The data file collegecost contains data on cost per student and related factors at four-year colleges in the U.S., covering the period from 1987 to 2011. In this exercise, we explore a minimalist model predicting cost per student. Specify the model to be

\[\ln \left(T C_{i t}\right)=\beta_{1}+\beta_{2} F T E S T U_{i t}+\beta_{3} F T G R A D_{i t}+\beta_{4} T T_{i t}+\beta_{5} G A_{i t}+\beta_{6} C_{i t}+\sum_{t=2}^{8} \delta_{t} D_{t}+u_{i}+e_{i t}\]

where \(T C\) is the total cost per student, FTESTU is number of full-time equivalent students, FTGRAD is number of full-time graduate students, \(T T\) is number of tenure track faculty per 100 students, \(G A\) is number of graduate assistants per 100 students, and \(C F\) is the number of contract faculty per 100 students, which are hired on a year to year basis. The \(D_{t}\) are indicator variables for the years \(1989,1991,1999,2005,2008,2010\), and 2011. The base year is 1987.

a. Calculate the summary statistics for the model variables for the years 1987 and 2011 separately for public and private universities. What do you observe about the sample averages of these variables? In particular, what is the increase in \(T C\) between 1987 and 2011 for each type of university. What has happened to the number of tenure track faculty and the number of contract faculty?

b. Using OLS, estimate the model for public universities using conventional and cluster-robust standard errors. Are the standard errors noticeably different?

c. Using OLS, estimate the model for private universities using conventional and cluster-robust standard errors. Are the standard errors noticeably different? How do the coefficient estimates for the private universities compare to those for the public universities?

d. Estimate the model using fixed effects with cluster-robust standard errors for the public universities. How do these estimates compare to the OLS estimates in (b)? What are the important differences?

e. Estimate the model using fixed effects with cluster-robust standard errors for the private universities. How do these estimates compare to the estimates for the public universities in part (d)? What are the important differences?