I want to predict a variable $Y$ given a set of variables $X_i$. To account for nonlinearity, my $X_i$ are put in several quantile dummies, so that I prefer transforming my $y$.
My $Y$ variable are different kinds of capital income, therefore always positive and skewed with high value variables. My explicative variables are part sociodemographic descriptors, and part other kind of incomes.
If I try to use $$ Y=Xb+u $$
If I use a log, $$ ln(Y)=Xb+u $$
I get this one, which is better, but with still some problems at the tails.
Are there any other common transformations ? I have tried with $Y^\alpha$ but it seems mostly worse, although I have not tested a lot of $\alpha$ yet. And specifically are there transformations useful in my case, that is the one in which residuals are still a bit more extreme after a log transformation.
Note that in the end I want to do prediction on another dataset, so I am also interested in the back transformation to get $Y$ back, which I have asked a bit about here