Do larger universities have lower cost per student or a higher cost per student? A university is many things and here we only focus on the effect of undergraduate full-time student enrollment (FTESTU) on average total cost per student (ACA). Consider the regression model \(A C A_{i t}=\beta_{1}+\beta_{2}\) FTESTU \(_{i t}+e_{i t}\) where the subscripts \(i\) denote the university and \(t\) refers to the time period, and \(e_{i t}\) is the usual random error term.

a. Using the 2010-2011 data on 141 public universities, we estimate the model above. The estimate of \(\beta_{2}\) is \(b_{2}=0.28\). The \(95 \%\) interval estimate is [0.169, 0.392]. What is the estimated effect of increasing enrollment on average cost per student? Is there a statistically significant relationship?

b. There are many other factors affecting average cost per student besides enrollment. Some of them can be characterized as the university "identity" or "image." Let us denote these largely unobservable individual characteristics attributes as \(u_{i}\). If we add this feature to the model, it becomes \(A C A_{i t}=\beta_{1}+\beta_{2}\) FTESTU \(U_{i t}+\left(u_{i}+e_{i t}\right)=\beta_{1}+\beta_{2}\) FTESTU \(U_{i t}+v_{i t}\). As long as \(v_{i t}\) is statistically independent of full-time student enrollment, then the least squares estimator is BLUE. Is that true or false? Explain your answer.

c. The combined error is \(v_{i t}=u_{i}+e_{i t}\). Let \(\hat{v}_{i t}\) be the least squares residual from the regression in (a). We then estimate a simple regression with dependent variable \(\hat{v}_{i, 2011}\) and explanatory variable \(\hat{v}_{i, 2010}\). The estimated coefficient is 0.93 and very significant. Is this evidence in support of the presence of unobservable individual attributes \(u_{i}\), or against them? Explain your logic.

d. With our 2 years of data, we can take "first differences" of the model in (b). Subtracting the model in 2010 from the model in 2011, we have \(\Delta A C A_{i}=\beta_{2} \Delta F T E S T U_{i}+\Delta v_{i}\), where

\[\begin{aligned}\Delta A C A_{i} & =A C A_{i, 2011}-A C A_{i, 2010} \\\Delta F T E S T U_{i} & =F T E S T U_{i, 2011}-F T E S T U_{i, 2010} \\\text { and } \Delta v_{i} & =v_{i, 2011}-v_{i, 2010}\end{aligned}\]

Using the first-difference model, and given the results in (c), will there be serial correlation in the error \(\Delta v_{i}\) ? Explain your reasoning.

e. Using OLS, we estimate the model in

(d) and the resulting estimate of \(\beta_{2}\) is \(b_{F D}=-0.574\) with standard error \(\operatorname{se}\left(b_{F D}\right)=0.107\). What now is the estimated effect of increasing enrollment on average cost per student? Explain why the result of this regression is so different from the pooled regression result in (a). Which set of estimates do you believe are more plausible? Why?