The Same Distribution, a Different Direction: Temporal Irreversibility in Global Equity Returns

Abstract

Reversing a return series leaves its empirical distribution unchanged. Its mean, variance, skewness, kurtosis, quantiles, and Wasserstein distance from the original sample are therefore identical. Yet the risk process need not be unchanged, because negative returns may precede volatility expansions more often than volatility expansions precede negative returns. We use this exact distribution-preserving transformation to study daily S&P 500, FTSE 100, Nikkei 225, and Hang Seng returns from January 2000 through June 2026. Temporal direction is measured by the difference between two mixed third moments, averaged over lags one through five. The forward-time statistic is positive in all four markets—0.339, 0.317, 0.320, and 0.364—and remains significant after Holm correction across the four primary tests. Reversing each series changes the statistic's sign while leaving marginal Wasserstein distance exactly zero. Randomizing return signs while preserving the magnitude sequence centers the statistic at zero in 2,000 replications (p = 0.0005 in every market). The direction remains positive across bandwidth, bootstrap, past-only standardization, lag, episode-deletion, subsample, and winsorization checks. Of 36 prespecified robustness tests, 31 pass a 5% Benjamini–Hochberg false-discovery-rate threshold and 13 survive the stricter Holm family-wise correction. Distributional risk measurement is therefore incomplete without an ordering-sensitive layer: the same return distribution can encode a different direction of risk propagation.

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