How to manually calculate Variance Inflation Factor (VIF)?

Question

I would like to manually calculate the Variance Inflation Factor (VIF) for mtcars dataset. When I do it using the function from package I have the following result:

install.packages('car') library(car)

fit <- lm(mtcars[,1] ~ ., mtcars[,-1])
summary(fit)

      cyl      disp        hp      drat        wt      qsec 
15.373833 21.620241  9.832037  3.374620 15.164887  7.527958 
       vs        am      gear      carb 
 4.965873  4.648487  5.357452  7.908747

Then I do it one by one for each column: For the first one cyl it's perfectly ok:

fit <- lm(mtcars[,2] ~ ., mtcars[,- c(1:2)])
summary(fit)$r.squared 
1/(1-summary(fit)$r.squared)

[1] 15.37383

While for disp there's a difference already.

fit2 <- lm(mtcars[,3] ~ ., mtcars[,- c(1:3)])
summary(fit2)$r.squared 
1/(1-summary(fit2)$r.squared)

[1] 20.08864 # but it must be 21.620241

What's wrong?

Answer 1

The correct formula for VIF is:

fit2 <- lm(mtcars[,3] ~ ., mtcars[,- c(1,3)])
1/(1-summary(fit2)$r.squared)

21.62024

Note that 1:3 means c(1,2,3). We don't want to exclude the second column in the design matrix. Therefore, we should specify c(1,3) instead.