  We propose a penalized least-squares method to fit the linear regression
model with fitted values that are invariant to invertible linear
transformations of the design matrix. This invariance is important, for
example, when practitioners have categorical predictors and interactions. Our
method has the same computational cost as ridge-penalized least squares, which
lacks this invariance. We derive the expected squared distance between the
vector of population fitted values and its shrinkage estimator as well as the
tuning parameter value that minimizes this expectation. In addition to using
cross validation, we construct two estimators of this optimal tuning parameter
value and study their asymptotic properties. Our numerical experiments and data
examples show that our method performs similarly to ridge-penalized
least-squares.
