524 Discrete Choice Modeling

that occurper unit of time. It can be shown that this discrete random variable has
thePoisson probability distribution:

f(y)=exp(−λ)λy/y!, λ= 1 /θ >0,y=0, 1,...

The expected value of this discrete random variable isE[y]= 1 /θ. ThePoisson
regression modelarises from the specification:

E[yi|xi]=λi=exp(x′iβ).

The log-linear form is used to ensure that the mean is positive. Estimation
of the Poisson model by ML is straightforward owing to the simplicity of the
log-likelihood and its derivatives:

lnL =

∑n

i= 1 −λi+yi(x

′

iβ)−ln(yi+^1 )

∂lnL/∂β =

∑n

i= 1 (yi−λi)xi

∂^2 lnL/∂β∂β′ =

∑n

i= 1 −λixix

′

i.

Inference about parameters is based on either the actual (and expected) Hessian:

V=

[∑n

i= 1

λˆixix′i

]− 1

=

[

X′%ˆX

]− 1

,

or the BHHH estimator, which is:

VBHHH=

[∑n

i= 1

(yi−ˆλi)^2 xixi′

]− 1

=

[∑n

i= 1

ε^2 ixix′i

]− 1

=

[

X′Eˆ^2 X

]− 1

.

Hypothesis tests about the parameters may be based on the likelihood ratio or Wald
statistics, or the LM statistic, which is particularly convenient here:

λLM=

[∑n

i= 1

ˆε^0 ix

]′

VBHHH

[∑n

i= 1

εˆi^0 x

]

,

where the residuals are computed at the restricted estimates. For example, under
the null hypothesis that all coefficients are zero save for the constant term,λˆ^0 i=y,

εˆi^0 =yi−yand:

λLM=

[∑n

i= 1 (yi−y)xi

]′[∑n

i= 1 (yi−y)

(^2) x
ix
′
i
]− 1 [∑n
i= 1 (yi−y)xi
]
.
The Poisson model is one in which the MLE is robust to certain misspecifications
of the model, such as the failure to incorporate latent heterogeneity into the mean
(i.e., one fits the Poisson model when the negative binomial is appropriate.) In this
case, the robust (sandwich) covariance matrix:
Robust Est.Asy.Var
[
βˆ
]

[
X′ˆX
]− 1 [
X′Eˆ^2 X
][
X′ˆX
]− 1
,
is appropriate to accommodate this failure of the model. It has become common
to employ this estimator with all specifications, including the negative binomial.
One might question the virtue of this. Since the negative binomial model already
accounts for the latent heterogeneity, it is unclear whatadditionalfailure of the
assumptions of the model this estimator would be robust to.