I have a question about negative binomial versus poisson in the presence of an over-dispersed dependent count variable, but with the need to include fixed effects for the cross-sectional component. My understanding is that one of the benefits of the Stata command xtpoisson, fe is that it does not suffer from incidental parameters while xtnbreg, fe in reality is still subject to the incidental parameters problem. Is my understanding correct? And if so what would be my best option given the over-dispersion and the need for fixed effects?
xtpoisson, fe computes the conditional fixed-effects estimator, as well as xtnbreg. So none should suffer from the incidental paramreter problems. However, as pointed out by Allison and Waterman (2004), xtnbreg is not what one usually calls a fixed-effect model. Their article* suggests among other things to use the conditional nbreg model with a correction on standard errors and by hoping that the incidental parameters problem is not so important. (In a simulation they perform, it is not so important. But is this also the case in your application? Maybe if it is close to their simulation.)
Depending on your problem, your best choice might be to go for the xtpoisson - notably in the case where you are not interested in estimating probability. This is stressed in this stata-forum discussion**. Conclusion of the discussion is "Basically, if you do not want to estimate probabilities, it is unlikely that you need to worry about over-dispersion [and you should use xtpoisson]. If you need to estimate probabilities, negative binomial models may still be too restrictive."***
*Allison, Paul D, and Richard P Waterman. 2002. “Fixed–effects negative binomial regression models.” Sociological methodology, 32(1): 247–265 https://onlinelibrary.wiley.com/doi/abs/10.1111/1467-9531.00117 ** https://www.statalist.org/forums/forum/general-stata-discussion/general/1323497-choosing-between-xtnbreg-fe-bootstrap-and-xtpoisson-fe-cluster-robust ***The provided reference in the stata-forum is: Wooldridge, J. M., “Distribution-Free Estimation of Some Nonlinear Panel Data Models,” Journal of Econometrics 90 (1999), 77–97. I would also suggest: Wooldridge, Jeffrey M. (1999) “Quasi-likelihood methods for count data.”Handbook of applied econometrics