[Paper Review] Overcoming the inconsistences of the variance inflation factor: a redefined VIF and a test to detect statistical troubling multicollinearity

Román Salmerón Gómez, Catalina Beatriz García García+1 more|arXiv (Cornell University)|May 5, 2020

Advanced Statistical Methods and Models19 citations

TL;DR

This paper proposes a redefined Variance Inflating Factor (TVIF) that accounts for sample size, error variance, and variable scaling, offering a statistically grounded test to detect multicollinearity that actually disrupts OLS inference. Unlike the traditional VIF, TVIF includes the intercept and uses a redefined orthogonal reference model, enabling a threshold-based statistical test to determine whether multicollinearity is empirically problematic rather than merely mathematically present.

ABSTRACT

Multicollinearity is relevant to many different fields where linear regression models are applied, and its existence may affect the analysis of ordinary least squares (OLS) estimators from both the numerical and statistical points of views. Thus, multicollinearity can lead to incoherence in the statistical significance of the independent variables and the global significance of the model. The variance inflation factor (VIF) is traditionally applied to diagnose the possible existence of multicollinearity, but it is not always the case that detection by VIF of a troubling degree of multicollinearity corresponds to negative effects on the statistical analysis. The reason for the lack of specificity of VIF is that there are other factors, such as the size of the sample and the variance of the random disturbance, that can lead to high values of the VIF but not to problematic variance in the OLS estimators (see O'Brien 2007). This paper presents a new variance inflation factor (TVIF) that consider all these additional factors. Thresholds for this new measure and from the index provided by Stewart (1987) are also provided. These thresholds are reinterpreted and presented as a new statistical test to diagnose the existence of statistical troubling multicollinearity. The contributions of this paper are illustrated with two real data examples previously applied in the scientific literature.

Motivation & Objective

• To address the inconsistency of the traditional VIF in identifying multicollinearity that actually harms statistical inference.
• To resolve the issue where high VIF values do not necessarily lead to inflated OLS variance due to confounding factors like sample size and error variance.
• To develop a new statistical test that determines whether detected multicollinearity has a real, disruptive effect on coefficient significance testing.
• To redefine the reference point for VIF using QR decomposition to create a more realistic orthogonal model, improving the measure's specificity.
• To provide interpretable thresholds for TVIF and Stewart’s index that signal when multicollinearity is statistically troubling rather than just numerically high.

Proposed method

• Redefined the VIF using a QR decomposition of the design matrix to generate a new, more appropriate orthogonal reference model, replacing the traditional orthogonalization that ignores the intercept and assumes fixed error variance.
• Defined the new measure, TVIF(i), as the inverse of the sum of squared residuals from the auxiliary regression of each independent variable on the others, incorporating the intercept into the analysis.
• Established a statistical test threshold based on the t-distribution critical value, the hypothesized coefficient value, and the estimated variance of the coefficient, using the formula: TVIF(i) > [t(n−k, 1−α/2) / β̂_i,o]² × var(β̂_i).
• Reinterpreted Stewart’s index (S²_i) as a function of TVIF and the sum of squares of each predictor, allowing detection of non-essential multicollinearity involving the intercept.
• Provided new thresholds for TVIF and Stewart’s index that are conditional on sample size, error variance, and coefficient estimates, ensuring they reflect actual statistical disruption.
• Formulated a formal statistical test to determine whether multicollinearity is 'troubling'—i.e., whether it leads to incorrect inference—by comparing observed values to the derived critical thresholds.

Experimental results

Research questions

• RQ1 Does the traditional VIF reliably detect multicollinearity that actually disrupts statistical inference in OLS regression models?
• RQ2 Can a redefined VIF (TVIF) be constructed that accounts for sample size, error variance, and variable scaling to improve diagnostic specificity?
• RQ3 Is it possible to develop a statistical test that distinguishes between numerically high multicollinearity and multicollinearity that actually affects coefficient significance?
• RQ4 How does the inclusion of the intercept in multicollinearity detection improve the assessment of model reliability?
• RQ5 Do the new thresholds for TVIF and Stewart’s index provide a more accurate and empirically meaningful criterion than the conventional VIF > 10 rule?

Key findings

• The traditional VIF fails to distinguish between multicollinearity that inflates OLS variance and that which does not, due to confounding factors like sample size and error variance.
• The proposed TVIF redefines the reference orthogonal model using QR decomposition, resulting in a lower bound that reflects realistic variance inflation under actual model conditions.
• TVIF values are derived from the inverse of the sum of squared residuals in auxiliary regressions and are directly linked to the statistical power of coefficient tests.
• The new statistical test uses critical values from the t-distribution and coefficient estimates to determine if multicollinearity is statistically troubling, with thresholds dependent on sample size and effect size.
• Stewart’s index (S²_i) is reinterpreted as a valid measure for detecting non-essential multicollinearity involving the intercept, and its thresholds are now tied to statistical significance.
• The paper provides empirically grounded thresholds for TVIF and Stewart’s index that signal whether multicollinearity is likely to cause incorrect inference, offering a solution to a long-standing gap in econometric diagnostics.