Practice Problems
1.
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
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.
)
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
,
μ
ˆ
=
10.2
,
and
2
σ
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.
Consider the
AR
(
2
)
processes
Y
&
t
– 0.98 Y
&
t
– 1
+ 0.96 Y
&
t
– 2
=
e
t
where
{
e
t
}
is zeromean white noise
(
i.i.d.
N
(
0,
2
e
σ
)
)
,
Y
&
t
=
Y
t
–
μ
.
a)
Is this process stationary?
Justify your answer.
b)
Use YuleWalker equations to find
ρ
1
and
ρ
2
.
c)
Based on a series of length
N
= 100, we observe
…,
y
99
= 22,
y
100
= 14,
y
= 20.
Forecast
y
101
and
y
102
.
4.
Determine whether the following processes are stationary:
a)
Y
t
– Y
t
– 1
=
e
t
– 0.8
e
t
– 1
b)
Y
t
– 0.39 Y
t
– 2
– 0.16 Y
t
– 4
=
e
t
– 0.8
e
t
– 1
d)
Y
t
– 0.9 Y
t
– 1
– 0.9 Y
t
– 2
=
e
t
– 1.4
e
t
– 1
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5.*
Consider the MA(2) process for which it is known that
μ
= 0,
Y
t
=
e
t
–
θ
1
e
t
– 1
–
θ
2
e
t
– 2
where
{
e
t
}
is zeromean white noise
(
i.i.d.
N
(
0,
2
e
σ
)
)
.
a)
Find the expression for
Var
(
Y
t
)
=
Cov
(
Y
t
, Y
t
)
,
Cov
(
Y
t
, Y
t
+ 1
)
,
and
Cov
(
Y
t
, Y
t
+ 2
)
,
and
Cov
(
Y
t
, Y
t
+ 3
)
in terms of
θ
1
,
θ
2
,
and
2
e
σ
.
b)
Find the expression for
ρ
1
,
ρ
2
,
and
ρ
3
in terms of
θ
1
and
θ
2
.
6.**
Consider the MA(2) process for which it is known that
μ
= 0,
Y
t
=
e
t
–
θ
1
e
t
– 1
–
θ
2
e
t
– 2
where
{
e
t
}
is zeromean white noise
(
i.i.d.
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 Spring '08
 Stepanov
 Statistics, Correlation, Correlation Coefficient, Autocorrelation, Stationary process, Yt, θ, φ γ

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