Survey Statistics: exploded logit !

Two weeks ago we modeled vote choice with candidates C = {Left, Right, Other} as a multinomial logit: P[voter i chooses candidate c from C] = exp(f(X_ic)) / sum_c’ exp(f(X_ic’)) We saw this model implies independence from irrelevant alternatives (IIA): Another consequence of the multinomial logit model is a simple expression for ranked data: P[i ranks Other then Left then Right] = exp(f(X_iOther)) / sum_c’ exp(f(X_ic’)) * exp(f(X_iLeft)) / (exp(f(X_iLeft)) + exp(f(X_iRight))) Train (2009) Chapter 7 calls this an exploded logit. To derive the exploded logit: Train (2009) Chapter 3 explains that the multinomial logit model is equivalent to latent utilities with a Gumbel distribution. Powell (2023)* notes “The exponentials of the negated Gumbel random variables are Exponential random variables” and uses the memoryless property of the Exponential to derive the exploded logit. The exploded logit form implies that the ranking of 3 alternatives can be expressed as 2 pseudo-observations: 1)…