The logarithmic regression model can be expressed as:
\[ \log(P(t)) = a \cdot \log(t - t0) + b \]
Where:
The logarithmic regression or power law describes many processes in nature. Historical price action of bitcoin can in first order be described using a power law as well. The diagrams below show the historical price of bitcoin and a risk metric based on deviations from the power law fit.
The model was historically quite predictive of future price action. Try out the drop down bar above to select until when data should be used for the regression. For example, even with data from December 2020, the regression predicted the 2021 tops and the 2022 low quite well. The same is true for the 2017 bubble with data until December 2016. In fact, the data from 2016 - which includes only 6 years of price action - still provides a good estimate today (Dec. 2024).
Note that the fit requires selecting an inception time, \(t0\). While this parameter can be fitted, I chose this value manually to a time around the first bitcoin block. The resulting fit is then simplified to a standard linear fit on log-price and log-time. While later data is barely affected by this selection, earlier predictions are more sensitive to this choice and choosing the correct value in 2016 might have been difficult without the benefit of hindsight.