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.

翻译：我们提出了一种基于算子理论的框架，用于分析多元时间序列中的方向依赖关系，其核心思想是顺序约束的谱非不变性。方向影响被定义为二阶依赖算子对指定源分量的容许、保序时间变形的敏感性，并通过正交不变的谱泛函进行综合度量。我们证明，在此类诊断框架中，由此导出的上确界-下确界色散泛函是唯一满足顺序一致性、正交不变性、Loewner单调性、二阶充分性以及连续性的指标，且经典的Granger因果性、定向相干性以及Geweke频域因果性在适当限制条件下均可视为其特例。一项信息论不可行性结果表明，在分布式（非稀疏）场景下，逐项稳定的基于边的检验需要平方级的样本量，而谱检验则能在最优的线性尺度上检测因果关系。我们建立了在弱依赖条件下的一致性及有效的基于偏移的随机化推断。仿真实验证实，该框架在分布式和非线性替代假设下均能保持正确的检验尺度并具备强大的检验功效；一项实证应用展示了金融市场中系统层面的有向因果结构。