Solutions to HW 7
Exercise 12.1
VAR(1) model,
1
tt
−
=Φ
+
t
X
XZ
,
where
10
.
5
0.2
0.7
⎡
⎤
Φ=
⎢
⎥
⎣
⎦
,
i.e.,
.
,1
1,1
,2
1
.
5
0.2
0.7
XX
−
−
⎡⎤
⎡
⎤
⎡
=+
⎢⎥
⎢
⎥
⎢
⎣⎦
⎣
⎦
⎣
t
t
Z
Z
⎤
⎥
⎦
The model is nonstationary if there are roots of
()


0
BIB
Φ
=−
Φ =
incide the unit circle.
1
0
0.5
1
0.5
0
1
0.2
0.7
0.2
1
0.7
B
BB
B
B
B
−−
⎡
⎤
⎡
−Φ =
−
=
⎢
⎥
⎢
⎣
⎦
⎣
I
B
⎤
⎥
⎦
.
2
12
.
5
1 1.7
0.6
0
2,
0.8333
0.2
1
0.7
B
B
+
=
=
≈
.
One root is inside the unit circle, therefore the model is nonstationary.
Exercise 12.2
The model is stationary if all the roots of


Φ
are outside of the unit
circle.
Similar to 12.1,
(a)
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 Spring '09
 gUR
 Unit Circle

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