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Economics 120C
Name:
_________________________
Professor Yongil Jeon
Summer 2006
Student ID#: _________________________
Answer to Homework #1 – Summer 2006
(Midterm Exam, Summer 2005)
1.
(15 points)
Consider the simple model
i
i
i
u
X
Y
+
=
β
for which the variance of the error
term is known to be
2
2
2
i
i
X
σ
=
.
(a)
(5 points)
Is the OLS estimate of
unbiased, consistent, and efficient (carefully justify
the answers)?
Answer:
Because the errors still satisfy the MLR assumptions 1 through 4, including the zero
conditional mean assumption
0
)

(
=
i
i
X
u
E
. The OLS estimates are unbiased and
consistent. However, since the errors are heteroscedastic, estimates are inefficient, according
to the GaussMarkov Theorem.
(b)
(5 points)
Describe stepbystep how you will go about obtaining the weighted least
squares estimate of
.
Answer:
Divide both sides of the equation
i
i
i
u
X
Y
+
=
by
i
X
so that the transformed
equation is
i
i
i
i
X
u
X
Y
+
=
The variance of
i
i
X
u
is constant since
()
.
1
1
2
2
2
2
2
=
=
=
i
i
i
i
i
i
X
X
u
Var
X
X
u
Var
Hence, we can apply OLS to the above equation and obtain estimates that are BLUE.
(c)
(5 points)
Explain derive the WLS estimate of
and give a geometric interpretation of
the estimate.
Answer:
Because the model has just a constant term, the OLS estimate is simply the mean of
the transformed dependent variable. Thus
∑
=
=
n
i
i
i
X
Y
n
1
1
ˆ
. Geometrically, connect each of the
observation points (
)
,
i
i
X
Y
to the origin and compute its slope. The average of these slopes
is the WLS estimate of
.
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Answer to HW #1, ECON 120C, Summer 2006
2.
(25 points)
Consider the relationship between the expenditure on travel (
i
E
) and the total
income (
i
Y
) given by
i
i
i
u
Y
E
+
+
=
1
0
β
You are certain that the error term is heteroscedastic with
2
2
1
0
2
)
(
i
i
i
i
P
a
P
a
a
u
Var
+
+
=
=
σ
,
where
i
P
is the population. You have data on
E, Y,
and
P.
(a)
(5 points)
State the null and alternative hypotheses for no heteroscedasticity.
Answer:
,
0
:
2
1
0
=
=
a
a
H
vs
:
1
H
at least one of them is not zero.
(b)
(5 points)
Describe the regressions to be run for carrying out the test.
Answer
:
(1) Regress
i
E
against a constant and
i
Y
,
(2) compute the residuals
i
i
i
Y
E
u
1
0
ˆ
ˆ
ˆ
−
−
=
(3) regress
2
ˆ
i
u
against a constant,
i
P
and
2
i
P
.
(c)
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This note was uploaded on 07/17/2008 for the course ECON 120C taught by Professor Stohs during the Spring '08 term at UCSD.
 Spring '08
 Stohs
 Economics

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