Order-Constrained Spectral Causality in Multivariate Time Series

ArXiv ID: 2601.01216 “View on arXiv”

Authors: Alejandro Rodriguez Dominguez

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.

Keywords: Operator-Theoretic Framework, Causal Analysis, Spectral Non-invariance, Multivariate Time Series, Granger Causality, Multi-Asset

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced operator theory, spectral functionals, and non-smooth supremum-infimum statistics (high math complexity), and includes a high-dimensional financial panel application with statistical inference, simulations, and real-data interpretation (moderate empirical rigor).

  flowchart TD
    A["Research Goal: Develop operator-theoretic framework for causal analysis in multivariate time series"] --> B["Methodology: Order-constrained spectral non-invariance framework"]
    B --> C["Data: Daily financial returns across major asset classes (high-dimensional panel)"]
    C --> D["Computation: Shift-based randomization inference with orthogonally invariant spectral functionals"]
    D --> E["Comparison: Linear Granger causality as special case; nonlinear dependence detection"]
    E --> F["Key Findings: Episodic & stress-dependent causal organization, multi-channel propagation, sparse transmission channels"]