Condition Index — Belsley Collinearity Diagnostics
Also known as: Belsley Condition Index, Collinearity Condition Index, Singular Value Condition Index, Koşul İndeksi
The Condition Index, introduced by Belsley, Kuh, and Welsch (1980), is a scalar measure derived from singular value decomposition of the scaled regressor matrix. It quantifies the degree of near-linear dependence among predictors in ordinary least squares regression, enabling analysts to detect collinearity that inflates coefficient variance and destabilises parameter estimates. Widely used in economics, social sciences, and biomedical research wherever OLS regression is applied.
Key highlights
- Identifies the number and severity of distinct near-dependencies simultaneously, not just an aggregate measure
- Variance decomposition proportions pinpoint exactly which predictors participate in each collinear relationship
- Column equilibration ensures scale-invariant results, unlike raw eigenvalue-based indices
- Threshold values (15 and 30) are empirically grounded and widely accepted in applied econometrics
Intuition
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How it works
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When to use it
Apply the Condition Index whenever fitting OLS regression with two or more continuous or binary predictors and suspecting near-linear dependence. It is especially valuable when VIF values are inconclusive or when the analyst needs to identify which specific predictors are involved in a collinear relationship rather than merely confirming its presence. The method assumes that the design matrix has been column-scaled; it does not require normality of residuals. It is less suited to highly sparse or very high-dimensional settings where principal component regression or ridge regression may be preferable alternatives.
Strengths & limitations
- Identifies the number and severity of distinct near-dependencies simultaneously, not just an aggregate measure
- Variance decomposition proportions pinpoint exactly which predictors participate in each collinear relationship
- Column equilibration ensures scale-invariant results, unlike raw eigenvalue-based indices
- Threshold values (15 and 30) are empirically grounded and widely accepted in applied econometrics
- Requires subjective threshold interpretation; the 30-cutoff is a guideline, not a statistical test with a known Type I error rate
- Performance degrades in very high-dimensional settings where the number of predictors approaches the sample size
- Does not directly prescribe a remedy; analysts must separately consider centering, ridge regression, or principal component regression
- Sensitive to the inclusion of an intercept column; whether to include or exclude it from the design matrix affects results
Common pitfalls
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Applications
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Frequently asked
How does the Condition Index differ from the Variance Inflation Factor?
The VIF is a coefficient-level measure that quantifies how much a single predictor's variance is inflated by its linear relationships with all other predictors, but it does not reveal how many distinct near-dependencies exist or which groups of predictors are jointly involved. The Condition Index, together with variance decomposition proportions, identifies each near-dependency separately and names the specific predictors participating in it, offering substantially richer diagnostic information.
Should the intercept column be included when computing the Condition Index?
Belsley (1980) explicitly recommends including the intercept column in the scaled design matrix for a complete collinearity analysis, because centering or excluding the intercept can mask collinearities that involve the constant term, such as near-perfect multicollinearity in polynomial or interaction models. However, some software implementations exclude the intercept by default, so analysts should verify the implementation's convention before interpreting results.
What remedies are available when the Condition Index exceeds 30?
Common remedies include collecting more data to break the near-linear relationship, removing one of the collinear predictors if theory permits, applying ridge regression or LASSO to stabilise coefficients under collinearity, or using principal component regression to replace correlated predictors with orthogonal components. The variance decomposition proportions guide which predictors to target in any remediation strategy.
Sources
- 1.Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. John Wiley & Sons.ISBN 978-0-471-05856-4
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Cite this page
ScholarGate. (2026, June 2). Condition Index. ScholarGate. https://scholargate.app/econometrics/condition-index