Set-Valued Vector Autoregressions: Stationarity, Granger and Sims Causality, and Wald Inference

This paper develops stationarity conditions and predictive-causality inference for vector autoregressions of random compact convex sets. A support-function embedding provides a linear representation in which Granger and Sims non-causality can be formulated as restrictions on cross-set lag and lead operators. We establish finite-dimensional least-squares asymptotics under martingale-difference or weak-dependence conditions and derive asymptotic χ2 Wald tests under identification and nondegenerate covariance. For intervals in R, there is a decisive geometric advantage: the support function is determined exactly by K = 2 directions, +1 and −1, so no functional principal-component truncation or growing projection dimension is required. We derive explicit restrictions for flexible and parsimonious interval systems and decompose predictive effects into center-to-center, center-to-radius, radius-to-center, and radius-to-radius channels. An empirical analysis of daily and weekly high–low return intervals for the S&P 500, Nasdaq Composite, and CAC 40 illustrates these tests, including controls for the third market. Dependent wild-multiplier calculations and trading-clock shifts complement the primary HAC-Wald evidence as sensitivity diagnostics; the general bootstrap-consistency question remains open.