Bitcoin's Power Law: Weak Structure, Strong Forecasts

2026-05-20

AI Summary

This study investigates whether Bitcoin prices follow a temporal power law and evaluates its efficacy as both a structural regularity and a predictive tool. Employing an enhanced Clauset–Shalizi–Newman protocol within a time-domain-adapted power-law testing framework—including tail distribution tests, multi-component sigmoid fitting, AR(1) residual diagnostics, scale invariance checks, and a K=3 wave stability bootstrap—the authors systematically analyze Bitcoin price series, UTXO balances, and absolute returns, complemented by cross-asset comparisons and forward-looking forecast evaluations. The findings reveal that the power-law hypothesis is decisively rejected for UTXO and return distributions, and the estimated temporal exponent proves sensitive to the choice of time origin. Nevertheless, within 12–24 month forecasting horizons, a simple power-law model significantly outperforms benchmark models such as random walk, ARIMA, and ETS (p<0.05), highlighting a trade-off between model robustness in fit and predictive performance.

Abstract

Bitcoin's price has been described as following a power law (PL) in time, $P \sim t^β$ with $\hatβ\approx 5.7$ over 2010-2026. We test this claim using the Clauset-Shalizi-Newman protocol applied to Bitcoin's tail-relevant distributional series, and develop three principled time-domain adaptations of the protocol. We find that (i) the distributional power law is rejected on UTXO balances and daily |returns|, with lognormal preferred decisively; (ii) the fitted time-domain exponent varies by nearly a factor of three across reasonable shifts of the time origin -- it is not specification-robust in the sense required for a shift-invariant structural reading; (iii) standard residual diagnostics and scale-invariance tests proposed in earlier work cannot distinguish a power law from a multi-component sigmoid stack fit to the same data; (iv) Bitcoin price stands apart in a cross-asset comparison spanning Bitcoin on-chain metrics and traditional asset classes: it is the only series in the nine-series in-sample test where no single-component growth curve improves on the power law, and the quarterly $K=3$ wave-stability bootstrap rejects the PL+AR(1) null on Bitcoin at $p = 0.015$ (strict 15% CV threshold) -- a clear cross-asset separation, although not a Bonferroni-robust rejection; and (v) walk-forward Diebold-Mariano evaluation against ten candidates -- including standard time-series baselines (RW with drift, auto-ARIMA, ETS, local-linear-trend) -- shows the in-sample winner (multi-sigmoid) is among the worst long-horizon forecasters, while the simple power law dominates 12-24 month horizons against every standard baseline at $p < 0.05$, precisely because it does not commit to specific wave shapes. The fit-prediction tradeoff is the practical counterpart of the descriptive findings.

Problem

• power law
• Bitcoin price
• time-domain analysis
• forecasting
• structural robustness

Innovation

• power law
• time-series forecasting
• Clauset-Shalizi-Newman protocol
• Bitcoin price dynamics
• model robustness