Determinantal generalizations of instrumental variablesTools Weihs, Luca and Robinson, Bill and Dufresne, Emilie and Kenkel, Jennifer and Kubjas, Kaie and McGee, Reginald L. II and Nguyen, Nhan and Robeva, Elina and Drton, Mathias (2017) Determinantal generalizations of instrumental variables. Journal of Causal Inference . ISSN 21933685 (In Press)
AbstractLinear 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 halftrek 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.
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