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STAT 8260 Extra Credit Problems – Thursday, May 1, 2008*
SHOW ALL WORK
Name:
1. Consider the “seemingly unrelated regressions:”
y
1
=
X
1
β
1
+
e
1
,
e
1
∼
N
n
1
(
0
,σ
2
1
I
n
1
)
y
2
=
X
2
β
2
+
e
2
,
e
2
∼
N
n
2
(
0
,σ
2
2
I
n
2
)
for data
y
= (
y
T
1
,
y
T
2
)
T
, where cov(
e
1
,
e
2
) =
0
, and the design matrices
X
1
and
X
2
are
n
1
×
p
and
n
2
×
p
, respectively, each of full rank with more rows than
columns.
a.
(4 pts)
Write the two seemingly unrelated regressions as a single linear
model for
y
of the form
y
=
X
β
+
e
. Deﬁne all quantities and state the
assumptions on
e
.
*
These questions pertain to the fullrank linear model material and allow a maxi
mum of 26 points that can be applied to your exam # 2 score.
1
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(3 pts)
State the hypothesis
H
0
:
β
1
=
β
2
in the form of the general linear
hypothesis.
c.
(2 pts)
Is the ordinary least squares estimator in your model from part (a)
optimal in any sense (e.g., BLUE, MVUE)? Explain why or why not.
2
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