Testing Bitcoin Power Laws for Robustness and Forecasting Value
Summary
This study tests claims that Bitcoin price follows a power law over time and examines whether the pattern is structurally robust or useful for forecasting. It applies the Clauset–Shalizi–Newman protocol to Bitcoin’s tail-relevant distributions and develops three time-domain adaptations. The reported distributional tests reject power laws for UTXO balances and absolute daily returns, favoring lognormal models; fitted time exponents also shift substantially when the time origin changes. The authors find that common diagnostics cannot separate a power law from a stack of sigmoid curves on the same data.
In cross-asset comparisons, Bitcoin price is the only tested series for which no single-component growth curve improves on the power law. A quarterly wave-stability test rejects the power-law-plus-AR(1) null at the stated threshold, but the authors say this is not robust to a Bonferroni correction. In walk-forward comparisons, the multi-sigmoid in-sample winner forecasts poorly, while the power law outperforms standard baselines at 12–24 month horizons. These results are specific to the study’s data, candidate models, and evaluation design.
Key ideas
- The study evaluates both distributional power laws and time-domain power-law fits for Bitcoin.
- Power-law fits vary with the chosen time origin, limiting a shift-invariant structural interpretation.
- The reported distributional tests favor lognormal models for UTXO balances and absolute daily returns.
- In-sample fit quality does not guarantee stronger long-horizon forecasts in the tested candidates.
- The power law outperforms standard baselines at the stated forecast horizons, while one cross-asset rejection is not Bonferroni-robust.
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Full text
# Bitcoin's Power Law: Weak Structure, Strong Forecasts # Bitcoin's Power Law: Weak Structure, Strong Forecasts 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.
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