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Contents 1 Introduction 2 Disadvantages 3 Advantages 4 Proportional Substitution 5 Tests of IIA 5.1 References

Introduction

In econometrics, the independence of irrelevant alternatives (IIA) is a property which implies that the relative odds between two alternatives are the same no matter what other alternatives are available. In discrete choice models, the IIA property holds for certain types of models. Logit model is a good example of this. Consider two alternatives a and b; then, under the logit model, the IIA property tells us that the ratio of their logit probabilities does not depend on any other alternatives other than a and b i.e.

Since the ratio is independent from alternatives other than a and b, it is said to be independent from irrelevant alternatives. In the nested logit model, when two alternatives are in the same nest, the IIA property holds since the ratio of probabilities is independent of the existence of other alternatives. However, if two alternatives are placed in different nests, then the IIA property no longer holds. Also, under the probit model, the IIA property does not hold.

Disadvantages

The IIA property is appropriate in certain choice situations but unrealistic in others. In reality, there are a number of outcomes that violate the IIA property such as (i) predicting the outcome of multicandidate elections or (ii) when the choice made by humans is involved. This was first pointed out by Chipman [1] and Debreu [2]. An example of such a situation is given below:-

Let us assume an individual has the option of (i) riding his bike, the probability of which is P_bike or, (ii) taking a bus, the probability of which is P_bus, and that P_bike = P_bus = 1 / 2. Thus, the ratio of probabilities is 1. Now, if an alternative bus, the probability of which is P_bus2 and is the same as the original bus is presented as an option to the individual, then according to the logit model the introduction of a new alternative does not affect the original outcome. Thus, the ratio is still one and logit predicts the individual outcomes to be P_bus = P_bike = P_bus2 = 1 / 3.

However, in reality the introduction of the new bus implies that the original probability of taking the bus is now split such that P_bus = P_bus2 = 1 / 4. Thus, we see that logit overestimates the probability of taking the buses while underestimating the probability of taking the bike. Other examples are given in [3] and [4].

Advantages

In cases when the IIA property reflects reality as close as possible, numerous advantages can be obtained by its use.

• It is possible to consistently estimate model parameters by using only a subset of alternatives in the decision making process. An example is provided in [5] wherein amongst 100 alternatives, a researcher may choose to estimate parameters using only 10 alternatives for each sampled person. Since, under the IIA property, relative probabilities within this subset are unaffected by the attributes or existence of alternatives outside the subset, the consistency of the estimator is not affected by excluding a majority of the alternatives.

• If a researcher wishes to examine choices amongst only a subset of all the alternatives and not amongst all possible alternatives, then the IIA property is of immense use since it allows the inclusion of only the relevant subset and hence saves the researcher considerable money and time.

Proportional Substitution

The IIA property can also be expressed in terms of the [[cross elasticity] of discrete choice probabilities. The latter has do with the change in probabilities of all alternatives when a given alternative changes. Thus, for probabilistic models that employ the IIA property, an improvement in one alternative draws equally from all other alternatives. Similarly, a descrease in the utility of an alternative will result in an equal increase of the probabilities of all other alternatives. This particular pattern of substitution is termed proportional substitution. [5]

Tests of IIA

Testing of IIA is intended as a way to test if the IIA property holds in a certain situation. Mcfadden et al [6] were the first to develop tests of IIA. The latter is a test of the hypothesis (under the IIA) that the parameter estimates obtained on a subset of alternatives are the same as those on all the alternatives. A second test of the IIA involves confirming that the probabilistic ratio of two alternatives is independent of the existence of other alternatives. [5] offers more information on IIA tests.

References

1. Chipman, J. (1960), 'The foundations of utility', Econometrica 28, 193-224.
2. Debreu, G. (1960), 'Review of R.D. Luce individual choice behavior', American Economic Review 50, 186-188.
3. Ortuzar, J. (1983), 'Nested logit models for mixed-mode travel in urban corridors', Transportation Research A 17, 283-299.
4. Brownstone, D. and K. Small (1999), 'Forecasting new product penetration with flexible substitution patterns', Journal of Econometrics 89, 109-129.
5. K.E. Train, Discrete Chocie Methods with Simulation, 2003, Cambrige University Press.
6. McFadden, D. (1987), 'Regression-based specification tests for the multinomial logit mode', Journal of Econometrics 34, 63-82.