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

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

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 . Only use data on public universities in this exercise.

a. Calculate the summary statistics for the model variables for the years 1987 and 2011. What do you observe about the sample averages of these variables?

b. Estimate the model by random effects. Discuss the signs and significance of the estimated coefficients. What is the predicted percentage cost per student change if one additional tenure track faculty is hired, per 100 students? What does the estimated value of \(\delta_{8}\) suggest?

c. Using the random effects estimates, test the following hypotheses at the \(5 \%\) level: (i) \(H_{0}: \beta_{2} \geq \beta_{3}, H_{1}: \beta_{2}<\beta_{3}\); (ii) \(H_{0}: \beta_{4} \leq \beta_{6}, H_{1}: \beta_{4}>\beta_{6}\); and (iii) \(H_{0}: \beta_{5} \geq \beta_{6}, H_{1}: \beta_{5}<\beta_{6}\). What do these tests imply about the relative costs of undergraduate students versus graduate students, tenure track faculty relative to contract faculty, and contract faculty relative to graduate assistants?

d. Calculate the time averages of the explanatory variables other than the indicator variables, for example, \(\overline{F T E S T U}_{i}\). Add these variables to the model and test their joint significance at the \(1 \%\) level. What does the test result tell us about using the random effects estimator in this case? Which assumption is being tested?

e. Obtain the fixed effects estimates of the model. Discuss the signs and significance of the estimated coefficients. What is the predicted percentage cost per student change if one additional tenure track faculty is hired, per 100 students? What does the estimated value of \(\delta_{8}\) suggest? How do these estimates compare to the random effects estimates?

f. Using the fixed effects estimates, test the following hypotheses at the \(5 \%\) level: (i) \(H_{0}: \beta_{2} \geq \beta_{3}\), \(H_{1}: \beta_{2}<\beta_{3}\); (ii) \(H_{0}: \beta_{4} \leq \beta_{6}, H_{1}: \beta_{4}>\beta_{6}\); and (iii) \(H_{0}: \beta_{5} \geq \beta_{6}, H_{1}: \beta_{5}<\beta_{6}\). What do these tests imply about the relative costs of undergraduate students versus graduate students, tenure track faculty relative to contract faculty, and contract faculty relative to graduate assistants?