Re: st: nbreg with fixed effect vs xtnbreg,fe

["Brian P. Poi" <bpoi@stata.com>]

Richard Williams wrote:
> At 12:50 AM 2/8/2012, Muhammad Anees wrote:
> > Also the abstract in online from Guimarães, P (2008) is
> >
> > In this paper I show that the conditional fixed effects negative
> > binomial model for count panel data does not control for individual
> > fixed effects unless a very specific set of assumptions are met. I
> > also propose a score test to verify whether these assumptions are met.
> >
> > The full reference for the paper is
> > Guimarães, P., (2008), The fixed effects negative binomial model
> > revisited, Economics Letters, 99, pp63­66
> >
> > It, thus, indicates to take care when to choose the fixed effects
> > model while using Negative Binomial Regressions.
>
> William Greene also has some working papers on this, e.g.
>
> http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1281928
>
> I can't say that I fully understand his arguments, but he says things 
> like "The difference between the HHG and true FE models is that HHG 
> builds the effects into the variance of the random variable, not the 
> mean. Thus, we cannot conclude that the HHG estimator is a consistent 
> estimator of a model that contains a heterogeneous mean...it is 
> reasonable to conclude that the HHG estimator is at least potentially 
> problematic...In the HHG fixed effects NB model, the fixed effects enter 
> the model through the dispersion parameter rather than the conditional 
> mean function. This has the implication that time invariant variables 
> can coexist with the effects. This calls the interpretation of the 
> heterogeneity in the model into question."
>
> On the other hand he proposes some alternatives but notes that they have 
> problems too. At this point I am thinking the safest route is to make 
> sure you never study a problem that requires negative binomial 
> regression with fixed effects. ;-)

Recently Richard Williams mentioned research by Allison and Waterman (2002) indicated that the conditional fixed-effects negative binomial regression estimator due to Hall, Hausman, and Griliches (HHG, 1984) as implemented in Stata via the -xtnbreg, fe- command is not a true fixed-effects estimator.

Allison and Waterman motivate their argument by considering an unconditional fixed-effects estimator in which a set of dummy variables representing the panels in the dataset are included in the regression specification.  HHG show that the variance of the dependent variable in their model is a function of the with the panel-level heterogeneity terms.  Allison and Waterman then argue that because the panel dummies and their coefficients do not play the same role as the other regressors and their coefficients in the HHG model, that model is not a true fixed-effects model.

Whether one considers HHG's conditional estimator a "true" fixed-effects estimator really depends on how one defines a fixed-effects estimator.  If one thinks of fixed-effects estimators as extensions of pooled estimators with the inclusion of a set of dummy variables to allow for panel-specific constant terms, then Allison and Waterman's argument stands.  If, on the other hand, one thinks of fixed-effects estimators as estimators that allow for panel-level heterogeneity without making the strict exogeneity assumptions typically required for (more efficient) random-effects estimators, then HHG's estimator is a valid fixed-effects estimator.

HHG's conditional fixed-effects estimator avoids the incidental parameters problem by conditioning the likelihood function for each panel by the sum of the counts for that panel.  That eliminates the panel-level heterogeneity terms from the likelihood; see, for example, the Methods and Formulas of [XT] xtnbreg.  With the panel-level heterogeneity term eliminated, we can use standard asymptotic theory with fixed T and N tending to infinity to establish that HHG's estimator is consistent.

Allison and Waterman propose using a set of panel dummies to implement an unconditional fixed-effects estimator, and their simulation results suggest the estimator works well.  However, Greene (2007) claims that such an unconditional fixed-effects negative binomial estimator nevertheless does suffer from the incidental parameters problem.


   -- Brian Poi             -- Gustavo Sanchez         -- Kristin MacDonald
      bpoi@stata.com           gsanchez@stata.com         kmacdonald@stata.com


----
References

Allison, P. D. and Waterman, R. P. (2002). Fixed-effects negative binomial
     regression models.  Sociological Methodology, 32, 247--265.

Greene, W. (2007). Fixed and random effects models for count data.  Working
     paper, Department of Economics, Stern School of Business, New York University.

Hausman, J., Hall, B. H., and Griliches, Z. (1984). Econometric models for
     count data with an application to the patents-R&D relationship.
     Econometrica, 52, 909--938.

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