Logit-Poisson Trick Consider an analysis of discrete-choice logit,
where the independent variables vary with the
alternatives. Suppose that each individual is
represented once in the sample, and that different
individuals may have different choice sets
(eg individual 1 may face alternatives 1,2,4, while
individual 2 may face alternatives 3 and 4 only, etc).
Question: does the Poisson Trick work for this
case? There seems to be no problem when everyone
faces the same choice set, but the equivalence
between logit (I'm using R's mlogit for this case)
and the glm/Poisson analysis seems to break down
(empirically) when different choice sets are involved.
Can anyone settle the question, one way or the
other? If a demonstration could also be
provided, I'd be very grateful.

Actually the question was very clear. And he found the correct answer on his own. There is a very large literature on the Poisson Trick as there is for the Frisch-Waugh-Lovell Trick or the kernel Trick. The problem is that various literatures use different nomenclature and many eschew the use of the term "trick". The earliest variant of the trick, but not using the term, that I've seen is Baker, S. G. (1994), ‘The multinomial-poisson transformation’, Journal of the Royal Statistical Society: Series D 43(4), 495–504. But it appears all over now in business (Matt Taddy's Book on ML in business) biostats and so on. The question was in a discrete choice context where the expected utilities are represented by the same linear in parameters function: that is a conditional logit, any of us working in the area recognize the question and contest immediately.
