Improvements in medical care and technology and reductions in traffic-related fatalities in Great Britain

(1984) for estimating NB models with panel data. This method has the advantage of
factoring out the overdispersion parameters and accounting for heterogeneity in the data.

To account for the fixed individual effects in the NB model, we rewrite the Poisson parameter
as,

⅛ = им                                              (1)

where,

μit = E(n_it) = exp(â X_it)     i = 1, 2,....N t = 1, 2,........T (2)

in which μ_it is the Poisson parameter indicating expected numbers of casualties in an
observation unit i in a given time period t, n_it is the number of observed casualties in an
observation unit i during a given time period t, X_it is a vector of covariates which describe
the characteristics of an observation unit i during a given time period t, â is a vector of
estimable coefficients representing the effects of the covariates, and (X. is the individual-
specific fixed effect. To derive the joint probability of the fixed effect NB model, it is
necessary to find a convenient distribution for the sum of events for a given individual,
∑n_it . A detailed derivation can be found in Hausman et al. (1984). The resulting joint
probability of the i^th individual conditional on total years is

,niτ | ∑tnit) =

Γ( μit + n_lt )  '^

Γ( μit )Γ( ni_t ) + 1 J

which includes â via μi_t but does not include ц and the overdispersion parameter k. From
this the likelihood function can be derived as,

L(Σtnit | n_t,......,niτ) = ∏

t

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