# Multicollinearity and interaction terms

My dataset has 2734 observations (but in some speficitations, that number reduces to 1280). I also have interaction terms (in some specifications, even fourth-order terms). As far as I know, multicollinearity can be safely ignored as regards interaction terms.

If this is correct, should I test multicollinearity without taking into consideration interaction terms?

If it is not correct, should I standardized continuous variables used to generate interaction terms?

Is there anything else I could do?

The very idea of testing for multicollinearity is dubious or even plain wrong, see Dave Giles blog post.

However, you may still be interested in the level of multicollinearity observed in your sample for diagnostic purposes.

Should you include interaction terms in testing for multicollinearity? I would include all regressors regardless of whether they are original variables or were generated as interactions of original variables. I would do that because the effect of multicollinearity will be the same regardless of the interpretation of the particular regressors (variables' names and interpretations do not matter when multiplying or inverting matrices).

Will standardizing the variables help avoid multicollinearity? I doubt that standardizing variables (subtracting estimated mean and dividing by estimated standard deviation) will help. If two variables $y$ and $x$ are perfectly collinear, that means

$$y-\beta_0-\beta_1 x=0$$

If you define standardized variables $\tilde{y}=\frac{y-\bar{y}}{\hat \sigma_y}$ and $\tilde{x}=\frac{x-\bar{x}}{\hat \sigma_x}$, the linear relationship between the standardized variables will still hold:

$$(\hat \sigma_y \tilde y + \bar y)-\beta_0-\beta_1 (\hat \sigma_x \tilde x + \bar x)=0$$

and hence the multicollinearity will still be there. The same logic applies also for imperfect multicollinearity where equality is approximate rather than strict. However, see Remedies for multicollinearity point 6. for a special case when standardizing may help.

• Thanks, Richard. According to your answer, if my model is well founded on (economic) theory and previous works, I can just ignore potential multicollinearity (although being aware of its possible effects on my results). One last thing. If I consider the basic version of my model, individual VIFs are smaller than 10 and mean VIF is 3.11; nevertheless, the condition number is quite high (65.9693). How should I interpret this contradiction between tests? – madu Feb 18 '15 at 12:07
• I would need to do some research to answer this question, but for now I cannot give a good answer. Sorry about that. – Richard Hardy Feb 18 '15 at 12:17