Identification and Inference in Nonlinear Dynamic Network Models

Estimating contemporaneous bidirectional interactions from observational data is difficult because each outcome is endogenous to the other, while flexible regressions may capture only reduced-form dependence. This paper proposes SEM-DNN, a heteroscedastic neural simultaneous-equation estimator that learns reciprocal structural interactions without external instruments. Identification exploits conditional covariance diagonalization: when structural shocks have zero conditional means, are conditionally uncorrelated given predetermined covariates, and exhibit nonproportional conditional variances, only the true interaction coefficients diagonalize the conditional residual covariance across the feature space. The method jointly approximates nonlinear structural mean functions and feature-dependent variances using a diagonal Gaussian quasi-likelihood that incorporates the simultaneous-system Jacobian. We establish unique identification and positive-definite local curvature of the profiled population criterion and show that, under neural-profile compatibility conditions, the implemented neural criterion inherits this curvature despite nonunique network parameterizations. The coefficients admit a causal interpretation when the structural equations represent autonomous mechanisms that remain invariant under the relevant interventions. Monte Carlo experiments with nonlinear, high-dimensional nuisance functions and non-Gaussian shocks show that SEM-DNN recovers structural effects more reliably than parametric, kernel-based, and separate-equation neural alternatives as information increases, although at greater computational cost. An application to ready-to-eat cereal scanner data illustrates how the method can study contemporaneous price-sales feedback and assess identification strength, residual diagonalization, variance calibration, and optimization sensitivity.

Continuous-time stochastic differential equations (SDEs) provide a natural framework for modeling interacting dynamical systems. A central causal question is whether causal effects among observed variables can be identified from observational data drawn from the stationary distribution, given the underlying causal graph. For linear SDEs, existing identifiability results are largely restricted to fully observed systems whose causal graphs contain no directed 2-cycles. We first establish new graphical conditions for the generic identifiability of causal effects in systems that may contain directed 2-cycles, thereby allowing for reciprocal interactions. Next, we address the previously unexplored setting of latent variables by developing an instrumental-variable framework for identifying a target direct causal effect under partial observability.

We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.