What can we learn about correlations from multinomial probit estimates?

It is well known that, in a multinomial probit, only the covariance matrix of the location and scale normalized utilities are identified. In this study, we explore the relation between these identifiable parameters and the original elements of the covariance matrix, to find out what can be learnt about the correlations between the stochastic components of the non-normalized utilities. We show that, in certain circumstances, it is possible to obtain information on these behavioural parameters and define appropriate tools for inference. We illustrate the usefulness of our results in applied settings using an example.