# Universal Logit Model

In Section 9.2.1 we stated that for a binary QR model a given probability function G(xf, в) can be approximated by F[H(xf, в)] by choosing an appropriate H(xf, в) for a given F. When F is chosen to be the logistic function A, such a model is called a universal logit model. A similar fact holds in the multinomial case as well. Consider the trichotomous case of Example 9.3.2. A universal logit model is defined by

, exp (go)

° 1 + exp (&,) + exp (ga) ’

,_________ exp (g„)

n 1 +exp (gn) + exp (gay

and

Pi0 1 + exp (gn) + exp (ga) ’ (9’3’71)

where gn and ga are functions of all the explanatory variables of the model— z№, zn, za, and w,. Any arbitrary trichotomous model can be approximated by this model by choosing the functions gn and ga appropriately. As long as gn and ga depend on all the mode characteristics, the universal logit model does not satisfy the assumption of the independence from irrelevant alternatives. When the ^s are linear in the explanatory variables with coefficients that generally vary with the alternatives, the model is reduced to a multinomial logit model sometimes used in applications (see Cox, 1966, p. 65), which differs from the one defined in Section 9.3.3.

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