So I'm looking at salary differentials between two occupations, and I'm really interested in medians. Initially I just had the descriptive statistics up, and reported the difference but I really want to control for a few things. So I'm getting the differential with a quantile regression.
Now, that's all well and good for the difference itself - but if I'm interested in a percentage difference, is it legitimate to regress the natural log of the salary on the occupational dummy I'm interested in a quantile regression? Essentially, does the natural log behave the same way in a quantile regression that it does in OLS regression?
I ask because the coefficients for the level salaries are identical to the median differentials from the descriptive statistics (which I expected), but the coefficients for the regressions of the log salaries are not the same as the percent difference.