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(B) Likelihood Function for A Simple Case
We now discuss the likelihood-based inference of the above model in the case of a
sample of grouped duration data. To ease exposition we first take the simple case where
(i) There is no unobserved heterogeneity, that is, P(U n = 0) = 1.
(ii) All the covariates are time-invariant that is X n (t) = X n for all t
In this simple set-up the relevant hazard function is
h(t |X n (t), U) = h(t |X n )=(t)e Xn . (4.2)
The corresponding survivor function, S(t | X n ) = P(T n > t | X n ) can be easily derived,
using the proportional hazard assumption,
S(t |X n ) = exp{–(t)e Xn }. (4.3)
The contribution to the likelihood of an observation with a completed spell is the
probability that T n [K n , K n + 1) conditional on X n :
P(T n [K n ,K n + 1) |X n ) = S(K n |X n ) – S(K n + 1|X n ) (4.4)
= exp{–(K n )e X n } – exp{–(K n + 1)e X n }
Similarly, the contribution to the likelihood function of the right censored spell is
the probability that T n [K n , ) conditional on X n :
P(T n [K n , ) |X n ) = S(K n |X n ) = exp{–(K n )e Xn }. (4.5)
Clearly under data grouping A4, the sample likelihood function depends on the
unknown baseline hazard only through the discrete values of the integrated baseline
hazard function evaluated at integers - distinct values of {K n }’s.
The likelihood function for the entire sample will be the product of terms either (4.4)
or (4.5), depending upon the censoring condition for each individual in the sample.
Suppose there are m distinct integer values represented in the entire sample (n =1,
2, …, N) and, without loss of generality, suppose the set of integers are {1, 2, 3 …m}.
Let 0 = (0) = 0, 1 = (1) , …, m = (m).
NATIONALØKONOMISK TIDSSKRIFT 2005. NR. 1
The sample log likelihood function can then be expressed as a function of the regression
coefficients and the baseline-hazard parameters = ( 1 , 2 , …, m ). The