Besides the issue of collinearity - making coefficients kinda useless and p-values non interpretable, I was wondering, what would be the issue of including a continuous variable and the discretized version of that variable in the same model?
For example, let's say X1 is continuous. I want to include a categorical variable, X2, where X2 is the quartiles of X1. The reason for this is, I want to use X2 to interact with other categorical variables in the model, as opposed to X1.
I'd imagine the upside of doing this is, more observations for the interaction term itself. I also have the hypothesis, that maybe the 4th quartile has an especially strong interaction effect with other variables, as opposed to the other 3 quantiles - and I want to bring out the effect of that quartile.
My goal here is to improve prediction accuracy, and the coefficients themselves aren't as important. The only important thing about the coefficient is if the sign is positive or negative. I know for a fact X1 is significant based on the data I am working with.