Equilibrium Causal Effects

• Equilibrium causal effects are measures defined as the system-wide impacts aggregating direct and feedback-driven indirect responses after an intervention.
• They capture feedback loops and adjustment dynamics in networks, spatial economies, and strategic interactions while requiring precise regime definitions.
• Advanced estimation methods like IPW, DR/DML, and diffusion-based models enable robust inference by identifying the new fixed point after system shocks.

Equilibrium causal effects are causal estimands defined in environments where outcomes of interest are determined as the solution to a system of simultaneous equations capturing feedback and interdependence among units, markets, or agents. Unlike standard settings with no interference, equilibrium causal effects aggregate both direct and system-wide indirect responses to an intervention, requiring analysis of the new fixed point or steady-state after a shock. These effects are central in modern empirical work on networks, spatial economies, strategic interactions, market mechanisms, and nonlinear cyclic systems, where interventions propagate beyond experimental or policy boundaries and feedback loops crucially determine observed responses.

1. Definitions and Conceptual Regimes

The interpretation of causal effects in equilibrium settings depends on precisely articulating the counterfactual regime under consideration. In canonical network and spatial models, three regimes are defined (Mate, 1 Jan 2026):

1. Partial-Equilibrium (PE) Effect: The direct effect on a treated unit when outcomes and behaviors of all other units are held fixed at their pre-intervention levels. This isolates the immediate impact, ignoring spillovers and network feedback.
2. Local-Interaction (LI) Effect: Extends PE by allowing only first-order neighbors to adjust in response to the treated unit, while higher-order feedback loops remain suppressed. This regime captures immediate spillovers to direct connections but not systemic propagation.
3. Network-Consistent (NC) or General-Equilibrium Effect: All directly and indirectly connected entities fully adjust, so the treatment shock propagates and feeds back until a new system-wide equilibrium is reached. The equilibrium causal effect aggregates direct impacts, indirect paths, and all higher-order feedback.

In market and mechanism design contexts, an analogous "global treatment effect" (GTE) is defined as the counterfactual change in average outcome when all units are treated versus none, with joint allocations determined by the market-clearing equilibrium (Munro, 9 Apr 2025).

Strategic game-theoretic environments formalize equilibrium effects via solution concepts such as Nash or quantal-response equilibrium, where interventions affect best-response mappings and the induced stationary distributions over outcomes (Xiao, 17 Oct 2025, Toulis et al., 2014). In spatial economies subject to threshold spillovers, equilibrium effects are triggered when local propagation causes a regime shift into systemic general equilibrium (Kikuchi, 8 Aug 2025).

2. Structural Models and Identification

Equilibrium systems are formalized as vector-valued fixed point problems: $$\mathbf{Y} = \mathbf{h}(\mathbf{Z}, \mathbf{L}, \mathbf{Y}, \mathbf{U}),$$ where $\mathbf{Y}$ are endogenous outcomes, $\mathbf{Z}$ are treatments, $\mathbf{L}$ encodes network or structural connections, and $\mathbf{U}$ are unobserved heterogeneities (Menzel, 31 Jan 2025). Existence and uniqueness of equilibrium are generally required, commonly ensured by contraction mappings or stability of the Jacobian.

Causal identification hinges on:

• Parameterization of feedback: Linear-in-means (SAR), nonlinear cyclic SCMs, or general response functions (Mooij et al., 2013, Mate, 1 Jan 2026, Menzel, 31 Jan 2025).
• Nature of assignment and interference: Randomization of treatments, exogeneity of shocks, and SUTVA-type assumptions at the mechanism or submission level (Munro, 9 Apr 2025, Toulis et al., 2014).
• Availability of intervention diversity: For cyclic SCMs, identifiability of equilibrium causal effects requires a sufficient number of interventional and observational regimes to disentangle direct and feedback pathways (Mooij et al., 2013).

Key objects for inference are derivatives of the equilibrium mapping: $$\frac{\partial Y_i}{\partial Z_j} = [e_i' (I - H_Y)^{-1} H_Z e_j],$$ with $H_Y$ and $H_Z$ the Jacobians of $\mathbf{h}$ with respect to $\mathbf{Y}$ and $\mathbf{Z}$, respectively (Menzel, 31 Jan 2025). In strategic equilibrium settings, potential outcome mappings must respect endogenous adjustments to treatments made by rational agents (Xiao, 17 Oct 2025, Toulis et al., 2014).

3. Estimation Methodologies

a. Structural and Design-Based Estimation

• Inverse Probability Weighting (IPW): Used to estimate local and global equilibrium effects under randomization of treatments in network equilibrium models. Marginal causal effects are consistently estimated from a single network realization via incremental reweighting (Menzel, 31 Jan 2025).
• Doubly Robust and Machine Learning (DR/DML): In strategic equilibria, the Strategic Doubly Robust (SDR) estimator incorporates nuisance modeling of strategic propensities and equilibrium states to remain consistent when either the outcome or assignment model is correctly specified (Xiao, 17 Oct 2025). In designed markets, cross-fitted sample splitting and re-running the equilibrium mechanism under perturbed treatment weights yield $\sqrt{n}$-consistent estimates of GTE (Munro, 9 Apr 2025).

b. Generative and Boundary-Aware Modeling

• Diffusion-Based Approach: In spatial settings with stochastic transitions between regimes, a Lévy jump-diffusion process models spillover propagation, with denoising diffusion probabilistic models (DDPM) generating counterfactual trajectories across boundaries from partial to general equilibrium (Kikuchi, 8 Aug 2025).
• CUSUM Boundary Detection: Sequential likelihood ratio tests (CUSUM) are applied to detect first passage into systemic equilibrium, identifying when local spillovers cross critical thresholds and require system-wide causal analysis (Kikuchi, 8 Aug 2025).

c. Cyclic and Nonlinear Causal Discovery

• For nonlinear cyclic SCMs, equilibrium causal effects are estimated via coupled local linearizations, Bayesian structure search, and mechanism-label clustering across observational and interventional regimes (Mooij et al., 2013).

Method	Setting	Key Identification/Estimation Mechanism
SAR/IPW/Design-Based	General network equilibrium	Assignment randomization, neighborhood reweighting
SDR (Doubly Robust)	Strategic equilibrium	Strategic covariate modeling, DR scores, Nash/QRE computation
DDPM + CUSUM	Spatial regime shift	Jump-diffusion counterfactuals, real-time regime detection
Cutoff Mechanism Re-run	Centralized market	Mechanism reweighting, sample splitting, semi-parametric DML
Cyclic SCMs	Biochemical networks	Local linearization, multi-task priors, Bayesian search

4. Practical Guidance and Identification Challenges

Correct specification of the counterfactual regime is critical: different regimes—PE, LI, NC—target different combinations of direct and spillover effects, and require distinct exogeneity assumptions. PE effects need only weak independence of treatment and idiosyncratic shocks, while NC effects demand full independence of the treatment vector and the vector of all shocks, a much stronger assumption seldom met in observational data (Mate, 1 Jan 2026).

Standard SAR and network regression methods often ambiguously report "indirect effects" or "multipliers" without specifying the counterfactual regime, conflating one-step and full general equilibrium impacts. Without explicit regime specification and matched identification strategy, reported effect sizes can misrepresent both magnitude and mechanism of policy (Mate, 1 Jan 2026).

In empirical disciplines (e.g., market design, spatial policy), ignoring equilibrium regime transitions can induce severe bias—typically underestimating effects in densely connected economies or when systemic spillovers trigger general equilibrium feedback (Kikuchi, 8 Aug 2025, Munro, 9 Apr 2025).

5. Illustrations and Empirical Findings

• Monte Carlo (SAR): Even before estimation, the NC effect is mechanically amplified relative to PE by network feedback. Small violations of exogeneity can generate fourfold increases in bias in the equilibrium effect versus the direct effect (Mate, 1 Jan 2026).
• Strategic Systems: SDR estimation under Nash equilibria shrinks bias by 7.6–29.3% over non-strategic DR, with robustness to covariate dimensionality and strong gains when agent behavior is highly strategic (Xiao, 17 Oct 2025).
• Market Mechanisms: In Chilean school choice, equilibrium-aware estimation of information effects via mechanism re-running reduced naive ATE estimates by 61%, and policy learning under full interference obtained optimized rules at parametric rates (Munro, 9 Apr 2025).
• Spatial General Equilibrium: CUSUM-detected boundaries in AI adoption across Japanese prefectures delineate PE/GE transition at ≈35 km scales, with GE effects amplifying PE estimates by 42%. Neglect of boundaries can understate real effects by up to 67%, and GE-aware targeting increased welfare by 67% over PE-only targeting (Kikuchi, 8 Aug 2025).

6. Extensions, Open Problems, and Policy Implications

The correct estimation and reporting of equilibrium causal effects rest on formulating explicit, policy-relevant counterfactuals and matching identification and estimation approaches accordingly. The rapid development of generative (DDPM) and boundary detection methods for spatial and complex network data expands diagnostics for when policy must consider general equilibrium effects (Kikuchi, 8 Aug 2025). Extensions include:

• Dynamic and Multi-period Equilibria: Long-term effects in repeated play or multi-period mechanisms require further integration of equilibrium learning into causal inference (e.g., quantal response over time (Toulis et al., 2014)).
• Handling Multiple Equilibria: When systemic interventions induce nonunique equilibria, selection and averaging rules must be explicit in defining equilibrium effects (Xiao, 17 Oct 2025).
• Agnostic vs. Structural Interpretations: Even estimands defined without structural assumptions (agnostic) can recover model parameters when combined, but maintain robust causal interpretation if the structural model is misspecified (Menzel, 31 Jan 2025).
• Policy Learning: Recent advances allow integration of spillover-aware policy learning at classical statistical rates, enabling optimized targeting in the presence of full or partial equilibrium feedback (Munro, 9 Apr 2025).

In summary, equilibrium causal effects constitute the rigorous formalization and estimation of treatment impacts after all system-wide feedbacks occur. Their computation requires explicit regime specification, careful identification, and robust, often model-based estimation techniques tailored to the endogenous interdependence at the heart of the system under study. Neglecting these features risks severe misestimation and misguided policy in interconnected economic, social, and biological networks.