Some characterizations of the ribge trace

Abstract

Ridge regression is a popular method for estimating linear regression coefficients when the explanatory variables are highly correlated. Ridge regression defines a class of estimators from which a unique estimator may be chosen based on the behavior of the ridge trace, a plot of the regression coefficient estimates versus the nonnegative scalar ridge parameter. In this paper characteristics of the ridge trace are determined algebraically when the correlation matrix of the explanatory variables is assumed to have a special structure (so-called Toeplitz). Special structure was required because quantitative characteristics are intractable in general. The Toeplitz structure was chosen because of its tractability and because it differs from the restrictive cases which have already been studied. Specific properties characterized include: (i) the order of the individual ridge estimates; (ii) the order of the rate-of-change of these estimates; and (iii) the number of sign changes in an individual estimate.

Keywords:

• ridge regression
• Toeplitz correlation structure