One question I have carried around with me for a while is related to including quadratic terms for model specifications. I wonder why it is considered ok to include linear and quadratic terms into OLS analysis, despite the fact that the Variance Inflation Factor (VIF) gets high. A model specification that I was taught to be fine to use is the following:
$y = c + x + x^2 + \epsilon$
But this model has quite high numerical linear correlation between $x$ and $x^2$ and I do not understand why in this case the VIF does not matter.
An example for what I mean using R is the following:
# Load packages require(car) requre(lmtest) # Create artificial data dat <- data.frame(x = 1:1000, x2 = (1:1000)^2) dat$y <- dat$x2 + rnorm(1000, 0, 10) # Run model with linear term dat.lm1 <- lm(y ~ x, dat) # Run model with linear and quadratic term dat.lm2 <- lm(y ~ x + x2, dat)
In this setup, the RESET test suggests using quadratic terms but the VIF is high.
> reset(dat.lm1) RESET test data: dat.lm1 RESET = 27344126617, df1 = 2, df2 = 996, p-value < 2.2e-16 > reset(dat.lm2) RESET test data: dat.lm2 RESET = 0.4404, df1 = 2, df2 = 995, p-value = 0.6439 > vif(dat.lm2) x x2 16.03008 16.03008
So the question is: Is this really a model specification that is valid and if yes, why does the VIF not matter in that case?