I have a dependent variable Y which is continuous. I want to study the impact of X on Y using OLS in a linear model, but I suspect the impact of X is more important for observations with a high value of Y compared to those with a low value of Y.
I could run a first regression using observations with Y values above the median of Y for instance, and another regression using observations with Y values below the median, but it would of course lead to a sample selection bias.
I suspect I would have the same bias if I were to interact X with a dummy equal to 1 for observations with a value of Y greater than the median of Y for instance. Is it true?
What are the solutions to study the effect of X on Y depending on the values of Y (i.e. high values of Y vs low values of Y). Are quantiles regressions the only solution?