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Describe multicollinearity and set it apart from perfect collinearity.

FRM® Part I · Quantitative Analysis · Regression Diagnostics · FRM1.QA.9.c

The idea

Multicollinearity means two or more regressors are highly, but not perfectly, correlated. OLS can still be computed and the coefficients stay unbiased, but each is estimated imprecisely. The standard errors are inflated, t-statistics fall, and coefficients become unstable across samples, while R² and the joint F-test can still look strong. The variance inflation factor measures the damage for regressor j. VIF = 1 ÷ (1 − Rj²). Here Rj² is the R² from regressing regressor j on the other regressors. The standard error is inflated by the square root of the VIF. A common rule of thumb flags a VIF above 10. Perfect collinearity is different. One regressor is an exact linear combination of the others, as in the dummy variable trap of an intercept plus a dummy for every category, and OLS has no unique solution. Perfect collinearity is a specification error, while imperfect collinearity is a precision problem.

Formulas

Variance inflation factor

VIF = 1 ÷ (1 − Rⱼ²)

Rⱼ²
the R² from regressing regressor j on all the other regressors

Measures how much collinearity inflates the variance of coefficient j; the standard error rises by √VIF, not by VIF. With two regressors Rⱼ² is their squared correlation.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly describes (imperfect) multicollinearity?

  1. Regressors are highly, but not perfectly, correlated, so standard errors are inflated. - correct
  2. One regressor is an exact linear combination of the others, so OLS has no unique solution.Why not: This describes perfect collinearity, which prevents estimation; imperfect multicollinearity does not.
  3. A regressor is correlated with the error term, so the OLS coefficients are biased.Why not: This is endogeneity, such as omitted-variable bias, and is not what multicollinearity means.
  4. The variance of the errors differs across observations, so the usual t-tests are invalid.Why not: This is heteroskedasticity, which concerns the error term rather than the correlation between regressors.

Multicollinearity is a high correlation among regressors. The estimates remain unbiased but become imprecise, with large standard errors and unstable coefficients.

A harder one

When regressor x1 is regressed on the other regressors in a multiple regression, the R² is 0.84. By what factor is the standard error of the x1 coefficient inflated relative to the case where x1 is uncorrelated with the other regressors?

  1. 6.25Why not: Reports the variance inflation factor, which scales the variance; the standard error scales by its square root.
  2. 2.50 - correct
  3. 5.25Why not: Uses R² / (1 - R²) rather than 1 / (1 - R²).
  4. 1.19Why not: Takes 1 / R² instead of 1 / (1 - R²), which wrongly shrinks the inflation toward 1.

VIF = 1 / (1 - 0.84) = 6.25. The standard error is inflated by the square root of the VIF, sqrt(6.25) = 2.50.

The hardest tier

A hedge fund returns regression uses x1 = market return and x2 = size-factor return. The analyst considers adding x3 = 0.5 x x1 + 0.5 x x2, computed exactly as a blended benchmark, or x4 = x1 plus small noise, with correlation 0.98 with x1. Which statement is correct?

  1. Adding x4 makes the model perfectly collinear, while x3 only inflates the standard errors.Why not: Reverses the cases: the exact combination x3 is perfect collinearity, whereas x4, correlated at 0.98, is imperfect.
  2. Adding either x3 or x4 makes OLS impossible to compute.Why not: Only the exact linear combination removes the unique solution; a 0.98 correlation still permits estimation.
  3. Adding either x3 or x4 only inflates the standard errors; unique estimates exist in both cases.Why not: Ignores that x3 is an exact function of x1 and x2, so the regressor matrix is singular and no unique estimate exists.
  4. Adding x3 makes the model perfectly collinear; adding x4 leaves it estimable but imprecise. - correct

Exact dependence (x3) is perfect collinearity and the coefficients are not identified. A very high but imperfect correlation (x4) leaves OLS computable, but the standard errors of x1 and x4 become very large.

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