Bandwidth-Sensitive Persistence in S&P 500 Volatility from 1990 to 2026

Persistent volatility in equity markets affects risk measurement and long-horizon forecasts. This study examines daily S&P 500 returns from 3 January 1990 to 16 October 2026 (9,265 observations). It estimates long memory in the logarithm of absolute returns and squared returns with the Geweke-Porter-Hudak and local Whittle methods. Three bandwidth exponents are considered. A Student-t FIGARCH model provides a parametric comparison. Estimates are positive, but their magnitude depends on bandwidth. At alpha equal to 0.6, the estimates range from 0.406 to 0.422. At alpha equal to 0.7, they fall to 0.312-0.323. At alpha equal to 0.5, the local Whittle estimate reaches its upper search bound, and Geweke-Porter-Hudak intervals include 0.5. Middle-bandwidth subperiod estimates range from 0.350 to 0.381. Qu’s test rejects the null of genuine long memory for both daily proxies at both trimming settings in the refreshed 9,267-observation sample. This points to a spurious-memory alternative that may include unmodeled shifts or trends. A robustness analysis compares the proxies with five-minute realized variance from 3 January 2000 to 31 March 2020 (5,079 matched days). Qu’s test does not reject its genuine-memory null for log realized variance, while its local Whittle estimate reaches 0.499 and has no regular confidence interval. In the asymmetric FIEGARCH model, the fractional parameter is 0.596 (95% CI 0.533-0.659), above 0.5, and the asymmetry coefficient is 0.154 (0.128-0.181). Information criteria improve relative to FIGARCH, but residual tests retain serial dependence. Taken together, the results show persistent volatility, but do not establish that all persistence in daily proxies is genuinely fractional. The estimated strength depends on bandwidth and model, and structural change remains a plausible explanation.