If we plug the γjs back into the expression for pj/k, we obtain,
_ exp(Yj) exp(β0zjk)
pj/k " PJ=1 exp(β0Zjk + Yj )
_ __________nj__exp(β0Zjt)
PK=1 exp(βzjk + Ik) PJ=1 exP(αj + βzjk)
_ exp(β0Zjk + Ik )
PtT=1 exp(β0zjk + Ik)
and the concentrated log-likelihood is that of a logit model where the choices
are now the sectors with an alternative specific constant added to the model.
This log-likelihood is equivalent to that of a Poisson regression with fixed-
effects (see, for example, Cameron & Trivedi (1998)].
24
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