Local Independence of Irrelevant Alternatives - in Econometrics

In Econometrics

IIA is a property of the multinomial logit and the conditional logit models in econometrics; outcomes that could theoretically violate this IIA (such as the outcome of multicandidate elections or any choice made by humans) may make multinomial logit and conditional logit invalid estimators.

IIA implies that adding another alternative or changing the characteristics of a third alternative does not affect the relative odds between the two alternatives considered. This implication is not realistic for applications with similar alternatives. Many examples have been constructed to illustrate this problem.

Consider the Red Bus/Blue Bus example. Commuters initially face a decision between two modes of transportation: car and red bus. Suppose that a consumer chooses between these two options with equal probability, 0.5, so that the odds ratio equals 1. Now suppose a third mode, blue bus, is added. Assuming bus commuters do not care about the color of the bus, consumers are expected to choose between bus and car still with equal probability, so the probability of car is still 0.5, while the probabilities of each of the two bus types is 0.25. But IIA implies that this is not the case: for the odds ratio between car and red bus to be preserved, the new probabilities must be: car 0.33; red bus 0.33; blue bus 0.33. In intuitive terms, the problem with the IIA axiom is that it leads to a failure to take account of the fact that red bus and blue bus are very similar, and are "perfect substitutes".

Many modeling advances have been motivated by a desire to alleviate the concerns raised by IIA. Generalized extreme value, multinomial probit (also called conditional probit) and mixed logit are alternative models for nominal outcomes which relax IIA, but these models often have assumptions of their own that may be difficult to meet or are computationally infeasible. The multinomial probit model has as a disadvantage that it makes calculation of maximum likelihood infeasible for more than five alternatives as it involves multiple integrals. IIA can also be relaxed by specifying a hierarchical model, ranking the choice alternatives. The most popular of these is called the nested logit model.

Generalized extreme value and multinomial probit models possess another property, the Invariant Proportion of Substitution which suggests similarly counterintuitive individual choice behavior.

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