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 interrelated only through the observed correlation structure in the data. Alternatively, such interrelationships can be taken into account by using generalized ridge regression, motivated as a form of penalized likelihood estimation. When the form of the penalty function is chosen appropriately, it has the effect of smoothing the regression coefficients corresponding to successive x‐variables in a similar way to the use of spline functions for interpolation using a single x‐variable. Such a smoothing of the regression coefficients is demonstrated in an example in which the response variable is a single measure of the growth rate 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 month at each site. The smoothing used penalizes the sum of squared third differences of the regression coefficients and leads to a reduction in the average variance of the fitted values of two‐thirds when compared with the unsmoothed regression. Furthermore, the smoothed regression coefficients suggest a bimodal relationship between growth rate and monthly mean temperature which would probably be missed by using standard rank reducing techniques.