Amemiya (1975) Qualitative Response Models

Summary

This article gives an early systematic taxonomy of qualitative response (QR) models — models whose discrete dependent variables have a conditional distribution specified up to finitely many parameters — with special emphasis on the polytomous (multi-response) and multivariate cases that the then-dominant univariate-dichotomous (probit/logit) literature had neglected. Amemiya organizes the univariate dichotomous and trichotomous models, defines several new models (notably multivariate polytomous probit models), reinterprets existing ones, and draws the key methodological contrast for multivariate QR: whether to build the joint model by specifying marginal probabilities first or conditional probabilities first.

Key Claims

• What a QR model is. A model involving one or more discrete random variables whose conditional distribution given the covariates is specified up to a finite parameter vector; the univariate dichotomous probit/logit is the base case, extended here to polytomous and multivariate responses.
• Normal or logistic families. All models considered are of the probit (normal) or logit (logistic) type; realistic behavioral micro-foundations are desirable but often lead to computationally intractable estimation, so tractable normal/logistic transforms are used.
• Trichotomous and polytomous cases. The trichotomous variable already raises the interesting modeling choices; the general polytomous case extends the trichotomous results without new problems, so it is treated by extension.
• Multivariate polytomous probit. New multivariate QR models are defined, in particular multivariate polytomous probit, giving joint models for several correlated discrete responses.
• Marginal-first vs. conditional-first. The central formulation distinction for multivariate QR: one can specify the marginal response probabilities and then a dependence structure, or specify conditional probabilities and build up the joint — two "basically differing" routes with different tractability and interpretation.
• Estimation deferred. Estimation (maximum likelihood, minimum chi-square) is treated only briefly, the general treatment having been given in Amemiya (1974a,b).

Concepts Introduced or Extended

• Multivariate Probit
• Discrete Choice Models
• Multinomial Probit

Entities Mentioned

• Takeshi Amemiya

Quotes

"This article gives a systematic discussion of various qualitative response models, with a special emphasis on multi-response and multivariate models."

"The article discusses the relative merits of two basically differing ways to formulate multivariate models: the one that specifies marginal probabilities first and the one that specifies conditional probabilities first."

My Take

This is a foundational stock-taking of qualitative-response modeling just before the field exploded — the direct ancestor of Amemiya's much-cited 1981 survey and the Advanced Econometrics treatment — and its lasting idea is the marginal-vs-conditional dichotomy for multivariate discrete data, which is exactly the fork the wiki's multivariate probit (joint-latent-normal, a marginal/joint construction) and the conditional/autologistic traditions later took. Reading it now, the honest limitation it flags itself — that behaviorally motivated multivariate models are computationally intractable — is precisely the wall that simulation methods (GHK, MSM/MSL) and MCMC data augmentation were later built to breach, turning Amemiya's "remains to be seen" multivariate polytomous probit into something routinely estimable. For the wiki it anchors the discrete-choice family historically and names the modeling choice that still organizes multivariate categorical modeling.