Completeness of identification methods for hierarchical causal models

Establish a complete identification procedure for hierarchical causal models (HCMs) that guarantees: (i) every causal effect the procedure identifies is identifiable, and (ii) any effect the procedure fails to identify is in fact non‑identifiable. The procedure should overcome limitations introduced by collapsing HCMs to flat models with deterministic constraints (for example, in the instrument graph where the unit‑level outcome Y_i depends on Q^{a|z}_i and Q^z_i only through their induced marginal), which prevent direct application of the standard completeness of do‑calculus.

Background

The paper proposes hierarchical causal models (HCMs) and an identification strategy based on collapsing the hierarchical structure to a flat model, augmenting and marginalizing variables, and then applying do‑calculus. While do‑calculus is known to be complete for flat causal graphical models, the collapsed representation of an HCM can impose deterministic relationships among variables (e.g., outcomes depending on parents only via induced marginals). These constraints mean the collapsed model is not fully nonparametric even if the original HCM is, which can break the direct applicability of completeness results.

Consequently, although the presented approach successfully identifies many effects in HCMs, the authors highlight the absence of a general, complete identification theory tailored to hierarchical settings. A complete method would adjudicate identifiability for any effect in an HCM without being limited by artifacts of the collapse that are irrelevant to the original hierarchical structure.