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

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.

Citation

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), https://doi.org/10.1515/jci-2017-0009

• 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
• DOI: https://doi.org/10.1515/jci-2017-0009
• Keywords: trek separation; half-trek criterion; structural equation models; identifiability, generic identifiability
• Public URL: https://nottingham-repository.worktribe.com/output/921699
• Publisher URL: https://www.degruyter.com/view/j/jci.ahead-of-print/jci-2017-0009/jci-2017-0009.xml
• Contract Date: Nov 10, 2017