Kit has kindly posted my new Heterogeneous Canonical Negative Binomial
regression program to the SSC site. One can download the files by typing at the
Stata prompt:
. ssc install hcnbreg
Canonical Negative Binomial (NB-C) is a parameterization that derives the
negative binomial directly from the exponential family form of the Negative
Binomial PDF, and not as a Poisson-gamma mixture model. Thought of as a GLM
model, the canonical form is the canonical link. nbreg allows you to model either
the traditional NB-2 model, or NB-1, which has a constant dispersion. Both
have been traditionally derived as Poisson-gamma mixtures. With respect to
GLM, NB-2 is a log-linked negative binomial - which is not the canonical form.
I posted a maximum likelihood NB-C regression program to SSC back in late
2005. Like nbreg, it estimates the heterogeniety parameter, called alpha. The
heterogeneous NB-C model is analogous to Stata's gnbreg command, called
"Generalized Negative Binomial". Actually the use of "generalized" here is
unfortunate. There have been Generalized NB models discussed in the literature for
years -- that are quite unlike Stata's version. LIMDEP (W Greene) and other
sources refer to what Stata calls a generalized negative binomial as a
heterogeneous negative binomial (NB-H). The NB-H command allows parameterization of
the estimated heterogeniety parameter (alpha). This may provide the user with
information about which predictors significantly impact unexplained
correlation in the model, which in turn may help the user to determine how to further
adjust the model for a more optimal fit.
Anyhow, I have found that the canonical NB model can produce a better fitted
model for certain count response models than can the more traditional
negative binomial. Just remember that the NB-C model (as well as the heterogeneous
NB-C which provides additional information) does not have the same
relationship to Poisson as do NB-1 or NB-2. But it is a viable count model nonetheless.
Joseph Hilbe
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