Chapter 7 Structural causal models
Structural causal models are a form of structural equation model, where we (at least initially) do not make any assumption about parametric forms. For this reason they are referred to as nonparametric structural equation models (NPSEMs) by some authors, and—for reasons discussed below—NPSEMs with independent errors by James M. Robins and Richardson (2010).
7.1 Definition
A collection of variables \(X_V \in \mathcal{X}_V\) obeys a structural causal model with respect to a directed graph \(\mathcal{G}\) if there exist exogenous noise variables \(U_V \in {\cal U}_V\), and structural functions \(f_v : \mathcal{X}_{\mathop{\mathrm{pa}}(v)} \times {\cal U}_v \to \mathcal{X}_v\), such that each \(X_v = f_v(X_{\mathop{\mathrm{pa}}(v)}, U_v)\).
Note that we do not make any mention of \(\mathcal{G}\) being acyclic, so in principle SCMs also allow for cyclic models; however, the existence of a unique solution to the equations is not guaranteed in this case, so we will not consider any cyclic models.
7.2 Cross-world independences
A point of criticism towards SCMs is that they (at least implicitly) make assumptions that are cross-world; that is, they (may) assert—for instance—that \(Y(a) \mathbin{\perp\hspace{-3.2mm}\perp}M(a')\), even though this claim is not testable even in principle by any experiment. Indeed, for most purposes (other than mediation) causal inference does not require such assumptions. We will see this illustrated in Chapter 8.