Ridge regression and its degrees of freedom

For ridge regression the degrees of freedom are commonly calculated by the trace of the matrix that transforms the vector of observations on the dependent variable into the ridge regression estimate of its expected value. For a fixed ridge parameter this is unobjectionable. When the ridge parameter is optimized on the same data, by minimization of the generalized cross validation criterion or Mallows CL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {C}_{L}$$\end{document}, additional degrees of freedom are used however. We give formulae that take this into account. This allows of a proper assessment of ridge regression in competitions for the best predictor.