Abstract
Variance Inflation Factors (VIFs) are reexamined as conditioning diagnostics for models with intercept, with and without centering regressors to their means as oft debated. Conventional VIFs, both centered and uncentered, are flawed. To rectify matters, two types of orthogonality are noted: vector-space orthogonality and uncorrelated centered regressors. The key to our approach lies in feasible Reference models encoding orthogonalities of these types. For models with intercept it is found that (i) uncentered VIFs are not ratios of variances as claimed, owing to infeasible Reference models; (ii) instead they supply informative angles between subspaces of regressors; (iii) centered VIFs are incomplete if not misleading, masking collinearity of regressors with the intercept; and (iv)variance deflationmay occur, where ill-conditioned data yield smaller variances than their orthogonal surrogates. Conventional VIFs have all regressors linked, or none, often untenable in practice. Beyond these, our models enable the unlinking of regressors that can be unlinked, while preserving dependence among those intrinsically linked. Moreover, known collinearity indices are extended to encompass angles between subspaces of regressors. To reaccess ill-conditioned data, we consider case studies ranging from elementary examples to data from the literature.
Highlights
Given {Y = Xβ+ε} of full rank with zero-mean, uncorrelated and homoscedastic errors, the p equations {XXβ = XY} yield the OLS estimators βfor β = [β1, . . . , βp] as unbiased with dispersion matrix V(β) = σ2V and V= [vij] = (XX)−1
VFv(β0) = 0.7222/0.9375 = 0.7704, with V11 = 0.7222 from (1), and 0.9375 from (19). This contrasts with VIFu(β0) = 3.6111 in (3) as in Section 2.2 and Table 2, it is noteworthy that the nonorthogonal design D(1) yields uniformly smaller variances than the V⊥-orthogonal RV, namely, D(0)
Our goal is clarity in the actual workings of Variance Inflation Factors (VIFs) as illconditioning diagnostics, to identify and rectify misconceptions regarding the use of VIFs for models X0 = [1n, X] with intercept
Summary
Ill-conditioning diagnostics include the condition number c1(XX), the ratio of its largest to smallest eigenvalues, and the Variance Inflation Factors {VIF(βj) = ujjvjj; 1 ≤ j ≤ p} with U = XX, that is, ratios of actual (vjj) to “ideal” (1/ujj) variances, had the columns of X been orthogonal. The key to our studies is that VIFs, to be meaningful, must compare actual variances to those of an “ideal” second-moment matrix as reference, the latter to embody the conjectured type of orthogonality. This differs between centered and uncentered diagnostics and for both types requires the reference matrix to be feasible. Studies of ill-conditioned data from the literature are reexamined in light of these findings
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