What is the econometric unordered (alternative-specific) multinomial probit model?

Question

Possibly a very trivial question, however I am somehow unable to find any mathematical definition of this model. From other posts on this website I perfectly understand the difference between an unordered multinomial probit model and an ordered multinomial probit model, however I cannot seem to find any econometric definition of the first. Clearly, whether the model is alternative-specific or not matters little to me in this case; this specification only means that I will have one less assumption, that is, that the error terms do not have to be uncorrelated, and this I know how to write down.

I have seen definitions of the ordered multinomial probit model, one of those is as follows:

Consider a latent random variable $y_n$ for individual $n = 1$, ..., $N$ that linearly depends on $x_n$.

$y^*_n = x^{'}_n\beta + \epsilon_n$ with $\epsilon_i $ a normally distributed random variable with mean 0 and variance $\sigma^2$.

Is the unordered definition at all similar to this? Is there a similar way of writing it?

Answer 1

Let alternative $j$ have indirect utility $v_j$ for $j=1,...,J$ and let $\epsilon =(\epsilon_1,...,\epsilon_J)$ be multivariate normal with covariance $\Omega$ then choice probability is $$Pr(j=\arg \max_h \{v_h + \epsilon_h\})$$

defines choice probability of agent choosing alternative $j$ in a multinomial probit model. See page 111 of Kenneth Trains book Discrete Choice Methods with Simulation here