Spline Functions Fitted by Standard Regression Methods

IT sometimes happens that when a new mathematical or statistical procedure is adopted from one discipline into another, it arrives complete with terminology, usage and (these days) computer software devised for specialized application to problems in the parent area. This is probably inevitable, but until the new method is more broadly perceived its application may fall considerably short of its potential in the adopting discipline. A recent example is the spline function. Briefly put, spline functions are a device for approximating the shape of a curvilinear stochastic function without the necessity of pre-specifying the mathematical form of the function. That is, it is unnecessary to restrict the estimate to a straight line, a polynomial of pre-specified degree, an exponential, or any other particular form. Brought over from engineering and the mathematics of interpolation, spline functions have appeared in several places in economic statistics in recent years. Application to economic problems has been made by Barth, Kraft and Kraft (1976), McGee and Carlton (1970) and Poirier (1973, 1976). Buse and Lim (1977) have shown spline functions to be a special case of restricted least squares. Yet because the idea is still wrapped in its original packaging, it is frequently overlooked when it might be a powerful adjunct to research. Moreover, even in some of the work where the spline function has been employed, it has not always been used to best advantage. For example, Barth and others in the article cited above, although admitting that it might improve their analysis to employ a multivariate spline function, were constrained by the fact that the software package at their disposal unfortunately ... permits only bivariate specification. Yet, in fact, the procedure is readily adapted to bivariate or multivariate analysis. In this article we show that by use of appropriately defined composite variables, spline functions are easily fitted by any standard package for ordinary least squares regression. Some of the examples given below were fitted by the familiar SPSS package, others were fitted by members of an undergraduate class in econometrics at the University of Hawaii, using the TSP routine. In the presentation, piece-wise linear regression is employed as a general introduction to the procedure. This is followed by development of the bivariate and the multivariate spline functions. The procedures are then illustrated by their application to the relationship of interest rates to money supply and inflation. Once the spline function is understood as a least squares regression model, additional variations become possible. As an example we present a modified or truncated spline function and apply it to the relationship of fertility to per capita income. We conclude with a few general remarks on the limitations of the method.