Federally insured student loans constitute an area that is almost completely unexplored by researchers despite intense scrutiny that federally insured loans are receiving after the savings and loan collapse. Based on a probit model of default for two thousand guaranteed student loans, the authors find that individual characteristics (including parents' income, presence of two parents at home, student's graduation, and student's race) have a significant impact on default rates, while institutional characteristics (four year vs. two year college, private vs. public, school size, and individual school dummies) have little significant effect. The results imply that proposals to penalize colleges with high default rates are premature.

The Box-Cox difference transformation permits the selection of either the first difference or percentage change form of a time series regression model. Monte Carlo evidence on the small sample properties of the transformation parameter A indicates that the difference transformation works quite well even in samples of size 30. Likelihood ratio testing is compared to an asymptotically equivalent alternative Lagrange Multiplier test. It is shown that values of R2 can often be higher for the incorrect transformation.

A maximum likelihood method for estimating and testing for the proper functional form in first difference regression models is developed. The parametric transformation of the regression variables we propose includes simple first differences and percentage changes as special cases. The method has a simple relationship to the familiar Box-Cox test, and the coefficient estimation and LR testing are easily implemented with standard regression packages. We apply the new method to three published studies: the St. Louis equation, a money demand model, and a model relating poverty to economic growth.

The Box-Cox transformation is used to compare alternative functional forms of market value equations. Based on evidence from a panel of 480 publicly-traded U.S. manufacturing companies and two additional data sets used previously in the literature, the semilog form of a Tobin’s q equation is found to be strongly preferred to the commonly estimated linear form. We provide illustrations in which inferences can be affected by the choice of functional form. The authors thank Zvi Griliches, Hendrik Houthakker, and two anonymous referees for helpful discussion and suggestions, and Jerry Stevens for providing access to one of the data sets examined in Section III. Remaining errors are ours. A longer working paper version is available on request. 1

The authors obtain expressions for the restricted least squares estimator and its covariance matrix in the classical regression model when the matrix of regressors is not necessarily of full rank. The standard expressions for the restricted least squares estimator are not usable in the short rank case because they rely on the unrestricted estimator. But, in the presence of restrictions, the restricted least squares estimator may be computable even if the unrestricted estimator is not. The authors' derivation produces some additional, useful algebraic results for least squares computation.