ORIE 4630 — D. Ruppert
Homework #4
This assignment will not be collected, but
in will be covered in the Exam on Sep 24
To help you study for the Exam 1, I am giving the answer to these problems now.
1. Suppose you just ﬁt at AR(2) model to a time series
Y
t
, t
= 1
,...,n
, and the estimates
were
b
μ
= 100
.
1,
b
φ
1
= 0
.
5, and
b
φ
2
= 0
.
1. The last three observations were:
Y
n

2
=
101
.
0,
Y
n

1
= 99
.
5, and
Y
n
= 102
.
3. What are the forecasts of
Y
n
+1
,
Y
n
+2
, and
Y
n
+3
?
Answer:
b
Y
n
+1
= 101
.
14,
b
Y
n
+2
= 100
.
84,
b
Y
n
+3
= 100
.
574. Notice that
Y
n

2
is not
needed for forecasting with the AR(2) model. This problem can be done without the
use of
R
but it is a little faster in
R
than by hand. The following
R
code computed
the forecasts:
mu = 100.1
phi1 = 0.5
phi2 = 0.1
y = c(99.5,102.3)
yhat1 = mu + phi1*(y[2]mu) + phi2*(y[1]mu)
yhat2 = mu + phi1*(yhat1mu) + phi2*(y[2]mu)
yhat3 = mu + phi1*(yhat2mu) + phi2*(yhat1mu)
yhat1
yhat2
yhat3
> yhat1
[1] 101.14
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 '10
 RUPPERT
 Statistics, Forecasting, Autoregressive integrated moving average, Trend estimation, yellow colored regions

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