I have the following model: $$ |X| = B_0 +B_1 \cdot y + B_2 \cdot z , $$ where $z$ and $y$ are normally distributed random variables, and $B_1$ and $B_2$ denote the coefficients.
My dependent variable contains either the positive or negative values of $X$. The positive and negative values of $X$ together are normally distributed, with a mean and median of around zero. For statistical reasons I would like to separately test the positive and negative values. This splits the left and right side of the otherwise normally distributed variable $X$. This clearly violates the normality assumption used for linear regression models. If I want to compare groups I would use a nonparametric type of test, but in this case I want to run the equivalent of a linear regression. Is this possible?