Audit Collinearity with the Actual Regression Design Matrix

Identify the columns the model actually uses

Begin with the fitted sample and model specification. Expand categorical variables, interaction terms, transformations, and any intercept exactly as the analysis software does. Collinearity concerns relationships among these design-matrix columns, not simply whether two raw spreadsheet variables have similar names.

The official statsmodels variance-inflation documentation describes VIF as a design-matrix diagnostic. Pairwise correlations can provide useful clues, but several columns together can predict another closely even when no individual pair appears extreme.

Trace the auxiliary regression

For an ordinary non-intercept column in the conventional full-rank linear-model setting, regress that column on the remaining predictors with the appropriate intercept. If its auxiliary R-squared is R², the familiar VIF is 1/(1−R²). Record the column, auxiliary specification, and sample used rather than storing an unlabeled list of numbers.

Do not confuse this R² with the outcome model's overall R². The diagnostic asks how much information about a predictor is already contained in the other predictors; it does not measure how well the outcome is explained.

Inspect a fictional example

Suppose a fictional auxiliary regression has R² = 0.80. The corresponding VIF is 1/0.20 = 5. Under the usual linear-model comparison holding residual variance and the predictor's total variation fixed, that represents fivefold variance inflation and an SE factor of sqrt(5), approximately 2.236.

This calculation is not a prediction that every reported SE will double after changing a model. Altering variables can also change the fitted sample, residual variance, estimand, and other quantities. If R² is exactly one, the denominator vanishes and the column is linearly redundant rather than merely associated with the others.

Investigate the source before changing the model

Check for duplicate variables, incompatible coding, all category indicators entered with an intercept, and deterministic transformations that reproduce an existing column. Distinguish those errors from intentional related terms such as a polynomial or interaction. Record whether a flagged value comes from a single coefficient or a method designed for a multi-parameter term.

A large VIF is a reason to investigate precision and numerical stability, not a universal instruction to drop a confounder. Removing a scientifically required variable changes the adjustment and may introduce bias. Centering can clarify interpretation and help some interaction or polynomial parameterizations, but it does not create independent information from redundant measurements.

Finish with the model matrix version, diagnostics, identified coding issues, and any justified revision. Compare coefficients, uncertainty, and fitted-sample membership after changes. The useful audit explains why estimates are difficult to separate and what evidence supports the chosen specification; it does not select a model by chasing the smallest possible VIF.

Source checked on 4 November 2026; fictional VIF and SE factor independently recalculated.