Generalized Collinearity Diagnostics

Abstract Working in the context of the linear model y = Xβ + e, we generalize the concept of variance inflation as a measure of collinearity to a subset of parameters in β (denoted by β 1, with the associated columns of X given by X 1). The essential idea underlying this generalization is to examine the impact on the precision of estimation—in particular, the size of an ellipsoidal joint confidence region for β 1—of less-than-optimal selection of other columns of the design matrix (X 2), treating still other columns (X 0) as unalterable, even hypothetically. In typical applications, X 1 contains a set of dummy regressors coding categories of a qualitative variable or a set of polynomial regressors in a quantitative variable; X 2 contains all other regressors in the model, save the constant, which is in X 0. If σ 2 V denotes the realized variance of , and σ 2 U is the variance associated with an optimal selection of X 2, then the corresponding scaled dispersion ellipsoids to be compared are ℰ v = {x : x′V ...