Discrete-Time Structural Causal Process

• DSCP is a mathematical framework for modeling causal relationships in discrete-time multivariate time series using structural equations and interventions.
• It generalizes VAR and nonparametric models by supporting both hard and soft interventions, enabling estimation of interventional effects.
• The framework is applied for causal forecasting, counterfactual analysis, and spectral identification, enhancing simulation and inference methods.

A Discrete-Time Structural Causal Process (DSCP) is a mathematical and algorithmic framework for representing, reasoning about, and estimating the causal dynamics of multivariate time series within discrete time. The DSCP perspective generalizes standard time-series models by endowing them with structural equations—thereby enabling the formulation and identification of interventional (do-) effects, equilibrium mappings, and counterfactuals. DSCPs admit both parametric (e.g., VAR, SVAR) and nonparametric (neural or template-based) instantiations, support a rich class of interventions, and connect naturally to both time-unrolled graphical models and static SCMs in the equilibrium regime. Recent research synthesizes the DSCP framework with developments in causal discovery, spectral identification, and path-dependent SEMs, thereby providing a comprehensive toolkit for causal inference over time series (Cinquini et al., 14 Oct 2024).

1. Mathematical Formulation of DSCPs

A DSCP specifies the time-evolution of a set of random vectors $\{X_t \in \mathbb{R}^n:\ t=0,1,2,\dots\}$ by a system of stochastic difference equations (SDEs) or structural assignments. The general form is

$X_t = f(X_{t-1}, X_{t-2}, \ldots, X_{t-p}) + U_t,$

where $f$ is a (possibly nonlinear) vector-valued function, $p$ is the finite memory or lag order, and $U_t$ is a white-noise random vector (i.i.d. across $t$), often assumed exogenous and mutually independent across coordinates and time (Cinquini et al., 14 Oct 2024, Li et al., 14 Nov 2025). Each coordinate $i = 1,\ldots,n$ can be written as

$$X_t^{(i)} := f_i\big(X_{t-1}^{(\text{pa}_i)}, U_t^{(i)}\big),$$

with $\text{pa}_i$ denoting the set of parents for coordinate $i$. This induces a causal (possibly cyclic) graphical structure in the time-unrolled graph, where edges $(j \to i)$ indicate $X_{t-1}^{(j)}$ directly appears in $f_i$.

The four-tuple $(V, T, E, F)$ formalizes a general DSCP (Li et al., 14 Nov 2025):

• $V$ = set of variable indices,
• $T$ = set of discrete time points,
• $E$ = set of directed edges, decomposed by lag $\ell$ as $E = \bigcup_{\ell=0}^L E^{(\ell)}$,
• $F$ = collection $\{f_v\}$ of node-local structural equations.

Thus, the DSCP generalizes both standard time-series models (e.g., VAR/SVAR) and nonparametric SCMs to multivariate, lag-indexed, and potentially nonlinear structural systems.

2. Interventions and Causal Effects

DSCPs are uniquely amenable to the formalization of interventions over time. A hard (do-) intervention at time $t_0$ on a subset $I \subseteq \{1,\ldots,n\}$ replaces the original structural assignments for coordinates $i \in I$ with constants: $X_t^{(i)} := x^{(i)} \ \ \text{for %%%%22%%%%, %%%%23%%%%},$ and all other assignments remain as defined (Cinquini et al., 14 Oct 2024, Li et al., 14 Nov 2025). This "cuts" all incoming edges to $X_t^{(i)}$ for $t \geq t_0$, propagating effects downstream through the time-unrolled DAG.

The causal effect at time $t_0 + h$ (the $h$-step effect) is

$$\mathrm{CE}_h^I = \mathbb{E}\big[X_{t_0 + h}^{\mathrm{do}} - X_{t_0 + h}\,|\,X_{< t_0}\big].$$

DSCPs support both hard and soft interventions (including modifications to noise distributions $\varepsilon_{v,t}$ or additive shifts for modeling distribution shifts) (Li et al., 14 Nov 2025).

Simulation under intervention proceeds by recursively substituting the fixed value(s) and propagating all other equations forward, with exogenous noise terms sampled as specified.

3. DSCPs and Equilibrium SCMs: Long-Run Behavior

A central theorem establishes that, under standard stability conditions, the equilibrium (stationary) distribution of a DSCP coincides with the solution distribution of an associated static SCM (Cinquini et al., 14 Oct 2024). Assume $F$ is Lipschitz (contractive) and noise has compact support. Then, as $t \to \infty$, $X_t \to X_\infty$, and

$X_\infty = A X_\infty + \varepsilon$

for some matrix $A$ and $$\varepsilon \sim \mathcal{N}(0, \Sigma_\varepsilon)$$. Crucially:

• The SCM's structural graph matches the time-unrolled DSCP dependency structure.
• Interventional distributions in equilibrium are identical between the DSCP (with intervention from $t_0$ onward) and the corresponding SCM with the modified assignment.

For vector autoregressive DSCPs (see below), this manifests as the equivalence between VAR(p) equilibrium and linear Gaussian SCMs with the coefficient matrix $\tilde{A} = A_1 + \cdots + A_p$.

4. Parametric Realizations: VAR and SVAR-DSCPs

The VAR(p) process exemplifies a linear DSCP: $$X_t = \nu + A_1 X_{t-1} + \cdots + A_p X_{t-p} + \varepsilon_t, \ \varepsilon_t \sim \mathcal{N}(0, \Sigma_\varepsilon),$$ with stability enforced by the roots of $\det(I - A_1 z - \cdots - A_p z^p)$ lying outside the unit circle (Cinquini et al., 14 Oct 2024, Hochsprung et al., 15 Apr 2025). The mapping to a static SCM at equilibrium is explicit: $$X \text{ (static)} = \tilde{A} X + \eta, \qquad \tilde{A} = A_1 + \cdots + A_p.$$ For SVAR models with contemporaneous (instantaneous) effects, the full-time graph is constructed (Hochsprung et al., 15 Apr 2025):

• Inst. edges: nonzero entries of $A^{(0)}$,
• Lagged edges: $A^{(h)}$ for $h = 1,\ldots,p$.

Identifiability of direct causal effects from second moments requires trek/separation-based graphical criteria and can be achieved by linear systems involving covariance matrices over observed time series (Hochsprung et al., 15 Apr 2025). Wright’s path rule enables total-effect computation by path-product sums.

5. Nonlinear, Event-Based, and Path-Dependent DSCPs

DSCPs are not limited to linear-Gaussian or VAR/SVAR realizations. Multiple generalizations appear:

• Structural Hawkes Processes (SHP): DSCP instantiations for discrete-time event sequences where each node's conditional intensity depends on contemporaneous and lagged parent counts. Instantaneous effects are identifiable through non-Gaussian (Poisson) thinning—resolving failures of Granger-causal approaches at low time resolution (Qiao et al., 2023).
• Path-Dependent SEMs (PDSEM): DSCPs with latent Markovian “state” processes, allowing the structural equations and DAG/CDAG to vary by state; suitable for modeling systems with transitions between qualitatively different modes and time-varying causal relationships (Srinivasan et al., 2020).
• KarmaTS and Neural DSCPs: Systems such as KarmaTS let structural assignments $f_v$ be neural networks or expert-designed templates, supporting mixed data types, varying lags, and flexible interventions and simulation (Li et al., 14 Nov 2025).

6. Causal Graphs, Latent Structure, and Spectral Identification

DSCPs admit a systematic graphical representation:

• Time-unrolled DAGs: Nodes $X_{v, t}$, with edges determined by the dependency structure across lags. Lagged edges $X_{u, t - \ell} \to X_{v, t}$ for each nonzero entry in $E^{(\ell)}$.
• Process graphs: Each process is a node; edge filters (convolutions) encode lag structure (Reiter et al., 2023).
• Full-time graphs: Nodes $S^i_t$ for each process and time, edges following $A^{(h)}$.

In the case of latent (unobserved) components, the observed marginal process is described by a mixed graph with bidirected edges representing contemporaneous confounding or hidden variable-induced dependence (Boeken et al., 3 Jun 2024).

Causal structure and causal effect identification can be characterized by:

• $d$- and $t$-separation in the process/full-time graph (Reiter et al., 25 Jun 2024).
• Algebraic (rank, determinant) constraints on the observed (auto-, cross-)spectral density (Reiter et al., 2023, Reiter et al., 25 Jun 2024).
• Latent Factor Half-Trek Criterion (LF-HTC) for rational identifiability of effects under latent confounding in the frequency domain.

7. Applications, Inference Recipes, and Simulation

DSCP methods provide a broad recipe for causal inference in time series:

• Parameter Estimation: Standard approaches (e.g., OLS for VARs, MM/likelihood fitting for Hawkes-based DSCPs) estimate model parameters and innovation covariance matrices (Cinquini et al., 14 Oct 2024, Qiao et al., 2023).
• Causal Forecasting: Interventional forecasts (do-forecasts) use the standard recursion with added intervention terms (fixed values, shock augmentations, or prescribed noise distributions) (Cinquini et al., 14 Oct 2024, Li et al., 14 Nov 2025).
• Impulse Response Functions (IRF): Quantify the time-propagated causal effect of a one-unit shock to any coordinate, computed via the associated moving-average coefficients (Cinquini et al., 14 Oct 2024).
• Counterfactuals: Computed by abduction of realized noise, imposition of counterfactual intervention, and forward simulation.
• Graphical/Spectral Identification: Causal effects can be algebraically (or rationally) identified using spectral methods and path-based decompositions, subject to separation and rank criteria (Reiter et al., 2023, Reiter et al., 25 Jun 2024, Hochsprung et al., 15 Apr 2025).

DSCPs enable simulation and benchmarking of algorithms under known causal dynamics, as in KarmaTS (Li et al., 14 Nov 2025), and support both fully specified and data-driven inference for time-series-based causal reasoning over observed and hidden-variable structure.