Questions: Variance Inflation Factor and Multicollinearity Diagnosis

Question 1

You run a regression and find VIF = 25 for the coefficient on X₃. What does this concretely mean for your estimates?

• A. The coefficient on X₃ is biased upward by a factor of 25
• B. The variance of β̂₃ is 25 times larger than it would be if X₃ were uncorrelated with the other regressors, making the standard error 5 times wider
• C. X₃ explains 25% of the variation in Y, which exceeds acceptable limits
• D. The OLS estimator has broken down and estimates are no longer consistent

Answer: (B)

VIF = 1/(1 - Rⱼ²) measures the multiplicative inflation in variance: a VIF of 25 means Var(β̂₃) is 25× what it would be in an orthogonal design. Standard errors are √25 = 5× wider, which dramatically shrinks t-statistics. Critically, OLS estimates remain unbiased (option A is wrong) and consistent (option D is wrong) — multicollinearity inflates variance without introducing bias. Option C misreads what VIF measures: Rⱼ² here is from an auxiliary regression of X₃ on the other regressors, not X₃ on Y.

Question 2

A researcher includes both height_cm and height_inches in a regression predicting weight. Both variables have VIF > 1,000. What does the VIF indicate she should do?

• A. Drop both variables — VIF above 10 means the variables are irrelevant
• B. VIF diagnoses the severity of the problem but does not prescribe the fix; subject-matter reasoning determines whether to drop one, combine them, or accept wide standard errors
• C. Use GLS instead of OLS to eliminate the multicollinearity
• D. Increase the sample size until VIF falls below 10

Answer: (B)

VIF is a diagnostic, not a decision rule. It tells you how much variance inflation you have but not which variable to drop or how to fix the problem. In this case, height_cm and height_inches are perfect linear functions of each other — one should be dropped on conceptual grounds. But VIF alone doesn't tell you which. In other settings (X and X²), centering helps; in others, combining variables makes sense; in others still, you accept the imprecision because theory requires both. VIF quantifies the cost; the remedy requires judgment.

Question 3

A high VIF on a coefficient indicates that the OLS estimate is imprecise but not necessarily biased.

• T. True
• F. False

Answer: True

This is the crucial distinction: multicollinearity inflates the *variance* of OLS estimates without affecting their *expected value*. The Gauss-Markov theorem still applies — OLS remains BLUE (Best Linear Unbiased Estimator) under multicollinearity. What changes is the 'best' part: the minimum achievable variance is now very large because the data cannot cleanly separate the effects of collinear regressors. The estimates are centered on the truth but are spread widely around it.

Question 4

If two regressors have VIF > 10, their OLS coefficient estimates are biased.

• T. True
• F. False

Answer: False

Multicollinearity does not bias OLS estimates — it inflates their variance. Bias arises from omitted variables, measurement error, or endogeneity, not from correlation among included regressors. A common misconception is that high VIF 'breaks' OLS or introduces bias; the reality is that OLS still produces unbiased estimates, but with very large standard errors, making hypothesis tests unreliable and confidence intervals very wide. VIF above 10 means the estimates are imprecise, not wrong in expectation.

Question 5

VIF = 1/(1 - Rⱼ²). Explain what the auxiliary regression's Rⱼ² measures and why a high value inflates the variance of β̂ⱼ.

Model answer: The auxiliary regression regresses Xⱼ on all other regressors and records the R². This R² measures how much of Xⱼ's variation is already explained by the other regressors — in other words, how redundant Xⱼ is. When Rⱼ² is high (close to 1), the data contain very little independent variation in Xⱼ that isn't already captured by the other variables. OLS estimates coefficients by comparing how Y moves when Xⱼ changes while holding the others constant, but if Xⱼ rarely moves independently of the others, there are very few such comparisons available. The coefficient estimate becomes sensitive to small perturbations in the data, resulting in a large variance.

The formula 1/(1 - Rⱼ²) quantifies this directly: Rⱼ² = 0.9 means only 10% of Xⱼ's variation is 'independent,' so the variance is inflated tenfold. This connects VIF to the intuitive problem: with correlated regressors, the regression cannot determine how much of Y's movement to attribute to Xⱼ versus the others — it's like trying to measure the separate effects of temperature and humidity when they always move together.