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I understand the usual procedure to code categorical variables is to convert n categories into n-1 coded variables. For example, the categorical variable colour with levels red/green/blue could be coded as
V1 V2 red -> 1 0 blue -> 0 1 green -> 0 0
which in a regression setting means that the effect of green on the response is factored into the intercept.
I know that if we created an additional binary variable V3 such that green is coded
V1 V2 V3 red -> 1 0 0 blue -> 0 1 0 green -> 0 0 1
then we should fit a regression model with no intercept.
What happens if I take the latter coding (i.e. 3 variables V1, V2, V3 for 3 levels of colour) and fit a regression model with an intercept? I can't figure out why we shouldn't do this.