What it means
Penn State's STAT 462 course describes VIF as the factor by which the variance of an estimated coefficient is inflated by correlation among the predictors (the input variables in a regression). Each predictor in a multiple regression has its own VIF.
The formula is VIF for predictor j = 1 / (1 - R squared of j), where R squared of j comes from regressing predictor j on the remaining predictors. A high R squared in that side regression means the predictor is largely explained by the others.
Penn State says a VIF of 1 means the predictor has no correlation with the others, so its variance is not inflated at all. The general rule of thumb is that VIFs above 4 warrant further investigation, while VIFs above 10 are signs of serious multicollinearity that need correction.
The course works through blood pressure data from 20 patients, where three of the VIFs were 8.42, 5.33 and 4.41. The Weight predictor had a VIF of 8.42 and a side regression R squared of 0.8812.
Penn State notes that checking pairwise correlations alone is limiting, because three or more variables can be linearly dependent even when each pair has a small correlation. That is why analysts rely on VIFs, and the course adds that a common fix is to drop one of two highly correlated predictors, with the choice often practical.
A paper in Quality and Quantity cautions against treating thresholds as rules: it says a VIF of 10, 20, 40 or higher does not by itself discount the results of a regression, nor does it by itself require dropping a variable, using ridge regression or building an index. The author says thresholds need to be judged alongside other factors that influence the variance of coefficients.
In practice
Real-world examples.
Example
A fictional analyst models house prices using floor area, number of rooms and number of bathrooms. The three are strongly related, so the VIF for rooms comes out near 9. The analyst checks whether rooms and bathrooms are adding the same information before deciding which to keep.
Example
A fictional credit model uses income, monthly spend and card limit. The side regression for card limit has an R squared of 0.75, so its VIF is 1 / (1 - 0.75) = 4. That is at the level Penn State says warrants a closer look, though it is not a reason to delete the variable on its own.
Example
A fictional marketing model has two predictors that are almost copies of each other, such as ad spend in dollars and ad impressions bought. The side regression R squared is 0.98, so the VIF is 1 / 0.02 = 50. The coefficient standard errors are very wide, and the model cannot say which of the two drives sales.
Formula
Calculation
VIF(j) = 1 / (1 - R squared(j)).
Example from Penn State: R squared for Weight is 0.8812, so VIF = 1 / (1 - 0.8812) = 1 / 0.1188 = 8.42.
The VIF is a multiplier on variance, so the standard error of a coefficient is inflated by the square root of the VIF. A VIF of 8.42 therefore stretches the standard error by about 2.9 times, because the square root of 8.42 is about 2.9.
Quick reference points: at an R squared of 0.5 the VIF is 2, at 0.75 it is 4, and at 0.9 it is 10. A VIF of 4 roughly doubles the standard error, since the square root of 4 is 2, and a VIF of 9 triples it.Case study
Seen in the real world.
This case study is fictional and illustrative. A bank builds a regression to explain loan loss rates using borrower income, debt payments and loan size. The model fits well overall. The analyst sees that two coefficients are insignificant one at a time, even though the model's overall F-test is significant. She recalls that Penn State lists this as a sign of multicollinearity.
She computes VIFs. Income has 2.1, loan size has 6.3 and debt payments have 6.8. The last two exceed 4, so they deserve a closer look. The side regression shows that debt payments are largely explained by loan size. She drops loan size, since debt payments are easier to measure.
The VIFs fall below 2.5 and the remaining coefficients become stable. The lesson is to use VIF as a prompt to investigate, not an automatic order to delete a variable. In her write-up for the credit committee she also records the alternative she rejected: keeping both variables and accepting wider confidence intervals, which would have been reasonable if the goal were only prediction. She notes that the choice depends on what the model is for, and files the VIF table alongside the model so a reviewer can see why loan size was removed.
Watch out
Common mistakes.
- Treating 10 as a hard line, when the Quality and Quantity paper says high VIFs alone do not discount a regression.
- Checking only pairwise correlations, when three or more predictors can be linearly dependent with small pairs.
- Dropping a variable without asking which of the correlated predictors is more useful or easier to measure.
Questions
People also ask.
What does a VIF of 1 mean?
It means the predictor has no correlation with the others. The variance of its coefficient is not inflated.
What VIF is too high?
Penn State gives a rule of thumb. Above 4 warrants a look and above 10 is a sign of serious multicollinearity. A paper cautions that these cutoffs are not strict rules.
How do I fix a high VIF?
One option is to remove one of the correlated predictors. Choose the one that is harder to measure or less useful.
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