Order-Constrained Spectral Causality for Multivariate Time Series

TL;DR

This paper develops a spectral operator framework for analyzing directional causal influence in multivariate time series, unifying classical causality measures and establishing optimal detection scales.

Contribution

It introduces an order-constrained spectral approach that uniquely satisfies key properties and unifies existing causality measures, with theoretical and empirical validation.

Findings

Spectral tests detect causality at linear sample size scale, outperforming quadratic-sample methods.

Classical causality measures are special cases of the proposed framework.

Simulations and empirical data demonstrate the method's effectiveness and robustness.

Abstract

We introduce an operator-theoretic framework for analyzing directional dependence in multivariate time series based on order-constrained spectral non-invariance. Directional influence is defined as the sensitivity of second-order dependence operators to admissible, order-preserving temporal deformations of a designated source component, summarized through orthogonally invariant spectral functionals. We show that the resulting supremum--infimum dispersion functional is the unique diagnostic within this class satisfying order consistency, orthogonal invariance, Loewner monotonicity, second-order sufficiency, and continuity, and that classical Granger causality, directed coherence, and Geweke frequency-domain causality arise as special cases under appropriate restrictions. An information-theoretic impossibility result establishes that entrywise-stable edge-based tests require quadratic sample size scaling in distributed (non-sparse) regimes, whereas spectral tests detect at the optimal linear scale. We establish uniform consistency and valid shift-based randomization inference under weak dependence. Simulations confirm correct size and strong power across distributed and nonlinear alternatives, and an empirical application illustrates system-level directional causal structure in financial markets.