Practice Problems
1.
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 zero-mean 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
.
2.
Consider the AR(2) process for which it is known that
μ
= 0,
Y
t
–
φ
1
Y
t
– 1
–
φ
2
Y
t
– 2
=
e
t
where
{
e
t
}
is zero-mean white noise
(
i.i.d.
N
(
0,
2
e
σ
)
)
.
Find the expression for
ρ
1
and
ρ
2
in terms of
φ
1
and
φ
2
.
3.
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 zero-mean white noise
(
i.i.d.
N
(
0,
2
e
σ
)
)
.
Based on a series of length
N
= 6, we observe
y
1
y
2
y
3
y
4
y
5
y
6
4.4
–
2.0
–
6.3
4.1
5.6
–
6.1
a)
Using
e
0
= 0,
e
–
1
= 0, calculate S
(
θ
1
,
θ
2
)
=
°
=
N
t
t
e
1
2
for
θ
1
= 0.3,
θ
2
= 0.4.
b)
For
θ
1
= 0.3,
θ
2
= 0.4, forecast
y
7
,
y
8
,
y
9
, and
y
10
.
c)*
For
θ
1
= 0.3,
θ
2
= 0.4, given
2
ˆ
e
σ
= 16.3, calculate 95% probability limits for
y
7
,
y
8
,
y
9
, and
y
10
.