Identification by Edge Contraction in Linear Structural Equation Models

We extend graph-based identification methods for linear models by allowing background knowledge in the form of externally evaluated parameters. Such information could be obtained, for example, from a previously conducted randomized experiment, from substantive understanding of the domain, or even from another identification technique. To manage such information systematically, we propose an operation called edge-contraction (EC), which transforms the original model into a new one where the value of the known coefficient would be equal to zero. The transformed graph can then admit conventional methods of identification (e.g., single-door criterion, instrumental variables, half-trek criterion) and model testing (e.g., d-separation, over-identification). By iteratively alternating steps of identification and contraction, the EC-operation can improve the power of existing identification methods, even without additional knowledge. We operationalize this general approach for instrumental sets (a generalization of instrumental variables) and show that the resulting method subsumes the most general identification method for linear systems known to date. We further discuss the application of the EC-operation in the tasks of model testing and z-identification.