In econometric analysis in some cases, such as models with interaction terms, multicollinearity between independent variables may exist. In such cases, some researchers suggest "mean-centering" strategy (subtract mean from the variables which appear as main effects and constitute the interaction terms). Usually, in the literature this methodology is applied when the dependent variable is normally distributed (e.g., Paper). Does mean centering remove multicollinearity if the variables are not normally distributed? Why is normal distribution required?
The correlation with interaction terms does not depend on having a normal distribution.
Here is a example based on sampling from two independent exponential distributions (so with positive means) where the correlation with the interaction terms is high, but is largely removed by subtracting the means. It also shows that the correlation between the samples themselves is unaffected by subtracting the means.
set.seed(2023) X <- rexp(10^6, 1) Y <- rexp(10^6, 2) mean(X) # 0.9984457 mean(Y) # 0.5001369 cor(X, Y) # -0.001121004 cor(X, X*Y) # 0.5781138 cor(Y, X*Y) # 0.5757716 cor(X-mean(X), Y-mean(X)) # -0.001121004 cor(X-mean(X), (X-mean(X))*(Y-mean(Y))) # 0.002229947 cor(Y-mean(Y), (X-mean(X))*(Y-mean(Y))) # -0.0002722061
You need to subtract the means before taking the product, as subtracting a constant at the end has no impact on the correlation. Compare these with the earlier results:
cor(X-mean(X), X*Y-mean(X*Y)) # 0.5781138 cor(X, (X-mean(X))*(Y-mean(Y))) # 0.002229947