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Determinantal generalizations of instrumental variables

Weihs, Luca; Robinson, Bill; Dufresne, Emilie; Kenkel, Jennifer; Kubjas, Kaie; McGee, Reginald L. II; Nguyen, Nhan; Robeva, Elina; Drton, Mathias

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Luca Weihs

Bill Robinson

Emilie Dufresne

Jennifer Kenkel

Kaie Kubjas

Reginald L. II McGee

Nhan Nguyen

Elina Robeva

Mathias Drton


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.


Weihs, L., Robinson, B., Dufresne, E., Kenkel, J., Kubjas, K., McGee, R. L. I., …Drton, M. (2018). Determinantal generalizations of instrumental variables. Journal of Causal Inference, 6(1),

Journal Article Type Article
Acceptance Date Nov 8, 2017
Online Publication Date Dec 8, 2017
Publication Date Mar 26, 2018
Deposit Date Nov 10, 2017
Publicly Available Date Dec 9, 2018
Journal Journal of Causal Inference
Print ISSN 2193-3685
Electronic ISSN 2193-3685
Publisher De Gruyter
Peer Reviewed Peer Reviewed
Volume 6
Issue 1
Keywords trek separation; half-trek criterion; structural equation models; identifiability, generic identifiability
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