Distributional Granger Causality: Identification, Sequential Inference, and Adaptive Testing

Predictive dependence in time series need not be confined to the conditional mean. Outside the Gaussian setting, causal content may arise through conditional scale, tail behavior, asymmetry, or other distributional features, implying that no single Granger-type test provides a complete characterization of predictive dependence. This paper develops a framework for distributional Granger causality based on a finite collection of channel-specific restrictions. Under suitable determinacy conditions, the channel menu is shown to be complete, yielding an identification result that links distributional Granger non-causality to a finite set of testable hypotheses. Building on this representation, we develop an adaptive sequential testing procedure that allocates inferential resources across channels while maintaining familywise error control through an alpha-investing mechanism. A policy-invariant validity theorem establishes finite-sample size control under arbitrary admissible selection rules, while an asymptotic efficiency theorem shows that a confidence-bound allocation rule achieves power equivalent to that of an infeasible oracle benchmark. The theoretical guarantees are derived from primitive mixing and moment conditions together with a circular-block permutation scheme.