R2 0.72, Ser 3,773.35

Problem Set 4
ECN 140 Econometrics

Professor Oscar Jorda
Name___________________________________
1) The cost of attending your college has once again gone up. Although you have been told that education is
investment in human capital, which carries a return of roughly 10% a year, you (and your parents) are not
pleased. One of the administrators at your university/college does not make the situation better by telling
you that you pay more because the reputation of your institution is better than that of others. To
investigate this hypothesis, you collect data randomly for 100 national universities and liberal arts colleges
from the 2000-2001 U.S. News and World Report annual rankings. Next you perform the following
regression
= 7,311.17 + 3,985.20˛Reputation - 0.20Size
(2,058.63) (664.58) (0.13)
+ 8,406.79˛Dpriv - 416.38˛Dlibart - 2,376.51˛Dreligion

(2,154.85) (1,121.92) (1,007.86)
R2=0.72, SER = 3,773.35
where Cost is Tuition, Fees, Room and Board in dollars, Reputation is the index used in U.S. News and World
Report (based on a survey of university presidents and chief academic officers), which ranges from 1
("marginal") to 5 ("distinguished"), Size is the number of undergraduate students, and Dpriv, Dlibart, and
Dreligion are binary variables indicating whether the institution is private, a liberal arts college, and has a
religious affiliation. The numbers in parentheses are heteroskedasticity-robust standard errors.
(a) Interpret the results and indicate whether or not the coefficients are significantly different from zero.
Do the coefficients have the expected sign?
(b) What is the forecasted cost for a liberal arts college, which has no religious affiliation, a size of 1,500
students and a reputation level of 4.5? (All liberal arts colleges are private.)
(c) To save money, you are willing to switch from a private university to a public university, which has a
ranking of 0.5 less and 10,000 more students. What is the effect on your cost? Is it substantial?
(d) What is the p-value for the null hypothesis that the coefficient on Size is equal to zero? Based on this,
should you eliminate the variable from the regression? Why or why not?
(e) You want to test simultaneously the hypotheses that . Your regression
package returns the F-statistic of 1.23. Can you reject the null hypothesis?
(f) Eliminating the Size and Dlibart variables from your regression, the estimation regression becomes
= 5,450.35 + 3,538.84˛Reputation + 10,935.70˛Dpriv - 2,783.31˛Dreligion;
(1,772.35) (590.49) (875.51) (1,180.57)
1
R2=0.72, SER = 3,792.68
Why do you think that the effect of attending a private institution has increased now?
(g) You give a final attempt to bring the effect of Size back into the equation by forcing the assumption of
homoskedasticity onto your estimation. The results are as follows:
= 7,311.17 + 3,985.20˛Reputation - 0.20Size

(1,985.17) (593.65) (0.07)
+ 8,406.79˛Dpriv - 416.38˛Dlibart - 2,376.51˛Dreligion

(1,423.59) (1,096.49) (989.23)
R2=0.72, SER = 3,682.02
Calculate the t-statistic on the Size coefficient and perform the hypothesis test that its coefficient is zero. Is
this test reliable? Explain.
(h) What can you say about causation in the above relationship? Is it possible that Cost affects Reputation
rather than the other way around?
2) (Requires Appendix Material) Consider the following multiple regression model
You want to consider certain hypotheses involving more than one parameter, and you know that the
regression error is homoskedastic. You decide to test the joint hypotheses using the rule-of-thumb
F-statistics. For each of the cases below specify a restricted model and indicate how you would compute
the F-statistic to test for the validity of the restrictions.
(a)
(b)
(c)
(d)
2
3) Write the following four restrictions in the form , where the hypotheses are to be tested
simultaneously.
Can you write the following restriction in the same format? Why not?
3
Answer Key
Testname: PS4.TST
1) (a) An increase in reputation by one category, increases the cost by roughly $3,985. The larger the size of the
college/university, the lower the cost. An increase of 10,000 students results in a $2,000 lower cost. Private
schools charge roughly $8,406 more than public schools. A school with a religious affiliation is approximately
$2,376 cheaper, presumably due to subsidies, and a liberal arts college also charges roughly $416 less. There are
no observations close to the origin, so there is no direct interpretation of the intercept. Other than perhaps the
coefficient on liberal arts colleges, all coefficients have the expected sign, although that coefficient is not
significantly different from zero. All other coefficients are statistically significant at conventional levels, with the
exception of the size coefficient, which carries a t-statistic of 1.54, and hence is not statistically significant at the
5% level (using a one-sided alternative hypothesis).
(b) $ 32,935.
(c) Roughly $ 12,400. Since over the four years of education, this implies approximately $50,000, it is a substantial
amount of money for the average household.
(d) Using a one-sided alternative hypothesis, the p-value is 6.2 percent. Variables should not be eliminated
simply on grounds of a statistical test. The sign of the coefficient is as expected, and its magnitude makes it
important. It is best to leave the variable in the regression and let the reader decide whether or not this is
convincing evidence that the size of the university matters.
(e) The critical value for is 3.00 (5% level) and 4.61 (1% level). Hence you cannot reject the null hypothesis in
this case.
(f) Private institutions are smaller, on average, and some of these are liberal arts colleges. Both of these variables
had negative coefficients.
(g) Although the coefficient would be statistically significant in this case, the test is unreliable and should not be
used for statistical inference. There is no theoretical suggestion here that the errors might be homoskedastic. Since
the standard errors are quite different here, you should use the more reliable ones, i.e., the
heteroskedasticity-robust.
(h) It is very possible that the university president and chief academic officer are influenced by the cost variable
in answering the U.S. News and World Report survey. If this were the case, then the above equation suffers from
simultaneous causality bias, a topic that will be covered in a later chapter. However, this poses a serious threat to
the internal validity of the study.
1
Answer Key
Testname: PS4.TST
2) (a) The restricted model is and the rule-of-thumb F-statistic would be
(b) and the rule-of-thumb F-statistic would be
(c) This is not a linear restriction. Hence you cannot use the F-test to test for its validity.
(d) and the rule-of-thumb F-statistic would be
3)
The restriction cannot be written in the same format because it is nonlinear.

R2 0.72, Ser 3,773.35

Uploaded by

Copyright:

Available Formats

R2 0.72, Ser 3,773.35

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

R2 0.72, Ser 3,773.35

Uploaded by

Copyright:

Available Formats

Problem Set 4

ECN 140 Econometrics

= 7,311.17 + 3,985.20˛Reputation - 0.20Size

(2,058.63) (664.58) (0.13)

+ 8,406.79˛Dpriv - 416.38˛Dlibart - 2,376.51˛Dreligion

R2=0.72, SER = 3,773.35

= 5,450.35 + 3,538.84˛Reputation + 10,935.70˛Dpriv - 2,783.31˛Dreligion;

(1,772.35) (590.49) (875.51) (1,180.57)

= 7,311.17 + 3,985.20˛Reputation - 0.20Size

+ 8,406.79˛Dpriv - 416.38˛Dlibart - 2,376.51˛Dreligion

R2=0.72, SER = 3,682.02

2) (Requires Appendix Material) Consider the following multiple regression model

2) (a) The restricted model is and the rule-of-thumb F-statistic would be

(b) and the rule-of-thumb F-statistic would be

(d) and the rule-of-thumb F-statistic would be

The restriction cannot be written in the same format because it is nonlinear.

You might also like