–1–
Econ 220B, Winter 2010
Midterm Exam
DIRECTIONS: No books or notes of any kind are allowed.
Answer all questions on
separate paper.
150 points are possible on this exam.
1.) (20 points) Consider the following equations:
y
=
X
β
+
ε
b
= (
X
0
X
)
−
1
X
0
y
.
Here
y
is a
(
T
×
1)
vector of observations on the dependent variable,
X
is a
(
T
×
k
)
matrix
of observations on the explanatory variables with rank
k
, and
β
is the true value of a
(
k
×
1)
vector of parameters.
Let
b
2
denote the second element of the vector
b
and let
β
2
denote
the second element of the vector
β
.
Consider the following theorem:
Let
ˆ
β
2
be any unbiased estimator of
β
2
that is constructed from some function
of
X
and
y
.
If condition
A
holds, then Var(
ˆ
β
2

X
)
≥
Var
(
b
2

X
)
.
a.) Write down an equation for what it means to say that
ˆ
β
2
is an unbiased estimator of
β
2
.
b.) State a su
ﬃ
cient condition or conditions
A
for which the above theorem is true.
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 Spring '08
 Hamilton,J

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