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What is the difference between unconditional and conditional (fixed effects negative binomial) regression models?

A similar question was asked for quantile regression here: What is the difference between conditional and unconditional quantile regression? But I am still not sure whether the same logic applies for all kinds of regression scenarios.

I stumbled upon a paper that dealt with un/conditional fixed effects: Van Ommeren, J. N., & Gutiérrez-i-Puigarnau, E. (2011). Are workers with a long commute less productive? An empirical analysis of absenteeism. Regional Science and Urban Economics, 41(1), 1-8. They claim that it has been criticised because the conditional negative binomial model is not a ‘true’ fixed-effects model as it does not condition out the fixed effects (Allison and Waterman, 2002; Greene, 2007),

Unfortunately I don't get the story behind the true and untrue FE. What I do know from their estimations is that this robustness check includes an overdispersion parameter. So I assume overdisperson might be a problem.

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Further, I managed to get some code from Stata:

Conditional FE negative binomial is:

xtnbreg outcome IV, i(id) fe

And unconditional FE negative binomial is:

nbreg outcome IV i.id, dispersion (constant)

The first command uses a fixed effects package (xt) whether the second one includes the id directly as dummies (as Least Square Dummy Variable estimation in normal FE). Does it have to do with fixed effects anyway or is it rooted in the negative binomal regression? Can overdispersion be a problem in standard fixed effects?

Perhaps someone can shed more light on these differences.

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