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Problem Set 3
Introduction to Econometrics  Fall 2006
Due Date: Thursday, October 26th, BEFORE CLASS.
Each student is responsible for handing one handwritten solution. If working in groups, indicate the
members of the group (to facilitate grading).
PLEASE SHOW YOUR WORK
.
1. Consider the following OLS regression line:
b
Y
i
=4
.
52
(1
.
04)
+1
.
53
(0
.
86)
X
i
R
2
=0
.
25
where the numbers in parentheses are the heteroskedasticityrobust standard errors. Assume that the
three least squares assumptions hold.
(a) Test the null hypothesis
H
0
:
β
1
=0
at the 5% signi
f
cance level. Do you reject the null? Compute
the 95% twosided con
f
dence interval for
β
1
.
(b) Now you decide to run the regression again and compute the homoskedasticityonly standard
errors. You obtain the following estimates:
b
Y
i
=4
.
52
(0
.
96)
+1
.
53
(0
.
75)
X
i
R
2
=0
.
25
Why didn’t the estimated coe
ﬃ
cients change? Are the OLS estimators unbiased and consistent
in this case?
(c) Test the null hypothesis
H
0
:
β
1
=0
at the 5% signi
f
cance level using the homoskedasticity
only standard errors, and compute the 95% twosided con
f
dence interval for
β
1
.
Are your results
di
f
erent from those in
(
a
)
?
(d) Assume that
Y
i
is yearly earnings in thousand of dollars and
X
i
is years of schooling. Which
results are more reliable, those in
(
a
)
or those in
(
c
)
? Explain.
2. When the least squares assumptions hold, the heteroskedasticityrobust variance of
b
β
0
is:
σ
2
b
β
0
=
Var
(
H
i
u
i
)
n
[
E
(
H
2
i
)]
2
where:
H
i
=1
−
µ
X
E
(
X
2
i
)
X
i
In this problem, you will use this formula to derive the variance of the OLS estimator
b
β
0
in the case
the error term is homoskedastic, that is when
Var
(
u
i

X
i
)=
σ
2
u
.
(a) Write
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 Fall '08
 Staff
 Econometrics

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