The approach I was considering was Stata’s clogit (page 15 on methods for estimation). Any thoughts on that? The package already has variable absorption so high dimensional features are not an issue. However, I will need to check how code up the gradient of the conditional log-likelihood (“The derivatives of the conditional log-likelihood can also be computed recursively by taking derivatives of the recursive formula f”, basically homework). If it is in any of the textbooks like (Stata cites, Greene 2018, chap. 18) it might be easier to find it.
Same for McFadden’s.