Paper: arXiv 2611.13962

Authors: Carlos Baquero, Raquel Menezes

Abstract

Plotted against time since its first block on logarithmic axes, the price of Bitcoin lies close to a straight line, and a power law in time has been proposed as its long-run trend. A good fit is weak evidence: when deviations from a trend are as persistent as a random walk, regressions on time find trends and high $R^2$ where none exist. We first bound what fifteen years of weekly prices can reveal, then test only what is answerable. Using a few low-frequency averages, invariant to the noise scale and calibrated over a declared family of persistent deviations with a four-year cycle and declining volatility, we test constant drift in expected log price against an added log-age term. The procedure was fixed, validated and publicly hashed before it was applied once. Constant drift is rejected under this family at nominal 5%, and a log-age coefficient of 5.7, close to the published exponent, is not rejected. The rejection does not survive allowing persistence beyond a random walk, and a Gompertz curve with a ceiling fits about as well as the power law. Whether deviations revert to the trend cannot be decided. Registered forecasts for 2027-2030 will be scored by future prices.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper demonstrates strong empirical rigor through its careful statistical methodology and pre-registered analysis, combined with a high level of mathematical sophistication in handling persistent noise. The novelty lies in its robust approach to trend testing under challenging time series conditions, making it a ‘Holy Grail’ candidate.

Research Flowchart

  flowchart TD
    A[Research Goal: Test Power-Law Trend in Bitcoin Prices under Persistent Noise] --> B{Initial Observation: Bitcoin Price ~ Power Law (log-log linear)};
    B --> C[Methodology: Bound what 15 years of weekly prices can reveal; Test what is answerable];
    C --> D{Data/Inputs: 15 years of weekly Bitcoin prices; Low-frequency averages (invariant to noise scale)};
    D --> E[Computational Processes: Calibrated over persistent deviations with 4-year cycle & declining volatility; Test constant drift vs. added log-age term];
    E --> F{Key Findings/Outcomes};
    F --> G[1. Constant drift rejected at nominal 5% (under declared family of persistent deviations)];
    F --> H[2. Log-age coefficient of 5.7 not rejected];
    F --> I[3. Rejection of constant drift does not survive allowing persistence beyond a random walk];
    F --> J[4. Gompertz curve fits about as well as power law];
    F --> K[5. Whether deviations revert to trend cannot be decided; Forecasts to be scored];