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I am trying to compute predicted probabilities from a multinomial probit model. Researching the internet has given me two ways how this can be done.

The first one is basically just applying the probit link for every group independently, as seen here. This probably requires some normalizing of the predicted group probabilities.

However, I also found this slide set where it says on slide 5 that the probabilities in a multinomial probit framework have to be simulated by numerical integration as there is no closed form solution available.

I appreciate any clarification on which solution is correct in which setting.

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That's right, you would need to simulate the probabilities. When you estimate the MNP model, you already use simulation procedures (More or less sophisticated depending on the size of the integral). In Kenneth Train's book [Discrete choice methods with simulation] there is a chapter on simulation methods and post-estimation procedures (such as obtaining predicted probabilities given a sequence of observed choices).

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