Palgrave Handbook of Econometrics: Applied Econometrics

526 Discrete Choice Modeling

then the unconditional distribution is:

f(yi|xi)=

∫∞

0

f(yi,vi|xi)dvi=

∫∞

0

f(yi|xi,vi)f(vi)dvi.

The integral can be obtained in closed form; the result is thenegative binomial
model:

Prob(Y=yi|xi)=

(θ+yi)

(yi+ 1 )(θ)

riyi( 1 −ri)θ,

λi=exp(x′iβ),

ri=λi/(θ+λi).

The recent literature, mostly associating the result with Cameron and Trivedi
(1986, 1998), defines this form of the negative binomial model as theNegbin 2
(NB2) form of the probability. This is the default form of the model in the received
econometrics packages that provide an estimator for this model. TheNegbin 1
(NB1) form of the model results ifθis replaced withθi=θλi. Then,rireduces to
r= 1 /( 1 +θ), and the density becomes:

Prob(Y=yi|xi)=

(θλi+yi)

(yi+ 1 )(θλi)

ryi( 1 −r)θλi.

This is not a simple reparameterization of the model. The results in the example
below demonstrate that the log-likelihood functions of the two forms are not equal
at the maxima, and their parameters are not simple transformations of each other.
We are not aware of a theory that justifies using one form or the other for the nega-
tive binomial model. Neither is a restricted version of the other, so we cannot carry
out a nested likelihood ratio test. The more generalNegbin P(NBP) family (Greene,
2008b) does nest both of them, so this may provide a more general, encompassing
approach to finding the right specification. The NBP model is obtained by replac-

ingθin the NB2 form withθλ^2 i−P. We have examined the cases ofP=1 andP= 2
above and, for generalP:

Prob(Y=yi|xi)=

(θλQi +yi)

(yi+ 1 )(θλQi)

⎛

⎝ λ

θλQi +λi

⎞

⎠

yi⎛

⎝

θλQi

θλQi +λi

⎞

⎠

θλQi

, Q= 2 −P.

The conditional mean function for the three cases considered is:

E[yi|xi]=exp(x′iβ)×θ^2 −P=αP−^2 λi, whereα= 1 /θ.

The parameterPis picking up the scaling. A general result is that, for all three
variants of the model:

Var[yi|xi]=λi( 1 +αλPi−^1 ).

Thus, the NB2 form has a variance function that is quadratic in the mean, while
the NB1 form’s variance is a simple multiple of the mean. There have been many