In a regression model, the VIF for variable X₃ is 25. A researcher concludes that the OLS estimate of the coefficient on X₃ is biased. Is this conclusion correct?
AYes — high VIF means the OLS estimator is systematically biased away from the true effect
BNo — multicollinearity inflates the variance of β̂₃, making it imprecise, but OLS remains unbiased; bias is not the problem
CYes — collinearity causes the coefficient to systematically underestimate the true effect of X₃
DNo — VIF only detects nonlinearity between predictors, not collinearity
This is the central misconception about multicollinearity. OLS remains unbiased under collinearity — the Gauss-Markov conditions do not require orthogonal predictors. What collinearity does is inflate the variance (standard error) of the coefficient estimate, making it imprecise and statistically unreliable. A VIF of 25 means the variance of β̂₃ is 25 times larger than it would be with no collinearity — you have a noisy estimate, not a biased one.
Question 2 Multiple Choice
The auxiliary regression of predictor X₂ on all other predictors yields R² = 0.96. What is the VIF for X₂, and what does it mean?
AVIF = 0.04; X₂ has very little collinearity because only 4% of its variation is explained by the others
BVIF = 25; X₂'s coefficient variance is 25 times larger than it would be if X₂ were orthogonal to all other predictors
CVIF = 0.96; X₂ is 96% collinear, which is a moderate concern
DVIF = 4; there is mild multicollinearity requiring attention
VIF = 1/(1 − R²) = 1/(1 − 0.96) = 1/0.04 = 25. The auxiliary R² of 0.96 means 96% of X₂'s variation is explained by the other predictors — X₂ is highly redundant. OLS can barely distinguish X₂'s independent contribution, inflating its coefficient's variance 25-fold. A VIF of 25 far exceeds the common threshold of 10, indicating severe multicollinearity.
Question 3 True / False
A VIF of 1 for a predictor means it is perfectly orthogonal to all other predictors in the model, so no variance inflation is occurring for that coefficient.
TTrue
FFalse
Answer: True
VIF = 1/(1 − Rⱼ²). When Rⱼ² = 0, the auxiliary regression explains none of predictor j's variation — meaning the other predictors share no information with j. VIF = 1/1 = 1 corresponds to a multiplicative inflation of exactly 1 (no inflation). This is the baseline case of perfectly orthogonal design.
Question 4 True / False
High VIF values are typically a critical problem requiring remediation before a regression model can be used for any purpose.
TTrue
FFalse
Answer: False
Whether high VIF is a problem depends on the goal. For prediction, multicollinearity is largely benign: OLS predictions can remain accurate even with high VIFs, as long as the collinearity structure in the training data mirrors what will be seen in prediction. High VIF is a serious problem for causal interpretation, because it means individual coefficient estimates are noisy and unstable. Remediation (dropping variables, PCA, collecting more data) is necessary when you need to make causal claims about specific predictors.
Question 5 Short Answer
Explain what the auxiliary regression underlying VIF measures and why it captures the severity of multicollinearity for a specific predictor.
Think about your answer, then reveal below.
Model answer: The auxiliary regression regresses predictor j on all other predictors in the model. Its R² measures how much of predictor j's variation is already 'explained' by the other predictors — in other words, how redundant j is. If R² is high, the other predictors contain most of j's information, so OLS cannot isolate j's independent effect without large variance in the estimate. VIF = 1/(1 − R²) formalizes this: as j becomes more redundant (R² → 1), VIF → ∞, reflecting the impossibility of separately identifying j's contribution. A VIF close to 1 means j adds genuinely new information.
The auxiliary regression is a regression within a regression — it tests whether one predictor can be predicted from the others. High predictability means low independent information, which means high variance in the main regression's coefficient for that predictor.