I am trying to include several income types into a regression model (specifically a logit model). This variable has the particularity to have numerous zero for some type of incomes (typically some capital incomes). How do I choose between
- including a dummy variable of whether the individual has this particular type of income or not
- including a variable that accounts for the amount of income the person has (either the plain value or perhaps some quantiles).
I cannot include both due to collinearity, so how to choose? Should I use rather use a criterion like $R^2$, or AIC, BIC, or something like whether one is significant or not, etc?
And additionaly, is there a way to include both in a model?