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Econ 620 Spring 2005
Professor N. Kiefer
TA H. Choi
Suggested Solutions to the midterm exam
1. We have
Z
=
XR,
where
R
=
1
1
1
1

1 1
0
0
1
.
Therefore we have
Rα
=
β
from
Ey
=
Zα
=
XRα
=
Xβ.
This implies
R
ˆ
a
LS
=
ˆ
β
LS
.
2. We have
ˆ
β
2
for a) (
X
0
2
M
1
X
2
)

1
X
0
2
M
1
y,
b) (
X
0
2
X
2
)

1
X
0
2
P
1
y,
c) (
X
0
2
P
1
X
2
)

1
X
0
2
P
1
y,
d) (
X
0
2
X
2
)

1
X
0
2
M
1
y,
e) (
X
0
2
M
1
X
2
)

1
X
0
2
M
1
y,
f) (
X
0
2
M
1
X
2
)

1
X
0
2
M
1
y,
g) (
X
0
2
M
1
X
2
)

1
X
0
2
M
1
y,
h) (
X
0
2
M
1
X
2
)

1
X
0
2
M
1
y.
Hence we have 4
diﬀerent estimators. (a)(d) are all diﬀerent and (e)(h) are the same as (a).
3. The prediction
y
p
is 10
×
1 vector and has the form
y
p
=
Cy,
where
y
= (
y
1
,...,y
n
)
0
. Under LS assumption, y
p
is
given by the relationship
Cy
=
X
p
ˆ
β,
where
X
p
is known and
ˆ
β
is OLS estimator. We have
C
=
X
p
(
X
0
X
)

1
X
0
. The
variance of the prediction errors is Var(
Cε

ε
p
) =
C
Var(
ε
)
C
0
+
V ar
(
ε
p
) =
σ
2
(
CC
0
+
I
) =
σ
2
‡
X
p
(
X
0
X
)

1
X
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 Spring '07
 KIEFER
 Econometrics

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