  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.
