I am asked to perform a regression on a dependent variable that only takes positive values. Intuitively one can think of Poisson or negative binomial, but unfortunately the response variable that I am dealing with is not a count variable. How can I perform a regression on such response variable? any ideas?
On this information, many distributions could make sense: gamma, lognormal, etc., etc. In general, generalized linear models with various links (logarithm first and foremost) and various families could all apply.
By the way, the usefulness of Poisson is not limited to count variables; this is a common myth. See for example for a brisk introduction to the question http://blog.stata.com/tag/poisson-regression/