I would like to ask for your help with the following issue. I am trying to estimate the model where dependent variable is a share of total (e.g. share of the US economy in terms of GDP on total world GDP). I use the panel data for 27 cross sections (countries) and 8 time dimensions (years). By definition, for each year the values of dependent variables across countries sum up to one. The distribution of the dependent variable is highly positively skewed with lots of observations being close to zero (or even zeros).
Usually, authors propose logistic transformation of the dependent variable in terms of $y*=\ln(y/(1-y))$ for y being the original dependent variable. However, even after such a transformation the dependent variable remains highly positively skewed. The pooled OLS regression or random-effect panel regression delivers residuals that are not normally distributed and again highly positively skewed. The possible option, I guess, is to go for beta distribution of dependent variable, but not sure whether appropriate for panel data and what type of model then to use to estimate the regression (would rather have random-effect panel regression as by nature of the independent variables the fixed-effect model is not the preferred one).