Set-Valued Vector Autoregressive Models and Inference

This paper develops a vector-autoregressive framework for time series whose primitive observations are random compact convex sets. Support functions and a Rådström-type embedding place centered set-valued observations in a linear Hilbert-space representation, allowing stable Set-VAR dynamics, a one-sided moving-average representation, and predictive non-causality restrictions to be formulated directly at the set level. For a fixed exact finitedimensional representation, we establish an OLS central limit theorem under martingaledifference or weak-dependence/HAC conditions, derive the corresponding Wald limits, and give identification conditions under which intrinsic Set-Granger non-causality is equivalent to nullity of the projected cross-lag blocks. For interval-valued observations the representation is exact because the one-dimensional unit sphere contains only the two directions +1 and −1. We derive explicit restriction matrices for fully flexible, direction-separable, and commondirection interval Set-VARs, yielding χ24p, χ22p, and χ2p Granger-Wald limits, respectively, and show that full block non-causality is invariant to the support-to-center–radius transformation. The empirical application uses daily and weekly low–high return intervals for the S&P 500, Nasdaq Composite, and CAC 40 over 2016–2026. Pairwise estimates are strongly channeldependent: daily S&P 500 → Nasdaq predictability is significant whereas Nasdaq → S&P 500 is not, and the cross-Atlantic systems display substantial range-to-range transmission. A joint six-dimensional Set-VAR then conditions every directional test on the complete center–radius history of the third market. At the daily frequency, S&P 500 → Nasdaq, S&P 500 → CAC 40, CAC 40 → S&P 500, and CAC 40 → Nasdaq remain significant, while the pairwise Nasdaq → CAC 40 relation disappears. At the weekly frequency only S&P 500 → Nasdaq and CAC 40 → S&P 500 remain significant in the joint system. A dependent wild-bootstrap calibration of the HAC-Wald statistics and a one-session trading-clock stress test leave the central classifications intact. The conditional evidence separates incremental predictability from relations absorbed by a common-market channel and confirms that interval amplitude can carry predictive information missed by midpoint reductions.