Selecting principal components in regression

Criteria for the deletion of principal components in regression are usually based on one of two indicators of components effects: (i) the magnitude of the eigenvalues of the predictor-variable correlation matrix or (ii) statistical tests of the significance of the components. Advocates of the first criterion cite guaranteed variance reduction properties as a rational for their proposals whereas proponents of inferential criteria point out that deletion solely on the basis of the magnitude of the eigenvalues ignores the potentials for bias. In this note we discuss the liminations of the second approach.