Identification and Overidentification of Linear Structural Equation Models
In this paper, we address the problems of identifying linear structural equation models and discovering the constraints they imply. We first extend the half-trek criterion to cover a broader class of models and apply our extension to finding testable constraints implied by the model. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification and constraint discovery power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification and constraint discovery algorithms for non-parametric models.
This paper establishes a generalization of the half-trek criterion (HTC) for identifying coefficients in linear structural equation models (LSEMs). The generalized criterion (g-HTC) is designed to identify the coefficients associated with a subset of edges into a variable when the set of all edges into that variable is not identified by the original HTC. It turns out, moreover, that there are models that are identifiable by the g-HTC but not by the HTC. In connection to the g-HTC, this paper also proposes a way to derive constraints implied by a linear structural equation model. Furthermore, it is shown that when combined with decomposition of semi-Markovian models into sub-models over its c-components and marginalization of descendant sets, the proposed method of identification can identify every coefficient in a semi-Markovian LSEM that is identifiable according to non-parametric criteria (such as the do-calculus), and the method of deriving constraints subsumes some non-parametric methods in the literature. 
This is a solid contribution to the literature on the identification of linear models. The paper is well organized and for the most part well written. Due to limited space, the technical details (even including the supplementary material) are unfortunately not fully self-contained and not easy to follow for non-experts. It would have been more readable if more background (such as the original HTC and more information on the proof of the theorem on HTC-identifiability) had been included in the supplementary material. Still, most proofs look credible (though I did not have time to check all of them carefully; sorry about that.) I am puzzled, however, by the last part of the proof of Theorem 4; I would appreciate some elaboration of how HT constraints express the conditional independence constraints in question.