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
This preview has intentionally blurred sections. Sign up to view the full version.
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
)
´
?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Zivot
 Maximum likelihood, Estimation theory, CAPM Revisited, A. Tiwari, S. Ray

Click to edit the document details