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Econ 220B
James Hamilton
Final Exam
Winter 2004
DIRECTIONS: No books or notes of any kind are allowed.
Answer all questions on
separate paper. 250 points are possible on this exam
1.) (25 points total) State (but do not prove) the GaussMarkov Theorem.
2.) (30 points total) Consider the following regression model:
y
=
X
β
+
ε
where
y
is a
(
T
×
1)
vector,
X
is a
(
T
×
k
)
matrix,
ε
is a
(
T
×
1)
vector with the property that
ε

X
∼
N
(
0
,
σ
2
I
T
)
. Suppose you wanted to test the single nonlinear hypothesis
g
(
β
)=0
.
Give the formula for a statistic you could use to test this hypothesis and state (but do not
prove) the distribution you would use to interpret this statistic.
3.) (40 points total) Consider the following regression, which includes a lagged dependent
variable:
y
t
=
α
+
β
y
t
−
1
+
γ
z
t
+
ε
t
.
Let
ε
=(
ε
1
,
ε
2
, ...,
ε
T
)
0
be the
(
T
×
1)
vector containing all the epsilons where
T
is the sample
size and let
z
=(
z
1
,z
2
,...,z
T
)
0
.
Suppose that
ε

z
∼
N
(
0
,
σ
2
I
T
)
and
y
0
=0
. Suppose further
that the true value of
β
satis
f
es
0
<
β
<
1
and that
{
z
t
,y
t
}
are stationary and ergodic with
E
1
y
t
−
1
z
t
£
1
y
t
−
1
z
t
¤
=
Q
where
Q
hasrank3w
itha
l
le
lemen
tso
f
Q
strictly positive. Let
ˆ
α
,
ˆ
β
,and
ˆ
γ
be the OLS
estimates de
f
ned by
ˆ
α
ˆ
β
ˆ
γ
=
T
Σ
y
t
−
1
Σ
z
t
Σ
y
t
−
1
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This note was uploaded on 03/02/2012 for the course ECON 220b taught by Professor Hamilton,j during the Spring '08 term at UCSD.
 Spring '08
 Hamilton,J

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