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Econ 220B Final Exam
Winter 2009
DIRECTIONS: No books or notes of any kind are allowed.
Answer all questions on
separate paper. 250 points are possible on this exam.
1.) (60 points total) The leastsquares regression model
y
=
X
β
+
ε
makes a lot of
assumptions, including:
(i)
X
0
X
is of full rank
k
(ii)
E
(
ε
2
t
)=
σ
2
(iii) plim(
X
0
X
/T
)=
Q
of full rank
k
(iv)
T
−
1
/
2
P
T
t
=1
x
t
ε
t
L
→
N
(
0
,σ
2
Q
)
.
In addition, we often make one or more of the following assumptions:
(v)
E
(
ε

X
)=
0
;
(vi)
E
(
εε
0

X
)=
σ
2
I
T
;
(vii)
ε

X
∼
N
(
0
,σ
2
I
T
)
.
(viii)
x
t
is strictly deterministic
Questions (a)(f) below involve certain statements about properties of the leastsquares
regression coe
ﬃ
cient
b
=(
X
0
X
)
−
1
X
0
y
and variance estimate
s
2
=(
T
−
k
)
−
1
e
0
e
for
e
=
y
−
Xb
.
In each of the questions, you can assume that (i)(iv) are satis
f
ed. The question is, which
further assumptions besides (i)(iv) would you need in order to prove the stated result?
For example, if you think assumptions (i), (ii), and (v) alone are su
ﬃ
cient to prove that
E
(
b
)=
β
, then your answer to part (a) should be “v”. If you think that (i), (iii), (v) and
(vi) are needed to prove that
E
(
b
)=
β
, then your answer to part (a) should be “v and vi”.
Note that you should not give any proofs or arguments— your answer in each case should be
some combination of the numerals “v”, “vi”, “vii”, or “viii”.
a.)
E
(
b
)=
β
b.)
b
is the best linear unbiased estimator of
β
c.)
E
[(
b
−
β
)(
b
−
β
)
0

X
]=
σ
2
(
X
0
X
)
−
1
d.)
m
−
1
(
Rb
−
r
)
0
[
s
2
R
(
X
0
X
)
−
1
R
0
]
−
1
(
Rb
−
r
)
∼
F
(
m, T
−
k
)
when
R
β
=
r
for
R
an
(
m
×
k
)
matrix
e.)
b
p
→
β
f.)
(
Rb
−
r
)
0
[
s
2
R
(
X
0
X
)
−
1
R
0
]
−
1
(
Rb
−
r
)
L
→
χ
2
m
when
R
β
=
r
for
R
an
(
m
×
k
)
matrix
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2.) (50 points total) Consider a regression of a scalar
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 Spring '08
 Hamilton,J

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