Stata 1  This problem works with Wooldridge Computer Exercise 4.C6. We use the data in WAGE2 for this exercise.

As usual, open a new do file and remember to save your work in a log file.

a)   Estimate the standard wage equation

ln(wage) =  F0  + F1 age +  F2 educ + F3 expeT + F4 tenuTe + u.  (EQ1)

b)   State and test the null hypothesis that age has no effect on the wage (log wage) in (EQ1). Use a 5% level of significance. Conduct the test without using any additional commands

– that is, just use the regression output.

c)   Now use the post-estimation command test to confirm the same conclusion of your test in b). What do you notice about the F-stat that Stata reports in relation to the t-stat reported in the regression output?

d)   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 in (EQ1).

e)   Now, we will test the null hypothesis in part d) against a two-sided alternative, at the 5% significance level. First, do so by using the post-estimation command test. What do you conclude?

f)    Now conduct the test of the null hypothesis in part d) by defining a new parameter e = F3  − F4  and re-estimating a reparametrized model, so that you can conduct your test using the 95% confidence interval reported in the regression output. Can you confirm the conclusion you reached in e)?

g)   Now use the if option to estimate (EQ1) for the sub-sample of workers that have at least 12 years of education.

Q1.  For each of the following, state whether it can cause OLS estimators to be biased?

a.   Heteroskedasticity.

b.   Omitting an important variable.

c.   A  sample  correlation  coefficient  of  .95  between  two  independent  variables  both included in the model.

For each, if your answer is no, then say why it does not cause bias in the OLS estimator. If yes, explain the source of the bias.

Q2. For each of the following, state whether (and how or why) it can cause the usual OLS t-statistics to be invalid (that is, not to have t distributions under H0 )?

a.   Heteroskedasticity.

b.   A sample correlation coefficient of .95 between two independent variables that are in the model.

c.   Omitting an important explanatory variable

Q3. This problem is concerned with the properties of a random sample from a population with mean µ and variance σ2 . Consider , the sample mean of {X1, ··· , Xn}.

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

b)   Derive the variance of  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?

d)   Consider another alternative estimator of µ,  = ∑ kiXi  . 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 =  , k2 =  , and k3 =  . Find the variance of  .

f)    On the basis of your results above, why is  the preferred estimator of µ ?

Q4. (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(salary) =  F0  + F1 log(sales) + F2roe + F3ros + 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(sry) = 4.32 + 0.280 loges) + 0.0174roe + 0.00024ros

(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.

Extra problems (more practice if you would like it)

1)   Wooldridge Computer Exercise 4.C1

The following  model can  be  used to study whether campaign expenditures affect election outcomes:

voteA  =  F0  + F1 log(expendA) + F2 log(expendB) + F3pTtystTA +  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 F1 ?

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)

2)   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.