Measuring the Memory Structure of Intraday Returns: Evidence from E-mini S&P 500 Futures

Abstract

Standard intraday trading systems treat consecutive returns as approximately independent, ignoring long-range dependence, fractional integration, volatility persistence, and multifractal regime mixing-properties that are measurable and consequential at the one-minute frequency. This paper implements and empirically validates a complete memory-structure measurement toolkit applied to E-mini S&P; 500 (ES) futures on the one-minute timeframe. The toolkit comprises three confidence-weighted Hurst estimators (R/S analysis, aggregated variance, absolute moments), the Geweke-Porter-Hudak (GPH) log-periodogram estimator for the ARFIMA fractional differencing parameter d, a Multifractal Detrended Fluctuation Analysis (MFDFA) spectrum estimator, and a volatility suite including realized, fractional, and FIGARCH-approximated conditional variance with leverage and jump diagnostics. All components are implemented in MQL5 as a single modular header file suitable for real-time use in indicators, Expert Advisors, and machine learning feature pipelines; the open-source implementation is documented in a companion article series (Brown, 2026a-d). Empirical analysis of 514 New York trading sessions (April 2024-March 2026, approximately 196,000 one-minute observations) yields four principal findings. First, the confidence-weighted Hurst blend produces H = 0.660 (median 0.662)-a strongly upward-biased estimate driven by short-range autocorrelation in the R/S statistic, as predicted by Lo (1991). Second, the GPH estimator, which is robust to this bias, gives pooled d =-0.057 and session-level mean d =-0.006 (SD = 0.121), consistent with near-random-walk behavior. The H-d discrepancy exceeds 0.1 in 75.1% of sessions, directly quantifying the R/S bias in real market data. Third, MFDFA singularity spectrum widths (delta-alpha, Legendre-transformed) average 0.415 (median 0.359), with 44.7% of sessions showing strong multifractal structure, confirming that opposing memory regimes coexist within individual sessions. Fourth, volatility clustering is present but modest at the minute level (mean clustering index 0.071), with jump intensity averaging 0.86%. Together these findings establish that ES M1 intraday returns are near-memoryless in aggregate but heterogeneous in structure, and that the R/S Hurst estimator systematically overstates persistence at this frequency.

Keywords: market microstructure, Hurst exponent, R/S bias, ARFIMA, GPH estimator, fractional differencing, MFDFA, multifractal spectrum, FIGARCH, intraday volatility, E-mini S&P, 500, MQL5