1.   This problem is concerned with the properties of a random sample from a population with

mean µ and variance σ2. Consider X, the sample mean of {X1, ··· , Xn}.

a)   Show that the expected value of the sample mean X equals the population mean, µ .

b)   Derive the variance ofX(-) for a sample of size n = 3.

c)   Consider an alternative estimator of µ, X1  (yes, use X1  to estimate µ). Is this estimator unbiased?

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d)   Consider another alternative estimator of µ, X = 1 kiXi  . What restrictions on the ki

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constants will ensure that the expected value of X is also µ?

e)   Again, consider a random sample of size n = 3 where k1  =  , k2  =  , and k3  = . Find the

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variance of X.

f)    On the basis of your results above, why isX(-) 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(salaTY) = β0 + β1 log(sales) + β2TOe + β3TOs + u

(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(salaTY) = 4.32 + 0.280 log(sales) + 0.0174TOe + 0.00024TOs

(0.32)  (0.035)                   (0.0041)       (0.00054)

N = 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(wage) = β0 + β1 educ + β2 expeT + β3 tenuTe + u.

State the null hypothesis that another year of general workforce experience has the same effect on log (wage) 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:

voteA  = β0 + β1 log(expendA) + β2 log(expendB) + β3 log(pTtystTA) + u,

where voteA is the percentage of the vote received by Candidate A, expendA and expendB are campaign expenditures  by  Candidates A  and  B,  and  pTtystTA 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 β1 ?

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 PTice and assess [see equation (6.44)]. Here, we use a

level-level formulation.

a)   In the simple regression model

PTice  =  β0 + β1 assess + u,

the assessment is rational if β1  = 1 and β0  = 0. The estimated equation is

PTlce(—) = − 14.47 + .976 assess

(16.27)   (.049)

n = 88, SSR = 165,644.51, R2  = .820

First, test the hypothesis that H0 : β0  = 0 against the two-sided alternative. Then, test

H0 : β1 = 1 against the two-sided alternative. What do you conclude?

b)   To test the joint hypothesis that H0 : β0  = 0 and H0 : β1  = 1, we  need the SSR in the restricted model. This amounts to computing 1 (PTicei − assessi )2, where n = 88,  since  the  residuals  in  the  restricted  model  are  just   PTicei − assessi .  (No estimation is needed for the restricted model because both parameters are specified

under H0 . ) This turns out to yield SSR = 209,448.99.

Carry out the F test for the joint hypothesis.

c)    Now, test H0 : β2  = 0, H0 : β3  = 0 and H0 : β4  = 0 in the model