A condition for the identification of multivariate models with binary instruments -- with Corrigendum and Addendum

2026-07-01

AI Summary

This study addresses nonparametric point identification in multivariate instrumental variable models with continuous endogenous variables when only binary instruments are available. The authors generalize the rank invariance assumption from univariate settings to cyclic monotonicity in the first stage and construct multivariate ranks via the inverse Brenier map, thereby overcoming limitations of conventional approaches that rely on restrictive heterogeneity structures or large instrument support. This framework substantially broadens the class of identifiable distributions, accommodating non-quasiconcave and multimodal densities. Moreover, the paper establishes a verifiable nonparametric identification condition that holds generically across common parametric distribution families and is empirically testable under mild non-degeneracy assumptions.

Abstract

This article introduces an empirical condition for the nonparametric point-identification of multivariate instrumental variable models with continuous endogenous variables using binary instruments. Verifying this condition can confirm point-identification in settings in which traditional approaches are not applicable. In particular, it shows that nonlinear instrumental variable models with general heterogeneity can be point-identified with only a binary instrument. This generalizes existing identification results which either restrict the unobserved heterogeneity substantially or require the instrument to have a large support. The main assumption on the instrumental variable model is cyclic monotonicity of its first stage, a multivariate generalization of the classical rank-invariance assumption for univariate models. Asymptotic convergence results for the empirical observable distributions are derived that allow to check the condition in practice. The identification rests on a fixed-set convergence result of cyclically monotone maps between quasi-concave functions. The corrigendum corrects the proof of Lemma 1. The proof given there incorrectly identifies preservation of distributional level sets with preservation of the underlying probability measure via Brenier maps. We replace that argument by one based on inverse Brenier maps, which play the role of multivariate ranks. The corrected argument applies to a different but significantly more flexible class of distributions than the quasi-concave class considered in the original paper. In particular, it allows for smooth non-quasi-concave and multimodal densities on compact supports, provided the associated rank fixed set satisfies a nondegeneracy condition. Moreover, it is generically satisfied for smooth parmetric classes of distributions.

Problem

• instrumental variable
• point-identification
• binary instrument
• multivariate model
• nonparametric identification

Innovation

• binary instruments
• point identification
• cyclic monotonicity
• nonparametric IV models
• inverse Brenier maps