Determinantal Generalizations of Instrumental Variables

TL;DR

This paper extends existing criteria for identifying linear relationships in Gaussian structural equation models by introducing determinantal formulas that generalize instrumental variable methods, improving understanding of model identifiability.

Contribution

It provides new necessary and sufficient conditions for generic identifiability of edge effects, generalizing instrumental variable formulas through determinantal approaches.

Findings

New determinantal formulas for edge coefficient recovery.

Extension of half-trek criterion to individual edges.

Enhanced criteria for generic identifiability in structural models.

Abstract

Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification.