I have a question regarding interpretation of regression results based on data where I have some negative values. Since the residuals were positively skewed I needed to log-transform my data, and because of negative values my log transformation looked like $\ln(Y-\min(Y)+1)$. Normally, if we have $\ln$ in both sides of the equation, then our $b$ coefficient means a percentage change of DV for 1 % change of IV. But I have big doubts whether the same is valid for such log-transformed data, where I added a constant to my DV in order to make the minimum value equal to 1. I can't also find any literature on this topic. Does anyone know for sure what would be the correct interpretation in such a situation?
I would try either a glm approach with a log link function or something like a cube root transformation (any any other root whose power is the reciprocal of an odd positive integer and that accomplishes your goal). Some guidance on the latter here, along with some more exotic transformations. The glm approach would allow you to get the marginal effects in terms of elasticities. The root transform will give the marginal effects with a bit of calculus and algebra, but they really won't be elasticities.