IIA is a statistical, not counterfactual, assumption
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I again found myself pondering the meaning of the independence of irrelevant alternatives (IIA) assumption for multinomial logit models. The IIA assumption is often said to imply that given two options in the outcome, removing a third option would not change the relative probability of the first two options. This implication reads very counterfactual: under IIA, counterfactually designing a voting regime or a survey questionnaire excluding an option shouldn’t change the relative probability.
But how can multinomial logit, as a statistical model (i.e., a parametric functional form), command such a counterfactual structure? Well, it really doesn’t. The IIA assumption is NOT a counterfactual assumption.
Suppose we observe the choice set $Y \in {1,\ldots,J}$. The multinomial logit model assumes that
\[Pr(Y=j \mid X)=\frac{\exp(X\beta_j)}{\sum_{i=1}^J \exp(X\beta_i)}.\]For any two alternatives $j$ and $k$, and a third alternative $l$, this functional form assumption implies
\[\frac{Pr(Y=j \mid X)}{Pr(Y=k \mid X)}=\frac{Pr(Y=j \mid X, Y \neq l)}{Pr(Y=k \mid X, Y \neq l)}=\exp(X\beta_j - X\beta_k).\]This means that if we statistically condition on the outcome not being $l$, the relative probability of choosing $j$ over $k$ is the same as its unconditional counterpart (this assumed equality underlies Hausman and McFadden’s (1984) test). This is what the IIA assumption gives us. It says nothing about what would happen to the relative probability if we counterfactually changed the choice set. Consider a familiar analogue: it is one thing to assume $E(Y|T=1)=E(Y)$, it is another to assume that counterfactually assigning $T=1$ wouldn’t change the mean.
The standard counterfactual interpretation of the IIA assumption seems pretty misleading.
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