Applied Econometrics I, Summer, 2009
Wednesday, May 13, 2009
(due: Monday, May 18, 2009)
Homework Assignment 1 Solutions
E
(
u
i
j
X
i
) = 0
, implies that
E
(
Y
i
j
X
i
) =
&
0
+
1
X
i
.
Population Regression Function :
Y
i
=
&
0
+
1
X
i
+
u
i
:
Take expectation of both sides with respect to
X
.
E
(
Y
i
j
X
i
) =
E
(
&
0
+
1
X
i
+
u
i
j
X
i
)
E
(
Y
i
j
X
i
) =
E
(
&
0
j
X
i
) +
E
(
1
X
i
j
X
i
) +
E
(
u
i
j
X
i
)
E
(
Y
i
j
X
i
) =
&
0
+
1
E
(
X
i
j
X
i
) +
E
(
u
i
j
X
i
)
E
(
Y
i
j
X
i
) =
&
0
+
1
X
i
+
E
(
u
i
j
X
i
)
E
(
Y
i
j
X
i
) =
&
0
+
1
X
i
Step
2
follows from the fact that expectation is a linear operator (That basically means
you can take expectation of individual terms in a summation).
Step
3
uses the fact that expectation of a constant is itself.
Step
4
is result of the fact that expectation of a random variable wrt itself is equal to
that random variable .
Last step uses the OLS assumption given in the question
E
(
u
i
j
X
i
) = 0
.
2. Write down the expressions for the
R
2
for the regression of
Y
on
X
and then
X
on
Y
. Compare them.
Write the
R
2
of the regression of
Y
on
X
R
2
=
ESS
TSS
=
P
n
i
=1
(
^
Y
i
Y
)
2
P
n
i
=1
(
Y
i
Y
)
2
observe that
^
Y
i
=
^
&
0
+
^
1
X
i
=
P
n
i
=1
(
^
&
0
+
^
1
X
i
Y
i
)
2
P
n
i
=1
(
Y
i
Y
)
2
use
^
&
0
=
Y
^
1
X
=
P
n
i
=1
(
Y
^
1
X
+
^
1
X
i
Y
)
2
P
n
i
=1
(
Y
i
Y
)
2
=
P
n
i
=1
(
^
1
(
X
i
X
))
2
P
n
i
=1
(
Y
i
Y
)
2
=
^
2
1
P
n
i
=1
(
X
i
X
)
2
P
n
i
=1
(
Y
i
Y
)
2
use
^
1
=
P
n
i
=1
(
X
i
X
)(
Y
i
Y
)
P
n
i
=1
(
X
i
X
)
2
=
P
n
i
=1
(
X
i
X
)(
Y
i
Y
)
P
n
i
=1
(
X
i
X
)
2
±
2
P
n
i
=1
(
X
i
X
)
2
P
n
i
=1
(
Y
i
Y
)
2
=
(
P
n
i
=1
(
X
i
X
)(
Y
i
Y
)
)
2
P
n
i
=1
(
X
i
X
)
2
P
n
i
=1
(
Y
i
Y
)
2
=
(
1
n
1
P
n
i
=1
(
X
i
X
)(
Y
i
Y
)
)
2
1
n
1
P
n
i
=1
(
X
i
X
)
2
1
n
1
P
n
i
=1
(
Y
i
Y
)
2
=
²
1
n
1
P
n
i
=1
(
X
i
X
)(
Y
i
Y
)
p
1
n
1
P
n
i
=1
(
X
i
X
)
2
p
1
n
1
P
n
i
=1
(
Y
i
Y
)
2
³
2
=
²
cov
(
X;Y
)
p
var
(
X
)
p
var
(
Y
)
³
2
=
´
±
X;Y
µ
2
Symmetrically
R
2
of the regression of
X
on
Y
will be equal to
´
±
Y;X
µ
2
. Since from the
±
X;Y
=
±
Y;X
:
Therefore
R
2
from both regressions will
be equal.
3. Go to the web address http://wps.aw.com/aw_stock_ie_2/50/13016/3332253.cw/index.html. Down
load the data "California Test Score Data Used in Chapters 49".
±
Replicate the regression result we saw in the lecture. Basically run a regression of testscr on str
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 Summer '08
 Staff
 Statistics, Econometrics, Linear Regression, Regression Analysis, Null hypothesis, Yi j Xi, dence intervals

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