Determine whether the following processes are stationary:
a)
Y
t
+ 0.2 Y
t
– 1
=
e
t
– 0.8
e
t
– 1
+ 0.5
e
t
– 2
Φ
(
B
) = 1 + 0.2
B
The root of
Φ
(
z
) = 0 is
z
=
–
5, it is outside the unit circle.
This process is stationary.
b)
Y
t
– 0.1 Y
t
– 1
– 0.2 Y
t
– 2
=
e
t
– 0.6
e
t
– 1
Φ
(
B
) = 1 – 0.1
B
– 0.2
B
2
=
(
1 – 0.5
B
) (
1 + 0.4
B
)
The roots of
Φ
(
z
) = 0 are
z
1
=
2 and
z
2
=
–
2.5.
All roots of
Φ
(
z
) = 0 are outside the unit circle.
This process is stationary.
OR
An AR(2) model is stationary if
–
1 <
φ
2
< 1,
2
+
1
< 1,
2
–
1
< 1.
–
1 < 0.2 < 1,
0.2 + 0.1 < 1,
0.2 – 0.1 < 1.
This process is stationary.
c)
Y
t
– 0.7 Y
t
– 1
– 0.6 Y
t
– 2
=
e
t
+ 0.9
e
t
– 1
– 0.7
e
t
– 2
Φ
(
B
) = 1 – 0.7
B
– 0.6
B
2
=
(
1 – 1.2
B
) (
1 + 0.5
B
)
The roots of
Φ
(
z
) = 0 are
z
1
=
5
/
6
and
z
2
=
–
2.
z
1
=
5
/
6
is NOT outside
the unit circle, it is inside the unit circle. The roots of
Φ
(
z
) = 0 must ALL be
outside the unit circle for the process to be stationary.
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 Spring '07
 STEPANOV
 Unit Circle

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