Session-Discontinuity Bias in Parkinson-Type Range Statistics: Stratified Decomposition of the H ≈ 0.56 Anomaly

An earlier and unpublished version of this analysis obtained a Hurst-type exponent H ≈ 0.5624 for Parkinson-type range statistics on US equities and read it as evidence of anomalous scaling. That reading is withdrawn here. This paper decomposes the exponent by stratifying the regression across data regimes and shows the elevation to be an artefact of the trading-session boundary. Using 1-minute OHLC bars for 57 US large-capitalisation common stocks over 2021–2024, four estimates are computed that differ only in how bars are aggregated. Path B, an intraday-only regression in which no window crosses a session break, gives H = 0.5354 (naive SE 0.0012; 0.0055 after the cross-sectional correction of Section 3.1a). Path A′, a multi-day regression on gap-adjusted chained prices, gives H = 0.5090 (naive SE 0.0025; 0.0063 by block bootstrap). Path A, the same multi-day regression on raw prices, gives H = 0.5688 (naive SE 0.0069; 0.0101 by block bootstrap). Path C, a single regression pooling intraday and multi-day scales — the aggregation used in the earlier analysis — gives H = 0.5620 (naive SE 0.0031; 0.0142 after correction) with R² = 0.9995, reproducing the earlier figure to within 0.0004. The elevation in Path C is therefore not evidence of long memory. It is what a single straight line returns when fitted across a discontinuity at the session boundary: intraday windows contain no overnight period, multi-day windows necessarily do, and the vertical offset between the two regimes enters the fit as slope. A controlled experiment on index futures establishes the mechanism without relying on the gap-adjustment counterfactual. Because ES and NQ trade approximately twenty-three hours per day, the same contracts over the same period can be analysed under nested session definitions that differ only in how much of the trading day the analyst discards. Across five nested settings the exponent rises monotonically from 0.5401 to 0.5945 as the discarded portion grows from one hour to sixteen and a half (Spearman ρ = 1.000; +0.0033 per hour discarded), while the gap-adjusted series stays flat near 0.525 and the residual standard deviation falls monotonically as the discontinuity shrinks. One finding survives the decomposition. The intraday exponent for single-name equities is 0.5354, above the Brownian benchmark for every name in the sample — though, as Section 3.1a sets out, these are not independent observations. Microstructure noise is too small to account for a departure of this size. The corresponding figure for index futures, measured on an identical session and scale set, is 0.5052. This asymmetry between single names and index futures is the residual that the session-discontinuity account does not explain. JEL: G12, G14, G17, C58 · Keywords: Hurst exponent, Parkinson volatility, session discontinuity, overnight gaps, range statistics