processes
are given by
P57X57+h −4.035 +
1
jh
θ57+h−1,j
X57+h−j − ˆX57+h−j
⎧
⎨ −4.035 + θ57,1 X57 − ˆX57 ,
⎩
−4.035,
with mean squared error
if h 1,
if h>1,
2040.75r57, if h 1,
E(X57+h − P57X57+h) 2
2040.75(1 + (−.818) 2 ), if h>1,
where θ57,1 and r57 are computed recursively from (3.3.9) with θ −.818.
These calculations are performed with ITSM by fitting the maximum likelihood
model (5.4.1), selecting Forecasting>ARMA, and specifying the number of forecasts
required. The 1-step, 2-step, ...,and 7-step forecasts of Xt are shown in Table 5.1.
Notice that the predictor of Xt for t ≥ 59 is equal to the sample mean, since under
the MA(1) model {Xt,t ≥ 59} is uncorrelated with {Xt,t ≤ 57}.
Assuming that the innovations {Zt} are normally distributed, an approximate 95%
prediction interval for X64 is given by
−4.0351 ± 1.96 × 58.3602 (−118.42, 110.35).
The mean squared errors of prediction, as computed in Section 3.3 and the example
above, are based on the assumption that the fitted model is in fact the true model
for the data. As a result, they do not reflect the variability in the estimation of the
model parameters. To illustrate this point, suppose the data X1,...,Xn are generated
from the causal AR(1) model
Xt φXt−1 + Zt, {Zt} ∼iid 0,σ 2 .
Table 5.1 Forecasts of the next 7 observations
of the overshort data of Example
3.2.8 using model (5.4.1).
# XHAT SQRT(MSE) XHAT + MEAN
58 1.0097 45.1753 −3.0254
59 0.0000 58.3602 −4.0351
60 0.0000 58.3602 −4.0351
61 0.0000 58.3602 −4.0351
62 0.0000 58.3602 −4.0351
63 0.0000 58.3602 −4.0351
64 0.0000 58.3602 −4.0351