On generalized ridge regression estimators under collinearity and balanced loss

In regression analysis, ridge estimators are often used to alleviate the problem of multicollinearity. Ridge estimators have traditionally been evaluated using the risk under quadratic loss criterion, which places sole emphasis on estimators' precision. Here, we consider the balanced loss function (A. Zellner, in: S.S. Gupta, J.O. Berger (Eds.), Statistical Decision Theory and Related Topics, vol. V, Springer, New York, 1994, p. 377) which incorporates a measure for the goodness of fit of the model as well as estimation precision. By adopting this loss we derive and numerically evaluate the risks of the feasible generalized ridge and the almost unbiased feasible generalized ridge estimators. We show that in the case of severe multicollinearity, the feasible generalized ridge estimator often produces the greatest risk reductions, even if a relatively heavy weight is given to goodness of fit in the balanced loss function.