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timator
unbiased?
d) Consider another alternative estimator of µ, � = ∑ =1 . What restrictions on the ki
constants will ensure that the expected value of � is also µ?
e) Again, consider a random sample of size n = 3 where k1 = 1
4
, k2 =
1
2
, and k3 =
1
4
. Find the
variance of �.
f) On the basis of your results above, why is � the preferred estimator of µ?
2. (Wooldridge Question 4.2) Consider an equation to explain salaries of CEOs in terms of annual
firm sales, return on equity (roe, in percentage form), and return on the firm’s stock (ros, in
percentage form): log() = 0 + 1 log() + 2 + 3 +
(a) In terms of the model parameters, state the null hypothesis that, after controlling for
sales and roe, ros has no effect on CEO salary. State the alternative that better stock
market performance increases a CEO’s salary.
(b) Using the data in CEOSAL1, the following equation was obtained by OLS: log() =� 4.32 + 0.280 log() + 0.0174 + 0.00024� (0.32) (0.035) (0.0041) (0.00054)
= 209; R2=0.283
By what percentage is salary predicted to increase if ros increases by 50 points? Does
ros have a practically large effect on salary?
(c) Test the null hypothesis that ros has no effect on salary against the alternative that ros
has a positive effect. Carry out the test at the 10% significance level.
(d) Would you include ros in a final model explaining CEO compensation in terms of firm
performance? Explain.
3. (Wooldridge Computer Exercise 4.C6)
Use the data in WAGE2 for this exercise.
a) Consider the standard wage equation log() = 0 + 1 + 2 + 3 + .
State the null hypothesis that another year of general workforce experience has the
same effect on log () as another year of tenure with the current employer.
b) Test the null hypothesis in part a) against a two-sided alternative, at the 5%
significance level, by constructing a 95% confidence interval. What do you conclude?
4. (Wooldridge Computer Exercise 4.C1)
The following model can be used to study whether campaign expenditures affect election
outcomes:
= 0 + 1 log() + 2 log() + 3 log() + ,
where is the percentage of the vote received by Candidate A, and
are campaign expenditures by Candidates A and B, and is a measure of party
strength for Candidate A (the percentage of the most recent presidential vote that went to
A’s party).
a) What is the interpretation of 1?
b) In terms of parameters, state the null hypothesis that a 1% increase in A’s
expenditures is offset by a 1% increase in B’s expenditures.
c) Estimate the given model using the data in VOTE1 and report the results in usual form.
Do A’s expenditures affect the outcome? What about B’s expenditures? Can you use
these results to test the hypothesis in part b)?
d) Estimate a model that directly gives the t statistic for testing the hypothesis in part b).
What do you conclude? (use a two-sided alternative)
5. Use the data in MLB1 for this exercise.
(a) Use the model estimated in equation (4.31) from the textbook and drop the variable
rbisyr. What happens to the statistical significance of hrunsyr? What about the size of
the coefficient on hrunsyr?
(b) Add the variables runsyr (runs per year), fldperc (fielding percentage), and sbasesyr
(stolen bases per year) to the model from part (a). Which of these factors are
individually significant? For runsyr, write out a complete formal hypothesis test starting
from stating the hypothesis, and ending with making a conclusion in the context of the
data.
(c) In the model from part (b), test the joint significance of bavg, fldperc, and sbasesyr. [If
we have not yet discussed joint testing in lectures, you can leave this part til next week.]
6. (Wooldridge Question 4.6)
In lectures, we used as an example testing the rationality of assessments of housing prices.
There, we used a log-log model in and [see equation (6.44)]. Here, we use a
level-level formulation.
a) In the simple regression model
= 0 + 1 + ,
the assessment is rational if 1 = 1 and 0 = 0. The estimated equation is
� = −14.47 + .976 (16.27) (.049)
= 88, = 165,644.51,2 = .820
First, test the hypothesis that 0:0 = 0 against the two-sided alternative. Then, test
0:1 = 1 against the two-sided alternative. What do you conclude?
b) To test the joint hypothesis that 0:0 = 0 and 0:1 = 1, we need the SSR in the
restricted model. This amounts to computing ∑ ( − )2=1 , where =88, since the residuals in the restricted model are just − . (No
estimation is needed for the restricted model because both parameters are specified
under 0. ) This turns out to yield SSR = 209,448.99.
Carry out the F test for the joint hypothesis.
c) Now, test 0:2 = 0,0:3 = 0 and 0:4 = 0 in the model
= 0 + 1 + 2 + 3 + 4 + .
The R-squared from estimating this model using the same 88 houses is 0.829.
d) If the variance of changes with , , or , what can you
say about the F test from part c)?