On the Limits of GARCH in Event-Driven Markets: A Null-Universe Perspective with Evidence from the S&P 500 and Bitcoin

Abstract This study examines whether the Bitcoin (BTC) return process can be adequately described by a noise-driven conditional variance framework within the GARCH family. Rather than proposing additional GARCH extensions for cryptocurrency markets, we adopt a null-universe perspective: if a market is compatible with a GARCH-type specification, the standardized innovations implied by the fitted model should resemble stable noise; if not, extreme standardized events may occur more frequently than predicted under the null. Using the S&P 500 index as a benchmark market across two eras (1962–1985 and 1986–2026), and Bitcoin at four time scales (30-minute, 1-hour, 4-hour, daily), we evaluate tail-event exceedances under both normal and Student-t innovations. We find that while S&P 500 tail deviations are largely accommodated by moderate-tailed Student-t distributions (ν = 5.6–10.0), Bitcoin exhibits substantially heavier standardized residual tails across frequencies. Estimated degrees of freedom for BTC (ν ≈ 3.0–4.0) approach the lower boundary of finite higher moments, and residual tail rejection persists at higher frequencies even under the Student-t specification. These results suggest a meaningful distinction in the empirical compatibility of GARCH-type models across asset classes. We outline a diagnostic distinction between noise-dominant and event-sensitive market environments—as a diagnostic tool for assessing model applicability prior to specification refinement. JEL Classification: C22; C58; G10; G17