Maximum number of alternatives in a discrete choice model We are modeling a discrete choice scenario, with alternative-specific coefficients. We also break the assumption of independence of irrelevant alternatives. To model this, we are using an alternative-specific multinomial probit regression. This is implemented in Stata as asmprobit. However, as described in the documentation, there is a limit on the number of alternatives: 20! We have up to 120 alternatives per case! Uh oh.
Why is there this limit? Is there anything that can be done to increase it? Is this model (or a good alternative) implemented in R or elsewhere, presumably without this limit?
 A: The main issue with asmprobit is the flexibility it provides which relaxes the independence of irrelevant alternatives (IIA) assumption but it comes at the cost of increased computing power. In this sense you allow the odds of choosing one alternative over some other alternative to depend on the remaining alternative, though this involves evaluation of probabilities from the multivariate normal distribution. Since there is no closed form solution to those you have to rely on simulation techniques. That's the bottle neck.
The simulation method used by asmprobit in order to solve the simulated maximum likelihood is the Geweke-Hajivassiliou–Kean multivariate normal simulator (GHK documentation) which allows only for dimension $m\leq 20$. That's where the restriction in asmprobit comes from because for more alternatives the simulation time becomes unmanageable. For a detailed description of this you can also see the "Simulated Likelihood" part in the Methods and Formulas section of the asmprobit documentation.
Given that the reason for the limit is a computational one rather than one that is concerned with implementation I would not be too hopeful for a better estimation routine in R. If there was one then probably also Stata would have implemented it by now. By the way, this restriction is also a problem for other probit models of discrete choice (e.g. mprobit allows max 30 distinct choices) for the same reason as outlined above.
A useful reference for you should be


*

*Train, Kenneth E. 2007. Discrete Choice Models with Simulation. New York: Cambridge University Press.


which is probably the main reference on this topic. If I remember correctly he also discusses cases where the number of choices is very large. I'm not sure, however, if you can have the best of both worlds, i.e. relaxing IIA and allowing for many alternatives. Certainly probit models will not get you far because of the multivariate normal but perhaps other models that also relax the IIA assumption may be useful. For instance, the mixed logit model also relaxes this assumption so it might be worth to have a look at the Stata options for estimating these kinds of models (see for instance this presentation for an overview).
