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Solutions to HW 8
Exercise 12.4
All pure MA processes, whether univariate or multivariate, are stationary.
The model is
invertible if all the roots of
()


0
BIB
Θ
=+
Θ =
are outside of the unit circle.
1
0
0.6
0.4
1
.6
0.4
()
0
1
0.2
0.4
0.2
1
0.4
B
BB
B
B
+
B
⎡
⎤⎡
⎤
Θ=
+
Θ
=
+
=
⎢
⎥⎢
⎥
+
⎣
⎦⎣
⎦
I
.
2
11
1.
6
0
.
4
1
.16
0,
5,
1.25
0.2
1
0.4
B
B
+
= + +
=
=−
+
.
Both roots lie outside the unit circle. Thus the model is invertible.
Exercise 12.5
VAR(1) model,
1
tt
−
=Φ
+
t
X
XZ
,
where
0.9
0.5
0.1
0.3
⎡
⎤
Φ=
⎢
⎥
−
⎣
⎦
,
ˆ
h
th
t
+
X
X
,
1
ˆ
t
+
t
X
X
i.e.,
1, 1
, 1
,2
1, 2
ˆ
0.9
0.5
ˆ
0.1
0.3
+
+
⎡⎤
⎡
⎤
=
⎢⎥
⎢
⎥
−
⎣⎦
⎣
⎦
t
t
XX
X
X
, or,
1,1
, 1
, 2
1, 2
ˆ
0.9
0.5
,
ˆ
0.1
0.3
.
+
+
+
X
t
t
X
2
2
ˆ
+
X
X
i.e.,
2
2,1
,1
2, 2
ˆ
0.9
0.5
0.76
0.6
ˆ
0.1
0.3
0.12
0.04
t
X
+
+
t
X
⎡
⎡
⎤
==
⎤
⎢
⎢
⎥
−−
⎣
⎦
⎥
⎢
⎥
⎣
⎦
t
X
,
or,
2,1
, 2
2,2
, 2
ˆ
0.76
0.6
,
ˆ
0.12
0.04
.
+
+
+
t
X
Problem 1
1
10
0
.
7 0
.
7
7
.
7
0
1
0.7
0.7
.7
1 .7
x
xx
x
x
⎡
⎤
⎡
−
Φ
=
−
=
⎢
⎥
⎢
⎣
⎦
⎣
I
x
⎤
⎥
⎦
,
22
7
.
7
(1 .7 )
(.7 )
1 1.4
0,
1/1.4
.714
.7
1 .7
−
=
=
=
⇒
nonstationary.
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This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.
 Spring '09
 gUR
 Unit Circle

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