Volatility Persistence is not Distributional Memory: Separating Variance Memory from Empirical-distribution Memory

Abstract

Volatility persistence is commonly summarized by the persistence parameter of a conditional variance recursion or by the autocorrelation of realized volatility. Neither statistic directly answers whether an empirical return distribution remains aligned with the same distributional state. This paper separates variance memory from distributional memory using daily returns on the S&P 500, FTSE 100, Nikkei 225, and Hang Seng from 2000 through June 2026. Distributional states are defined from pre-2007 low- and high-volatility prototypes, and are studied through state duration, Wasserstein displacement, and tail-state recurrence. The analysis combines a four-specification GARCH-family grid with a nested circular-block bootstrap that re-estimates prototypes and thresholds in each replication. All sixteen volatility-model fits converge, and Student-t GJR-GARCH has the lowest BIC in every market. The state-switching pattern is positive across the first three markets under the primary nested bootstrap, but only the Nikkei 225 retains a positive lower bound under the conservative 264-day block. Wasserstein and realized-volatility half-life comparisons are formally inconclusive, while no tail-recurrence lift survives Holm correction. The evidence supports a distinction between the estimands and a model-robust but block-sensitive pattern of state turnover; it does not establish a universal or causal separation between variance persistence and distributional memory. The numerical thresholds and decision rules were not changed in response to this revision.

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