Identification and Inference in Nonlinear Dynamic Network Models

Diego Vallarino

stat.MLApr 3, 2026

Identification and Inference in Nonlinear Dynamic Network Models

Authors: Diego Vallarino

Organizations: Inter-American Development Bank (IDB), Washington, DC, United States

Abstract

We study identification and inference in nonlinear dynamic systems defined on unknown interaction networks. The system evolves through an unobserved dependence matrix governing cross-sectional shock propagation via a nonlinear operator. We show that the network structure is not generically identified, and that identification requires sufficient spectral heterogeneity. In particular, identification arises when the network induces non-exchangeable covariance patterns through heterogeneous amplification of eigenmodes. When the spectrum is concentrated, dependence becomes observationally equivalent to common shocks or scalar heterogeneity, leading to non-identification. We provide necessary and sufficient conditions for identification, characterize observational equivalence classes, and propose a semiparametric estimator with asymptotic theory. We also develop tests for network dependence whose power depends on spectral properties of the interaction matrix. The results apply to a broad class of economic models, including production networks, contagion models, and dynamic interaction systems.

Figures & tables

Figure 1: Empirical size under $H_{0}$ across $(n,T)$ .

$n$	$T$	Rejection rate	Std. error
25	50	0.043	0.009
25	100	0.067	0.011
25	200	0.060	0.010
25	400	0.028	0.008
50	50	0.040	0.009
50	100	0.062	0.010

Table 1: Empirical size under

H_{0}

Panel of Figure 2 — Figure 2: Finite-sample null distributions of $\|\hat{A}\|_{F}$ .

Figure 3: Frobenius and spectral estimation errors across sample sizes.

$n$	$T$	Frobenius error	Spectral error	Bias	RMSE
25	50	5.87	1.37	0.12	0.23
25	100	3.12	0.89	0.08	0.13
25	200	1.85	0.57	0.05	0.08
25	400	1.34	0.39	0.03	0.05
50	50	25.6	1.91	0.18	0.52
50	100	8.23	1.38	0.10	0.17

Table 2: Monte Carlo summary statistics

Figure 5: Power under regular network structures.

$n$	$T$	Rejection rate
25	50	0.03
25	100	0.62
25	200	1.00
25	400	1.00
50	50	0.09
50	100	0.04

Table 3: Power under regular networks

Figure 6: Power under local alternatives.

Figure 7: Rejection probabilities under degenerate network structures.

Network	$T=50$	$T=200$	$T=400$
Complete	0.10	0.76	1.00
Rank-1	0.09	0.80	1.00
Star	0.08	1.00	1.00

Table 4: Degenerate network structures

Panel of Figure 8 — Figure 8: Spectral dispersion and covariance heterogeneity.

Figure 9: Distributional separation between $H_{0}$ and $H_{1}$ .

Network	Spectral dispersion	Rejection probability
Sparse	1.12	0.442
Hub	1.25	0.459
Block	1.31	0.458
Chain	1.78	0.469

Table 5: Spectral dispersion and power

Network	Spectral range	Std. dev. of $Cov(U_{i},U_{j})$
Sparse	0.85	0.021
Hub	1.02	0.027
Block	1.11	0.029
Chain	1.54	0.041

Table 6: Spectral range and covariance heterogeneity

Parameter Identifiability

Parameter identifiability studies when observations and model assumptions uniquely determine a system’s unknown parameters.

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Learning Bidirectional Causal Interactions with Heteroscedastic Neural Networks

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Faculty of Economics, Fukuoka University, Fukuoka, Japan. Address: 8-19-1, Nanakuma, Jonan, Fukuoka, Japan 814-0180.

Causal Effect Estimation Structural Causal Models Causal Identifiability

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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.

Abstract

Parameter Identifiability

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