Dynamical Structural Equation Models (DSEMs)

• Dynamical Structural Equation Models (DSEMs) are frameworks that extend classical SEMs by incorporating time-dependent causal relationships and dynamic feedback loops.
• They utilize discrete, continuous, and stochastic formulations to model interventions, estimation, and causal inference across multivariate time series and network data.
• Applications span social network analysis, clinical psychology, and econometrics, with methods like quasi-likelihood estimation and sheaf theory enhancing robust, real-time prediction.

Dynamical Structural Equation Models (DSEMs) generalize classical Structural Equation Models by introducing explicit time-dependence into the modeling of causal relationships, thereby capturing the evolution, interventions, and stochasticity of dynamical systems. DSEMs are rigorously formalized across discrete time, continuous time (via ODEs, SDEs), and hybrid combinatorial-topological frameworks, supporting both deterministic and stochastic processes, multivariate time series, and feedback systems. This entry synthesizes core mathematical constructions, equivalence classes, model checking, estimation, and applications, as established across the literature (Gladyshev et al., 17 Jan 2025, Rubenstein et al., 2016, Peters et al., 2021, Kusano et al., 2023, Kusano et al., 2022, Valentini et al., 2013, Baingana et al., 2013, Zaman et al., 2020, Faleh et al., 18 Aug 2025, Robinson et al., 6 Nov 2025, Bongers et al., 2018, Boeken et al., 3 Jun 2024).

1. Mathematical Formulation of DSEMs

DSEMs are structurally richer than static SEMs due to their variable indexing over time and their recurrent (difference/differential) nature. In the discrete-time formulation (Gladyshev et al., 17 Jan 2025), a DSEM is defined as a tuple $(U, V, R, F)$ with exogenous processes $U_j(t)$ and endogenous processes $X_k(t)$, each evolving with the update rule: $X_k(t) = F_{X_k}(U(t-1), X(t-1))$ or, more generally,

$$X_k(t) = f_k\bigl(X_1(t-1), \dots, X_n(t-1), U_1(t-1), \dots, U_m(t-1)\bigr)$$

Thus, each $X_k$ is generated by a structural equation applied recursively, mapping past exogenous and endogenous variables to current values.

Continuous‐time DSEMs appear as generalized SEMs (GSEMs), where variables index continuous time, e.g., $X_i^t$, and the model outcome is a function space assignment $v:V \to \mathbb{R}$ consistent with ODE/SDE laws except where "do"-style interventions fix trajectories (Peters et al., 2021, Rubenstein et al., 2016).

Stochastic DSEMs (also SDCMs, DSCMs) (Bongers et al., 2018, Boeken et al., 3 Jun 2024) expand the setting:

• Endogenous: $X_i$ are stochastic processes (not just random variables).
• Structural equations may include derivatives (ODE/RDE/SDE), e.g., $$dX_i(t) = g_i(\bar X^{(n-1)}(t), X_{-i}(t), E(t)) dt$$.
• Graphical representation includes temporal and instantaneous edges and Markov properties via variants of $\sigma$-separation.

Key prerequisites for well-defined solutions include existence and uniqueness theorems (e.g., Picard-Lindelöf for ODEs/RDEs) and dynamic stability assumptions. Nonlinear and time-varying feedback is naturally accommodated—cyclic dependencies become temporally mediated feedback loops rather than algebraic cycles (Gladyshev et al., 17 Jan 2025, Bongers et al., 2018).

2. Causal Semantics, Interventions, and Temporal Logic

DSEMs support a temporal and counterfactual interpretation of causality analogous to the Halpern-Pearl/SCM framework but over time-indexed variables or trajectories (Gladyshev et al., 17 Jan 2025, Peters et al., 2021).

Discrete-time interventions override variables at specified times: an intervention $[Y_1(n_1),...,Y_k(n_k) \leftarrow y_1,...,y_k]$ replaces the ordinary temporal update for $Y_j$ at $n_j$.

Continuous-time do-operators (Peters et al., 2021, Bongers et al., 2018) act on trajectories or intervals (e.g., $X_i((a, b)) \gets k$), resulting in solution paths where the targeted variables follow prescribed values over designated time segments.

The Temporal Causal Past-and-Future LTL (CPLTL) framework couples interventions with rich temporal logic (Gladyshev et al., 17 Jan 2025). CPLTL supports formulas such as $[T(0)\gets1]\, \bigcirc^2\,(R = 1)$ ("if at $t = 0$ treatment is given, then at $t = 2$ the patient recovers"), with full use of LTL's temporal operators (next, until, previous, since).

Model equivalence is formalized via "temporal equivalence" (matching observable variable traces under all interventions and exogenous contexts) and "rescalable equivalence" (aligning models operating at different temporal resolutions via k-to-1 mapping) (Gladyshev et al., 17 Jan 2025).

3. Estimation, Identification, and Goodness-of-fit in DSEMs

Parameter estimation in DSEMs adheres to the structure of their time domain and data generation process. For high-frequency data and diffusion processes, quasi-likelihood methods are employed (Kusano et al., 2022, Kusano et al., 2023). Observed increments $∆X_i$ are modeled as (approximately) Gaussian with covariance $h_n\,Σ(θ)$, leading to: $$\theta_n = \arg\min_\theta \log\det Σ(θ) - \log\det Q_{XX} + \operatorname{tr}\left[Σ(θ)^{-1} Q_{XX}\right] - p$$ where $Q_{XX}$ is the realized sample covariance. Asymptotic normality and consistency results hold under regularity conditions.

Sparse estimation is achieved with adaptive $L_1$ penalties. Penalized Least Squares Approximation (PLSA) achieves both variable selection and oracle properties (recovering true sparsity in the high-frequency limit) (Kusano et al., 2023).

Goodness-of-fit is assessed using the quasi-likelihood ratio test, with test statistics converging to appropriate $\chi^2$-distributions under correct specification (Kusano et al., 2022, Kusano et al., 2023).

Measurement and structural equations admit latent variable models (e.g., continuous-time LISREL/SD-SEM) with block covariance structure, and the estimation pipeline leverages either Bayesian or frequentist algorithms depending on context (Faleh et al., 18 Aug 2025, Valentini et al., 2013).

4. Feedback, Cyclicality, and Dynamic Topologies

A crucial property of DSEMs is robust handling of recursion and feedback. Classical acyclicity requirements of SEMs are replaced by temporal mediation in DSEMs: cycles in the static graph translate to feedback loops operating across time steps, which no longer prevent well-defined computations (Gladyshev et al., 17 Jan 2025).

Stochastic or time-varying edge weights are modeled explicitly in time-varying network DSEMs for applications such as social network topology inference (Baingana et al., 2013, Zaman et al., 2020). Dynamic regression and online algorithms, including proximal gradient and FISTA, allow real-time tracking of sparse, evolving adjacency matrices, with dynamic regret analysis providing performance guarantees (Zaman et al., 2020).

DSEMs are further generalized via sheaf theory (Robinson et al., 6 Nov 2025), where the DSEM's directed temporal graph becomes a netlist, translated into a sheaf of dynamical systems whose global sections correspond to system solutions. This topological approach supports consistency testing, missing data imputation, uncertainty quantification, and subsystem extraction.

5. Graphical and Topological Representation

DSEM dynamics induce rich graphical models:

• Time-unfolded graphs with nodes $(x_j, t_k)$ and path coefficients $\gamma_{j_1, k_1; j_2, k_2}$.
• Sheaf-theoretic models where variables and update functions are "stalks" over a topological base defined by causal and computational relationships (Robinson et al., 6 Nov 2025).
• Mixed graphs (nodes for processes/intervals, directed and bidirected edges) supporting $\sigma$-separation, Markov properties, and causal discovery—both for finite and infinite systems (Boeken et al., 3 Jun 2024, Bongers et al., 2018).

Spatial DSEMs (SD-SEMs) integrate spatio-temporal dependencies and cointegration, modeling high-dimensional lattices (e.g., statewise housing prices) with spatially structured factor loadings and vector ARDL latent-factor time series (Valentini et al., 2013).

6. Inference, Prediction, and Validation

DSEMs allow for rigorous inference, counterfactual analysis, and predictive validation:

• Causal-effect identification carries over from static SCMs, with the do-calculus and ID-related algorithms applicable under appropriate Markov and structural assumptions (Boeken et al., 3 Jun 2024).
• Efficient polynomial-time algorithms are available for model checking in finite DSEMs with ultimately periodic exogenous contexts, despite the use of full past-and-future LTL logic (Gladyshev et al., 17 Jan 2025).
• Bayesian estimation pipelines are available for multilevel and latent-class DSEMs, supporting missing data, random effects, and measurement invariance testing (Faleh et al., 18 Aug 2025).

Practical implementations demonstrate sublinear dynamic regret and accurate tracking in simulated and real-world network inference. Applications are illustrated in fields ranging from social diffusion, clinical psychology (intensive longitudinal data), spatio-temporal econometrics (housing prices), to composite ecological food webs.

---

Domain	DSEM Framework	Key Features / Results
Discrete time	Recurrence/unfolded	Arbitrary feedback; efficient model-checking; CPLTL counterfactuals
Continuous time	ODE/SDE-based GSEMs	Interventions on intervals; HP actual causality extends verbatim
Stochastic	SDCM/DSCM	Markov property via $\sigma$-separation; Granger local independence
High-frequency	Quasi-likelihood SEM	Asymptotic normality; sparse adaptive estimation; $\chi^2$ goodness-of-fit
Network topology	Online DSEM	Proximal/fista updates; dynamic regret bounds; sparse recovery
Topological	Sheaf/netlist DSEM	Consistency radius; subsystem inference; imputation, uncertainty

7. Outlook and Extensions

Recent work emphasizes several frontiers:

• Integration with modern causal discovery algorithms in the constraint-based paradigm, exploiting $\sigma$-separation and subsampling operations (Boeken et al., 3 Jun 2024).
• Extension to infinite-dimensional and hybrid systems, with topological data analysis bridging DSEM representations and systems biology/ecology (Robinson et al., 6 Nov 2025).
• Model classes incorporating spatial, multilevel, or latent class heterogeneity (Faleh et al., 18 Aug 2025, Valentini et al., 2013).
• Algorithmic generalizations to nonlinear, non-Gaussian, or delay-differential DSEMs, and time-inhomogeneous settings (Zaman et al., 2020).

Contemporary DSEM models provide a unified, flexible framework for representing, interrogating, and predicting highly structured temporal causal systems, subsuming both static models and classical time-series approaches.