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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
ˆ
.
2.
Consider the AR(1) model:
(
Y
t
–
μ
) =
(
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
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.
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This note was uploaded on 02/10/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 STEPANOV
 Statistics

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