Negative Binomial Model: Fixed vs Random Effects How to choose between Fixed and Random Effects in panel negative binomial model?
Is Hausman test valid for this?
 A: Fixed effects vs Random effects is a common question and not limited to negative binomial model. 
Let check the fixed effect only generalized linear model. This model has long history in statistics and is used widely at present. These models are used to describe the relation between covariates and conditional mean of the response variable. For example, in linear model, $E(Y|X)=X\beta$; in logistic regression, $E(Y|X) = \mathrm{logit}^{-1}(X\beta)$. 
One important assumption for the fixed effect only generalized linear model is the independence among the response variables, i.e., conditional on given $X$, the $Y$ are independent from each other. 
When the assumption of independence is not true, the fixed effect only generalized linear model is not applicable anymore. One example is the repeated measurements (also called longitudinal data, panel data) from the same subject. Obviously, the response variable is not independent anymore because the i-th measurement from one subject can help you to guess the (i+1)-th measurement from the same subject, expert from the information provided by covariates. Another example is the multilevel data structure. The subjects from the same unit of a level have some kind of similarity, and it is correlation in statistics.
To deal with the dependence (correlation) between response variables, we have two approaches. 


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*Specify marginal correlation directly. In the linear mixed model, specify the variance-covariance of error term $\mathrm{Var}(\epsilon)$ directly. For other generalized model, use the GEE (generalized estimating equations). Advantage: specify the relationship directly, so it is easy to understand and maybe easy for calculation. Disadvantage: Cannot deal with complicated variance-covariance structure and has strong requirement to data structure.

*Random effect: This approach is to create the correlation by sharing the common random effects among the correlated response variable. So random effect is used indirectly to specify the variance-covariance matrix of response variable. So before specify the random effects, we need the structure of the variance-covariance matrix of response variable as start point, then specify the random effects to achieve the designed structure. For example, suppose the sample of students come from several different schools and the response variable is a test score. Then we can say that two test scores are correlated if they come from the two students in the same school. Also we can assume that correlations are the same for any pair students from the same school. Of course, the scores from the different school are independent. Then we need the school specific random intercept to create this structure.
In summary, the fixed effect is used for the conditional means of the response variable. The random effect is used to specify the structure of the response variable. If no correlation between response variable, no random effect.
A: Allison & Watermann (2002) stress that in most statistical software negative binomial fixed effects are modelled via the HHG negative binomial method, which allows for individual-specific variation in the dispersion parameter rather than in the conditional mean. Consequently, time-invariant variables often end up getting non-zero coefficients and being highly significant. So just applying fixed effects leads o biased results....
For solutions check out: https://statisticalhorizons.com/fe-nbreg/
