EC 420, Spring 2007
Midterm
Show all your work.
You will have the entire class period to complete this exam.
This
exam is “open book” but “closed note”; you may use the course text (Wooldridge), but
you cannot use notes, old homework questions and answers, etc.
Good luck.
1.
If I estimate a regression and find that the standard error of my slope estimate is 0.25,
what precisely does that mean?
If we just randomly threw out 80% of our data and
reran the same regression (on the smaller sample), would you expect the standard
error to be larger than 0.25, smaller than 0.25, or roughly the same? (4 points)
What that means is that the standard deviation of the sampling distribution of the
slope estimate is 0.25 (and the sampling distribution shows how the estimated
slope varies from one sample of size
n
to the next).
If we randomly threw out
80% of the data, we would expect the standard error to increase because we’re
throwing away data.
Students could also just appeal to the GaussMarkov
theorem, which says that the full sample OLS estimate is the BEST (i.e.,
minimum variance) estimate of the slope.
2.
A simple regression model that relates the final exam scores of students in EC 499
(
final
) to the number of hours they studied in EC 420 (
hours
) is
final = β
0
+ β
1
hours+
u
,
where
u
is the unobserved residual.
What is the meaning of
u
, and what kinds of variables are contained in
u
?
Are they
likely to be correlated with the number of hours studied in EC420? (3 points)
u
represents all factors that affect final exam scores in EC 499 other than the
number of hours they studied in EC 420.
This could include intelligence, grade
in EC 420, number of hours studied in EC 499, luck, motivation, how many
hours a week that someone goes out, etc.
3.
Provide proofs for the following. Provide an explicit description of each of the steps
you take. (2 points each)
a. Show that
0
)
(
1
=

∑
=
X
X
n
i
i
.
This one is essentially a gimme.
0
)
(
1
=

=

∑
=
X
n
X
n
X
X
n
i
i
b. Define
x
y

1
ˆ
β
to be the slope coefficient from a regression of
Y
on
X
,
x
y

0
ˆ
β
to be the intercept from a regression of
Y
on
X
, and
X
Y
x
y
x
y
x
y

1

0

ˆ
ˆ
ˆ
β
β
+
=
to be
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the predicted value for
Y
based on the regression of
Y
on
X
.
In the third
homework, you proved that
x
y
cx
y
x
y
cx
y
c

0

0

1

1
ˆ
ˆ
and
ˆ
ˆ
β
β
β
β
=
=
, where
cx
y
cx
y

1

0
ˆ
and
ˆ
β
β
are the intercept and slope from a regression of
Y
on
cX
,
where
c
is an arbitrary constant.
Based on these two results, prove that
x
y
cx
y
Y
Y


ˆ
ˆ
=
(this is important because it means that predicted values of
Y
do
not depend on the units we use to measure
X
).
Here’s the proof:
x
y
x
y
x
y
x
y
x
y
cx
y
cx
y
cx
y
Y
X
cX
c
cX
Y


1

0

1

0

1

0

ˆ
ˆ
ˆ
)
(
ˆ
ˆ
)
(
ˆ
ˆ
ˆ
=
+
=
+
=
+
=
β
β
β
β
β
β
c.
Show that in a random sample of size 5, the sample mean is an unbiased
estimator of the population mean (hint: first define what an unbiased
estimator is).
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 Spring '08
 ToddElder
 Statistics, Regression Analysis, Mean squared error, math test scores

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