Revision: Variance Inflation in Regression

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 deflation may 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.

Detection of Near-Nulticollinearity through Centered and Noncentered Regression

“…The distinction between essential and nonessential multicollinearity and its diagnosis has not been not been adequately treated in either the scientific literature or in statistical software and this lack of information has led to mistakes in some relevant papers, for example Velilla [3] or Jensen and Ramirez [13]. This paper analyzes the detection of essential and nonessential multicollinearity from auxiliary centered and noncentered regressions, obtaining two complementary measures between them that are able to detect both kinds of multicollinearity.…”

“…Thus, the algebraic contextualization provided by Salmerón Gómez et al [12] will be complemented from an econometric point of view. This question was also presented by Jensen and Ramirez [13], striving to commit to a clarification of the misuse given to the VIF over decades since its first use, who insinuated: To choose a model, with or without intercept, is substantive, is specific to each experimental paradigm and is beyond the scope of the present study. It was also stated that: This differs between centered and uncentered diagnostics.…”

“…where HICP is the Harmonized Index of Consumer Prices (100%), BC is the Balance of Payments to net current account (millions of euros) and u is a random disturbance (centered, homoscedastic, and uncorrelated). Table 10 presents the analysis of model (13) and its corresponding auxiliary regressions. The values of the VIFs which are very close to one will indicate that there is not essential multicollinearity.…”

“…Estimations by OLS of model(13) and its corresponding auxiliary regressions (estimated standard deviation in parenthesis and coefficients significantly different from zero in bold). VIFnc of auxiliary regressions associated to model(13).…”

Standardization of Variables and Collinearity Diagnostic in Ridge Regression

“…Dias & Castro () showed that the real impact on variance can be overestimated by the traditional VIF when the explanatory variables contain no redundant information about the dependent variable and stated that a corrected version of this multicollinearity indicator becomes necessary. In their diagnostic of collinearity, Jensen & Ramirez () made a distinction between centred and uncentred VIFs. Moreover, they pointed out the special relevance of uncentred VIF when the independent term plays an important role in the interpretation of the model.…”