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st: Interpretation of coefficients in the Fixed Effects Negative Binomial Model

Subject   st: Interpretation of coefficients in the Fixed Effects Negative Binomial Model
Date   Mon, 7 Mar 2005 10:55:17 -0600


Last week I posted a question regarding a fixed effects negative binomial
model and asked about the interpretation of coefficients on covariates that
do not vary over time. Nick Cox referred me to the FAQ which clearly points
out why and how these coefficients are estimated in the xtreg context.
However, this explanation does not carry over to the nonlinear negative
binomial models. Fortunately, Paulo Guimaraes referred me to a very
interesting article(Paul Allison and Richard Waterman. 2002. "Fixed Effects
Negative Binomial Regression Models" Sociological Methodology 32:247-265)
that explained why coefficients on time-invariant covariate can be estimated
in a fixed effects negative binomial model.

I have tried to explain this below along with Paulo's input. Hope this will
be useful to Stata users. Also, we would like to hear any comments that
people may have. 

The Fixed effects negative binomial (FENB) version that Stata implements
follows from the analysis of Hausman, Hall and Griliches (Econometrica,
1984) where the overdispersion (shape) parameter in the NB distribution is
modeled as a function of covariates while the individual level fixed effect
is modeled through the scale parameter that is assumed to be fixed across
time for a specific individual. Specifically,

Let 	E(Y(it)|X(i), Z(it)) = theta(i)*lambda(it) and 
	V(Y(i)|X(i), Z(it)) = (1+ theta(i))* E(Y(it)|X(i), Z(it))
Where theta(i) is the scale parameter, and
Lambda(it) is the shape parameter
X(i) are the time invariant covariates and Z(it) are the time variant

Therefore, 	ln{E(Y(it)|X(i))} 	
		= ln(theta(i)) + ln(lambda(it))
		= ln(theta(i)) + [alpha + beta*X(i) + gamma*(Z(it))]

The conditional likelihood for this model lets the theta(i) parameter drop
out thereby overcoming the incidental parameter problems with fixed effects.
However, the lambda parameter or any part of it is not eliminated in the
conditional likelihood. Therefore, parameters such as the common intercept
(alpha), those on time invariant covariates (betas) and the time variant
covariates (gammas) are all estimated in the FENB model, since the
covariates are assumed to affect lambda and not theta.
All the coefficients can still be interpreted in the conventional way as the
effect of covariates on the mean in a log-link model. However, the model
implements specific assumptions about how these covariates and the fixed
effects affect the variance parameter.


Anirban Basu PhD
Section of General Internal Medicine
Department of Medicine
University of Chicago
5841 S. Maryland Ave, MC-2007
AMD B201
Chicago IL 60637
Tel:  +1 773 834 1796
Fax: +1 773 834 2238

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