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.
Individual treatment effects are not point-identified from data. The Probability of Necessity and Sufficiency (PNS) circumvents this limitation by characterizing individual-level causality through intersection bounds derived from combined experimental and observational data. In finite samples, however, standard plug-in estimators systematically fail: they violate structural probability constraints and suffer from extremum bias induced by max-min operators, yielding spuriously narrow intervals. We propose a neural framework for finite-sample PNS estimation that resolves both pathologies. We introduce an anchored neural architecture that guarantees structural constraint satisfaction by construction. To correct extremum bias, we employ precision-corrected intersection-bound inference, leveraging Epistemic Neural Networks for scalable, high-dimensional uncertainty quantification. Empirical evaluations confirm that this approach maintains nominal coverage and exact constraint validity in high-dimensional regimes where standard estimators systematically undercover.
When a neural time-series model reports that one variable modulates another's effect on a target, is the discovered interaction a property of the data or an artifact of model flexibility? We argue that this is fundamentally a question of identifiability, governed by the geometry of the observed input support rather than by the specific neural architecture. We study the problem in a multiplicative-gating extension of neural additive vector autoregression (GNAVAR), in which source contributions are modulated by other lagged variables. We show that representational capacity is not identifiability: dependent inputs induce leakage between edge-specific interaction terms, and low-dimensional support permits distinct interaction decompositions that agree on the observed data while differing elsewhere. We then prove a population identifiability theorem for normalized minimal GNAVAR decompositions under explicit support conditions, including settings with shared modulators. The theory yields a simple practitioner-facing diagnostic: the effective rank of the joint lag-block covariance predicts, before fitting, whether interaction recovery is feasible for a given candidate set. When the candidate set is unknown, a two-seed stability check provides a practical operational test. The same support condition organizes empirical outcomes into the three states predicted by the theory. Our results show that interaction recoverability depends on support geometry, that effective rank provides a practical pre-fit diagnostic, and that instability across independent fits is a characteristic signature of non-identifiable interaction discovery. The identifiability phenomenon, the support condition, and the instability signature are model-agnostic; GNAVAR is the vehicle that makes them provable.
Valentina Kuskova, Dmitry Zaytsev, Michael Coppedge
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.