Practice Problems
1.
Consider the AR(1) model:
(
Y
t
–
μ
)
=
φ
(
Y
t
– 1
–
μ
)
+
e
t
where
e
t
is a mean zero white noise process.
Given the time series of 5 observations:
y
1
=
10.1
,
y
2
=
9.3
,
y
3
=
9.4
,
y
4
=
9.8
,
y
5
=
10.6
a)
Calculate the first two sample autocorrelation coefficients,
r
1
and
r
2
.
(
Note: In practice reliable autocorrelation estimates are only obtained from
series consisting of approximately 50 observations or more.
)
b)
Obtain the leastsquares estimates for the AR(1) model parameters,
°
ˆ
and
φ
ˆ
.
2.
Consider the AR(1) model:
(
Y
t
–
μ
)
=
φ
(
Y
t
– 1
–
μ
)
+
e
t
where
e
t
is a mean zero white noise process. The model has been fitted to a time
series giving
φ
ˆ
=
0.8
,
°
ˆ
=
10.2
, and
2
±
ˆ
e
=
0.25
.
The last five values of the series are
y
96
=
10.1
,
y
97
=
9.3
,
y
98
=
9.4
,
y
99
=
9.8
,
y
100
=
10.6
. Using the
t
=
N
= 100 as the forecast origin, forecasts the next three
observations. Calculate the 95% probability limits for the next three observations.
3.
(
Homework 13, Problem
4
)
Consider the ARMA
(
1,
1
)
model
(
Y
t
– 60
)
+ 0.3
(
Y
t
– 1
– 60
)
=
e
t
– 0.4
e
t
– 1
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 Spring '07
 STEPANOV
 Autocorrelation, Das Model, Stationary process, Time series analysis, Yt

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