Dear Stata Forum,

My question relates more with the theory of Poisson and Negative Binomial Poisson as it does Stata codes. I hope you are still able to help.

With regards to the theory, I would like clarification on the equidispersion assumption of the Poisson model. I understand that it often fails and, by extension, the Negative Binomial corrects for this by including a dispersion parameter which essentially controls for unobserved heterogeneity, which the Poisson model does not. But does a Poisson regression with fixed effects [xtpoisson Y x1 x2 x3, fe ] not deal with the over dispersion problem because the FE deals with unobserved heterogeneity? In which case would Negative Binomial model not just serve as an alternative specification for robustness purposes rather than explicitly correcting the shortcoming of the Poisson? I noticed that in the trade literature on the extensive margin, researchers tend to use i) a Poisson with bootstrapped standard errors and no NB model or ii ) a Poisson with robust standard errors with fixed effects and a NB model with bootstrapped errors.

Also, what is the difference between a Poisson, a Poisson-pseudo maximum likelihood, a Poisson quasi-maximum likelihood? (Any direction towards literature here would be great!) I ask because I am unsure whether to use poisson, xtpoisson or ppml codes.

Many thanks in advance.

Kind regards,
Ray