The data file college cost 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

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

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, 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 . Use data only on public universities for this question.

a. Create first differences of the variables. Using the 2011 data, estimate by OLS the first-difference model

\[\Delta \ln \left(T C_{i t}\right)=\beta_{2} \Delta \text { FTESTU }_{i t}+\beta_{3} \Delta F T G R A D_{i t}+\beta_{4} \Delta T T_{i t}+\beta_{5} \Delta G A_{i t}+\beta_{6} \Delta C F_{i t}+\Delta e_{i t}\]

b. Repeat the estimation in (a) adding an intercept term. What is the interpretation of the constant?

c. Repeat the estimation in (a) adding an intercept plus the 2011 observations on the variables FTESTU, FTGRAD, TT, GA, and CF. If the assumption of strict exogeneity holds none of the coefficients on these variables should be significant, and they should be jointly insignificant as well. What do you conclude? Why is this assumption important for the estimation of panel data regression models?

d. Create the one period future, or forward, value for each variable, \(x_{t+1}\). That is, for example, in year \(t\) create a new variable FTESTU \(_{i, t+1}\). Using data from 2008 and 2010, estimate the panel data regression model by fixed effects, including the forward values of FTESTU, FTGRAD, TT, GA, and \(C F\). If the assumption of strict exogeneity holds none of the coefficients on these variables should be significant, and they should be jointly insignificant as well. What do you conclude?