SMOOTHING REGRESSION-COEFFICIENTS IN AN OVERSPECIFIED REGRESSION-MODEL WITH INTERRELATED EXPLANATORY VARIABLES

Standard rank reducing methods such as principal components regression and partial least squares take account of the x-variables being inter related only through the observed correlation structure in the data. A lternatively, such interrelationships can be taken into account by usi ng generalized ridge regression, motivated as a form of penalized like lihood estimation. When the form of the penalty function is chosen app ropriately, it has the effect of smoothing the regression coefficients corresponding to successive x-variables in a similar way to the use o f spline functions for interpolation using a single x-variable. Such a smoothing of the regression coefficients is demonstrated in an exampl e in which the response variable is a single measure of the growth rat e of Sitka spruce trees at each of 363 sites in Scotland and the 12 x- variables available are 30-year mean temperatures for each calendar mo nth at each site. The smoothing used penalizes the sum of squared thir d differences of the regression coefficients and leads to a reduction in the average variance of the fitted values of two-thirds when compar ed with the unsmoothed regression. Furthermore, the smoothed regressio n coefficients suggest a bimodal relationship between growth rate and monthly mean temperature which would probably be missed by using stand ard rank reducing techniques.