Accuracy of Least Squares Computer Programs: Another Reminder: Comment

Abstract

In a recent issue of this Journal, Boehm, Menkhaus, and Penn present yet another warning concerning the accuracy of least squares programs. While their warning is valid and useful, the recommendations to users in the authors' conclusions are, at the very least, tedious. This is especially so when users have access to highly effective methods for coping with ill-conditioned data in least squares problems. This comment examines each of the recommendations of Boehm, Menkhaus, and Penn in the light of what can be done with a singular value decomposition algorithm, in particular demonstrating that this technique reveals collinearities and permits some assessment to be made of their seriousness. Statistical computing packages, for instance the TROLL system of the National Bureau of Economic Research, are beginning to incorporate such methods. While most packages do obtain reasonable answers to the least squares problems they attempt to solve, doubts can only be set aside if the computing techniques employed are able to reveal the structure of the problem. That such techniques are not more widely used is largely a matter of imperfect exchange of information between researchers in different disciplines, a fault which this note will try in part to correct. In the following sections the recommendations of Boehm, Menkhaus, and Penn are reviewed. The difficulties which may be inherent in linear least squares problems are then outlined briefly in the light of principal components solution methods. The family of computational procedures based on the singular value decomposition of a matrix is recommended as a practical way to implement such methods, which provide valuable information about the numerical stability of the regression problem whether or not principal components estimators are desirable from a statistical point of view. The fractional variation and explanation indices f and E, presented below are, it is believed, novel. The rest of the material is a summary of ideas taken from several sources outside the literature commonly consulted by economists. Recommendations