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Econ 583 Take Home Final Exam
Fall 2009
Eric Zivot
December 17, 2009
1 Asymptotics
Consider the simple AR(1) model
=
−
1
+
∼
(0
2
)
=1


1
0
is
f
xed.
1. Is
{
}
covariance stationary and ergodic? What are
[
]
and
var(
)?
2. Consider the sample mean
¯
=
−
1
P
=1
Show that
¯
is an unbiased and
consistent estimator for
[
]
For the consistency result, be sure to state the
appropriate LLN.
3. What is the asymptotic distribution of
√
¯
?
Be sure to state the appropriate
CLT to justify your result.
4. How would you estimate the asymptotic variance of
√
¯
?
5. The least squares estimator of
is
ˆ
=
³
P
=1
2
−
1
´
−
1
P
=2
−
1
Is
ˆ
an
unbiased estimator of
?
Brie
F
y explain.
6. Show that
ˆ
is a consistent estimator of
Be sure to state the appropriate LLN
to justify this result.
7. Let
=
−
1
and
=
{
−
1
0
}
Show that
{
}
is a MDS.
8. What is the asymptotic distribution of
√
(ˆ
−
)?
Be sure to state the appro
priate CLT to justify your result.
9. How would you estimate the asymptotic variance of
√
(ˆ
−
)?
1
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View Full Document2 Single Equation Linear GMM
Consider the regression model with a lagged dependent variable and autocorrelated
errors
=
0
+
1
−
1
+
2
1
+
=
z
0
δ
+
=
−
1
+
∼
(0
2
)
z
=(
1
−
1
1
)
0
δ
0
1
2
)
0

1

1


1
Assume that
1
is ergodicstationary and
[
1
]=0
for all values of
and
1. Show that
[
z
]
6
=0
and use this result to show that the least squares estimate
of
δ
is inconsistent.
2. GMM can be used to get consistent and asymptotically normal estimates of
δ
given suitable instruments
x
What instruments would you use to identify
and estimate
δ
?
Brie
f
y justify your choice of instruments by showing that your
choice of
x
satis
F
es
[
x
]=
0
and
[
x
0
Σ
has full rank
=3
3. For your choice of instruments
x
is
g
(
δ
)=
x
an ergodicstationary MDS
or is it a serially correlated process?
What is the asymptotic variance of
√
g
(
δ
³
1
√
´
P
x
?
That is, what is
S
=avar
³
√
g
(
δ
)
´
?
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 Fall '09
 Zivot

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