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Local Identification in Instrumental Variable Multivariate Quantile Regression Models

Haruki Kono

TL;DR

This paper extends the instrumental variable quantile regression (IVQR) framework to multivariate outcomes by introducing a nonlinear IV model with a vector-valued rank U_d and structural maps Y_d = q_d^*(U_d, X) whose derivative is symmetric and positive definite. It shows that when D and Z are discrete, the structural functions are locally identified under a sufficiently strong positive correlation between the treatment and the instrument, even with a binary instrument. The approach leverages optimal transport theory to interpret multivariate quantiles and yields a measure-valued identification condition that links observable distributions to the latent rank structure. The result enables joint analysis of several correlated outcomes under endogeneity, capturing substitutability and complementarity across outcomes while preserving a link to the univariate IVQR intuition.

Abstract

In the instrumental variable quantile regression (IVQR) model of Chernozhukov and Hansen (2005), a one-dimensional unobserved rank variable monotonically determines a single potential outcome. In practice, when researchers are interested in multiple outcomes, it is common to estimate separate IVQR models for each of them. This approach implicitly assumes that the rank variable in each regression affects only its associated outcome, without influencing others. In reality, however, outcomes are often jointly determined by multiple latent factors, inducing structural correlations across equations. To address this limitation, we propose a nonlinear instrumental variable model that accommodates multivariate unobserved heterogeneity, where each component of the latent vector acts as a rank variable corresponding to an observed outcome. When both the treatment and the instrument are discrete, we show that the structural function in our model is locally identified under a sufficiently strong positive correlation between the treatment and the instrument.

Local Identification in Instrumental Variable Multivariate Quantile Regression Models

TL;DR

This paper extends the instrumental variable quantile regression (IVQR) framework to multivariate outcomes by introducing a nonlinear IV model with a vector-valued rank U_d and structural maps Y_d = q_d^*(U_d, X) whose derivative is symmetric and positive definite. It shows that when D and Z are discrete, the structural functions are locally identified under a sufficiently strong positive correlation between the treatment and the instrument, even with a binary instrument. The approach leverages optimal transport theory to interpret multivariate quantiles and yields a measure-valued identification condition that links observable distributions to the latent rank structure. The result enables joint analysis of several correlated outcomes under endogeneity, capturing substitutability and complementarity across outcomes while preserving a link to the univariate IVQR intuition.

Abstract

In the instrumental variable quantile regression (IVQR) model of Chernozhukov and Hansen (2005), a one-dimensional unobserved rank variable monotonically determines a single potential outcome. In practice, when researchers are interested in multiple outcomes, it is common to estimate separate IVQR models for each of them. This approach implicitly assumes that the rank variable in each regression affects only its associated outcome, without influencing others. In reality, however, outcomes are often jointly determined by multiple latent factors, inducing structural correlations across equations. To address this limitation, we propose a nonlinear instrumental variable model that accommodates multivariate unobserved heterogeneity, where each component of the latent vector acts as a rank variable corresponding to an observed outcome. When both the treatment and the instrument are discrete, we show that the structural function in our model is locally identified under a sufficiently strong positive correlation between the treatment and the instrument.
Paper Structure (25 sections, 18 theorems, 81 equations)

This paper contains 25 sections, 18 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. IV multivariate quantile regression model
  3. Model
  4. Examples
  5. Comparison with chernozhukov2005iv
  6. Identification
  7. Outline of the proof of Theorem \ref{['thm:local-identification-IVQR-binary']}
  8. Concluding Remarks
  9. Preliminaries
  10. Local identification for system of equations
  11. Setup and theorem
  12. Proof of Theorem \ref{['thm:local-identification-general']}
  13. Proof of Theorem \ref{['thm:local-identification-IVQR-binary']}
  14. Generalization of Theorem \ref{['thm:local-identification-IVQR-binary']}
  15. Omitted proofs
  16. ...and 10 more sections

Key Result

Theorem 2.1

Suppose Assumption ass:ivqr holds. Then, it holds with probability one that for each measurable $B \subset \cU,$ where $q_D^\ast (B, X) = \{q_D^\ast (u, X) \mid u \in B\}.$ In particular, $U \mid X, Z \sim \mu$ holds.

Theorems & Definitions (38)

  • Theorem 2.1
  • Definition 3.1
  • Theorem 3.1
  • Proposition 3.1
  • Theorem A.1
  • Lemma A.1
  • Theorem A.2
  • Theorem B.1
  • Lemma B.1
  • proof : Proof of Theorem \ref{['thm:local-identification-general']}
  • ...and 28 more