[Paper Review] Order-Constrained Spectral Causality in Multivariate Time Series

Adriel Dominguez|arXiv (Cornell University)|Jan 3, 2026

Functional Brain Connectivity Studies0 citations

TL;DR

Introduces an operator-theoretic framework for directional causality in multivariate time series based on order-constrained spectral non-invariance of second-order dependence, extending beyond local predictability to capture collective and nonlinear directional effects.

ABSTRACT

We introduce an operator-theoretic framework for causal analysis in multivariate time series based on order-constrained spectral non-invariance. Directional influence is defined as sensitivity of second-order dependence operators to admissible, order-preserving temporal deformations of a designated source component, yielding an intrinsically multivariate causal notion summarized through orthogonally invariant spectral functionals. Under linear Gaussian assumptions, the criterion coincides with linear Granger causality, while beyond this regime it captures collective and nonlinear directional dependence not reflected in pairwise predictability. We establish existence, uniform consistency, and valid inference for the resulting non-smooth supremum--infimum statistics using shift-based randomization that exploits order-induced group invariance, yielding finite-sample exactness under exact invariance and asymptotic validity under weak dependence without parametric assumptions. Simulations demonstrate correct size and strong power against distributed and bulk-dominated alternatives, including nonlinear dependence missed by linear Granger tests with appropriate feature embeddings. An empirical application to a high-dimensional panel of daily financial return series spanning major asset classes illustrates system-level causal monitoring in practice. Directional organization is episodic and stress-dependent, causal propagation strengthens while remaining multi-channel, dominant causal hubs reallocate rapidly, and statistically robust transmission channels are sparse and horizon-heterogeneous even when aggregate lead--lag asymmetry is weak. The framework provides a scalable and interpretable complement to correlation-, factor-, and pairwise Granger-style analyses for complex systems.

Motivation & Objective

• Motivate causal analysis in high-dimensional time series where influence is distributed across modes rather than localized edges.
• Develop an order-constrained, invariance-based framework that defines causality via sensitivity of second-order dependence to admissible temporal deformations.
• Provide a spectral-functional approach that remains informative beyond linear Gaussian settings.
• Establish statistical inference procedures with finite-sample validity under weak dependence without parametric models.
• Demonstrate applicability to high-dimensional financial data for system-level causal monitoring.

Proposed method

• Define admissible, order-preserving temporal deformations of a designated source component.
• Summarize second-order dependence with orthogonally invariant spectral functionals of a directed coherence operator.
• Relate causality to non-invariance of these spectral summaries under deformation (sup–inf dispersion over the deformation set).
• Show equivalence to linear Granger causality under linear Gaussian VAR(p) assumptions, and potential divergence otherwise.
• Extend from scalar spectral summaries to the full spectral distribution to capture bulk and low-rank dependence changes.
• Propose nonparametric randomization-based inference exploiting order-induced invariance for finite-sample validity.

Experimental results

Research questions

• RQ1 Can causality be defined as non-invariance of second-order dependence geometry under admissible, order-preserving temporal deformations?
• RQ2 Under what conditions does the proposed order-constrained spectral causality coincide with linear Granger causality, and when does it reveal additional directional structure?
• RQ3 How does extending to the full spectral distribution improve detection of distributed or nonlinear directional dependence?
• RQ4 What are the statistical properties and valid inference procedures for the resulting non-smooth, supremum–infimum statistics?
• RQ5 Is the framework scalable and interpretable for high-dimensional financial time series?

Key findings

• The framework defines directional influence as sensitivity of second-order dependence to order-preserving temporal deformations of a source component.
• Under Gaussian VAR(p) assumptions, the criterion coincides with linear Granger causality; outside this regime, it detects nonlinear or distributed directional dependence missed by Granger tests.
• Extending to the full spectral distribution allows detection of bulk redistribution of dependence, not just edge effects.
• Existence, uniform consistency, and valid inference are established for the non-smooth supremum–infimum statistics using shift-based randomization exploiting order invariance.
• Simulations show correct size and power against both edge-dominated and bulk-dominated alternatives, including nonlinear dependencies.
• Empirical application to a high-dimensional financial panel demonstrates episodic, stress-driven, system-level directional organization with sparse robust transmission channels.