Linear Structural Equations in SEMs

• Linear Structural Equations are mathematical formulations in SEMs that model linear functional relationships among observed variables using mixed graphs and structured matrices.
• The framework utilizes structural coefficient matrices and error covariance representations to capture directed effects and latent confounding in complex data.
• Global identifiability conditions, derived from avoiding specific subgraph structures, ensure unique parameter recovery and reliable statistical inference.

Linear structural equations constitute the foundation of a broad class of multivariate statistical models known as linear structural equation models (SEMs). These models describe noisy linear functional relationships among observed variables, are encoded by mixed graphs (comprising both directed and bidirected edges), and are parametrized via matrices representing structural coefficients and error covariances. Their mathematical, graphical, and inferential properties have been the subject of substantial theoretical development, particularly in the context of identifiability, algorithmic recovery, and algebraic characterization.

1. Formal Definition and Model Parametrization

A linear SEM is built upon a mixed graph $G = (V, D, B)$ where $V$ indexes observed variables, $D$ is the set of directed edges specifying linear effects (e.g., $i \to j$ interpreted as $X_j \coloneqq \lambda_{ij} X_i + \ldots$), and $B$ is the set of bidirected edges $i \leftrightarrow j$ encoding potential correlation between exogenous errors (reflecting, for instance, latent confounding). The model, under Gaussian noise, is parametrized by $(\Lambda, \Omega)$:

• $\Lambda \in \mathbb{R}^D$ is the matrix of structural coefficients, with $\lambda_{ij} = 0$ whenever $i \to j \notin D$.
• $\Omega \in \mathrm{PD}(B)$ is a positive definite error covariance matrix, with off-diagonal entries $\omega_{ij} = 0$ unless $i \leftrightarrow j \in B$.

The induced covariance matrix over $X = (X_1, ..., X_m)$ is

$$\Sigma = (I - \Lambda)^{-T}\, \Omega\,(I - \Lambda)^{-1}$$

and the parametrization map is

$\varphi_G : (\Lambda, \Omega) \mapsto \Sigma.$

The invertibility of $I-\Lambda$ is ensured by requiring acyclicity in the directed part of $G$. The explicit rational form for $\Sigma$ is central to both identification theory and statistical estimation.

2. Global Identifiability: Injectivity and Graphical Characterization

Global identifiability is defined as the injectivity of the parametrization map $\varphi_G$, i.e., the property that $$\varphi_G(\Lambda, \Omega) = \varphi_G(\Lambda', \Omega')$$ implies $(\Lambda, \Omega) = (\Lambda', \Omega')$. This property is essential for valid statistical inference: when injectivity fails, parameter estimates may not be unique and the likelihood surface may possess singularities, resulting in nonstandard asymptotic properties (e.g., likelihood ratio test statistics deviating from $\chi^2$ behavior).

The main graphical result is a necessary and sufficient condition: $\varphi_G$ for an acyclic mixed graph $G$ fails to be injective if and only if $G$ contains an induced subgraph $G_A = (A, D_A, B_A)$ (with $A \subset V$) such that

• its directed part $(A, D_A)$ contains a converging arborescence (i.e., a directed tree spanning $A$ and converging at some node $i$),
• and the bidirected part $(A, B_A)$ is connected.

This dual condition (combining directed flow convergence and error correlation connectedness) pinpoints the precise structural situations in which non-identifiability arises.

3. Statistical Implications and Pathological Scenarios

When the global identifiability criterion is violated, inference procedures—such as maximum likelihood estimation and the assessment of significance via likelihood ratio tests—become unreliable. Simulations (see Figure 1 and 2 in (Drton et al., 2010)) demonstrate that in the non-injective case, the distribution of $p$-values under the null hypothesis is nonuniform (e.g., overly conservative), reflecting an irregular model geometry. For example, in a model where $\lambda_{12} = 3$, $\lambda_{23} = -1/2$, $\lambda_{34} = \lambda_{45} = 1$ and specific off-diagonal error covariances, the map $\varphi_G$ is not injective and the resulting hypothesis test does not behave as expected. Modifying the parameters (e.g., $\lambda_{23} = +1/2$) can restore injectivity and standard asymptotics.

4. Algebraic and Combinatorial Structure

The image of $\varphi_G$ is an algebraic variety, whose structure is shaped by the combinatorics of $G$. For acyclic graphs, inversion of $\varphi_G$ can be achieved via recursive (block-LDL) decomposition; for general graphs (especially with cycles), $\Sigma$'s entries remain rational functions of the parameters. The algebraic viewpoint reveals that $\Sigma$ satisfies polynomial constraints—many of which correspond to conditional independence and more intricate (non-determinantal) relations. These constraints are critical in model testing and structure learning.

A related combinatorial result is the relationship between treks (paths without colliding arrowheads) and matrix entries: each $\sigma_{ij}$ decomposes as a sum over treks between $i$ and $j$, with each trek monomial defined by products of $\lambda$'s and $\omega$'s. This "trek rule" facilitates both explicit parameter recovery and the computation of determinantal (rank) constraints on submatrices of $\Sigma$.

5. Practical and Methodological Considerations

Ensuring global identifiability at the model specification stage is paramount for reliable inference. The absence of the problematic induced subgraphs described above is a necessary and sufficient graphical criterion. In practical terms, the diagrammatic structure of the SEM must avoid any set of nodes for which the directed structure supports a converging arborescence and the error terms are fully connected via bidirected edges.

Applied researchers are thereby advised to inspect their mixed graph models for such substructures. Failure to do so risks inference anomalies, including nonuniqueness of parameter solutions, degeneracy of the likelihood function, and invalid asymptotics for classical test statistics.

6. Summary Table: Identifiability Condition

Graph Structural Feature	Identifiability Consequence	Inferential Implication
No induced subgraph with both converging	Global identifiability (injective)	Unique parameters; regular likelihood
arborescence and connected bidirected part		correct χ² inference
Some induced subgraph with both	Not globally identifiable	Nonunique parameters; irregular model
		Possibly nonstandard $p$-values

7. Conclusion

Linear structural equations, as formalized by mixed graph models and parametrized via explicit matrix functions, are globally identifiable if and only if their underlying graphs avoid specific substructures that couple directed arborescences and error covariance connectivity. This global identifiability is requisite for classical statistical inference, and its graphical characterization affords an effective and essential diagnostic for SEM construction and application. The interplay of linear algebra, combinatorics, and statistical theory thereby underpins both the power and the delicacy of inference in linear SEM frameworks (Drton et al., 2010).