For a multinomial response, let uij denote the utility of response outcome j for subject i. Suppose that uij = xi????j + ????ij, and the response outcome for subject i is the value of j having maximum utility. When {????ij} are assumed to be iid standard normal, this model is the simplest form of the multinomial probit model (Aitchison and Bennett 1970).

a. Explain why (????ak − ????bk) describes the effect of a 1-unit increase in explanatory variable k on the difference in mean utilities, as measured in terms of the number of standard deviations of the utility distribution.

b. For observation i, explain why the probability of outcome in category j is

????ij = ∫ ????(uij − xi????j)

∏

k≠j

Φ(uij − xi????k)duij, for the standard normal pdf ???? and cdf Φ. Explain how to form the likelihood function.