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I am running a regression of Y on X (both are continuous variables). I'd like to measure how the effect differs between two groups of individuals, coded by a dummy variable Z. The traditional way of doing it is, I believe, to run:
Y ~ X*Z + X + Z
However, I get much different results when I use an indicator function, meaning, I run:
Y ~ X*1(Z==0) + X*1(Z==1) + Z
Where 1(Z==0) is 1 if Z=0, and 0 otherwise. But I've never seen any regression like that. What is wrong with that approach?
Typically, the indicator variable, $Z$ in your terminology, is an indicator for $Z=1$. By using an indicator for $Z=0$, you are effectively switching which group is coded as $0$ and which is coded as $1$.
This is a strange way of coding your group labels and is likely to cause confusion. If you have a reason to consider group $1$ to be a “baseline” and group $0$ to be a group of interest (e.g., control and treatment groups, respectively), it might make more sense to stick with the convention where you code the baseline as $0$, though the math makes sense either way.
