In this exercise, we will explore some of the factors predicting costs at American universities using the data file poolcoll2. Let \(T C=\) the real (2008 dollars) total cost per student, \(F T U G=\) number of full-time undergraduate students, FTGRAD = number of full-time graduate students, FTEF = full-time faculty per 100 students, \(C F=\) number of contract faculty per 100 students, FTENAP = full time nonacademic professionals per 100 students, PRIVATE \(=1\) if the school is private, and 0 if it is public.

a. Estimate the regression of \(\ln (T C)\) on the remaining variables. What are the predicted effects of additional graduate students on total cost per student? What are the predicted effects of additional full-time faculty?

b. Include in the model not only PRIVATE but also PRIVATE \(\times\) FTEF. Are these variables individually and jointly significant at the \(5 \%\) level?

c. Use the \(N R^{2}\) test for heteroskedasticity that is possibly related to PRIVATE. What do you conclude at the \(1 \%\) level of significance?

d. Test the hypothesis in (b) using OLS estimates with robust standard errors.

e. Include in the model not only PRIVATE but also PRIVATE times all the other variables. Test the joint significance of PRIVATE and PRIVATE times all the other variables using an \(F\)-test. Use robust standard errors and carry out a robust \(F\)-test. Can we say "We reject the hypothesis that the models determining total cost per student are the same for public and private universities?"

f. Hypothesize \(\sigma_{i}^{2}=\exp \left(\alpha_{1}+\alpha_{2}\right.\) PRIVATE \()\). Obtain FGLS estimates of the model in (e) and carry out the F-test on PRIVATE and PRIVATE times all the other variables. What is the value of the \(F\)-test statistic? What is the \(1 \%\) critical value?