Bitcoin Power Law & log regression bands
No predictive value
Plotted with both axes logarithmic, Bitcoin’s price since 2010 falls close to a straight line. The power-law reading treats that line as a law of growth and extends it forward, sometimes with bands around it, to say where price belongs in any given year.
- Bitcoin price (monthly close)
- Power law fitted on data to Dec 2017, extended
Our own calculation on Coin Metrics Community Data (BTC reference rate, daily, through 14 August 2026). The dashed line is an ordinary least-squares fit of log(price) on log(days since the genesis block) using only data up to 31 December 2017, then extended without refitting — the honest out-of-sample test. Reproducible from the public series.
How it is built
log(price) is regressed against log(time since the genesis block). The result is a straight line in log-log space, and bands are added as offsets or percentiles around it. The only inputs are price and elapsed time.
Our verdict
No predictive value in the form it is usually shown. The underlying idea — that network growth drives value — is worth taking seriously; a straight line drawn against elapsed time is not that idea, it is a shape.
The record
Every line below is our own calculation on the public price series — reproducible where it is our own calculation, and linked to its primary source where it is quotation.
| When | What the chart says | What the data shows | Source |
|---|---|---|---|
| 2017-12 | Fitted on all data through 2017, the power law describes that history well: R² of 0.909 on log-log axes. | We generated 2,000 pure random walks with the same drift and volatility as Bitcoin over that period, containing no power law of any kind. Their median R² in the identical regression was 0.917 — higher than Bitcoin’s. The real data sits at the 44th percentile of pure noise. | Moonkelp’s own analysis |
| 2017-12 → 2026-05 | The fitted curve, extended without adjustment, is the model’s forecast. | It read $24,361 for November 2021, when Bitcoin was $64,756 — 2.7 times too low. It read $137,390 for May 2026, when Bitcoin was $76,620 — 44 per cent too high. The same line was wrong in both directions. | Moonkelp’s own analysis |
| 2026-08 | The bands drawn around the curve are said to contain the price. | They do, and that is the problem: a ±2σ band from the 2017 fit spans $26,456 to $827,933 today — a factor of 31 — and has contained 100 per cent of the 3,148 out-of-sample days since. A range that excludes nothing cannot inform. | Moonkelp’s own analysis |
| 2026-08-14 | The curve is the fair value price returns to. | It read $148,000 for 14 August 2026. Bitcoin was $62,925 — 57 per cent below the line, the largest gap the 2017 fit has produced in either direction. | Moonkelp’s own analysis |
| 2021-12 / 2026-08 | The exponent is a property of Bitcoin. | Refitting the identical regression at different cut-offs gives 5.728 (2017), 5.889 (2021) and 5.652 (2026). It moves with the data it is shown — and it has now moved twice in opposite directions. | Moonkelp’s own analysis |
Why the method does not hold
01
On a series that trends upward over orders of magnitude, a log-log fit will always look excellent. A high R² here is a property of the axes, not evidence about Bitcoin.
02
Power-law fits are notoriously hard to distinguish from log-normal alternatives on limited data — the standard reference on this is Clauset, Shalizi and Newman’s work on power-law distributions in empirical data. Eyeballing a straight line is not a test.
03
Time is not a cause. A model whose only explanatory variable is “how long Bitcoin has existed” contains no mechanism that could break, which is another way of saying it makes no risky prediction.
The strongest case for it
This is the most serious candidate in the study, and it deserves a serious hearing: proponents argue the line reflects network growth — adoption, hash rate and users compounding together, in the spirit of Metcalfe’s law — rather than time as such. That is a real mechanism and would be a genuine model. The test is whether the mechanism is measured or merely invoked: if the fit is performed against elapsed time and only narrated as network growth, the mechanism is doing no work in the arithmetic.
What to look at instead
Network growth can be measured directly rather than assumed: fees paid, DEX volume and value settled on-chain say what a network is actually being used for, and they can fall as well as rise.
Questions
Does Bitcoin really follow a power law?
Its price plotted on log-log axes has stayed close to a straight line so far, and that is a real observation. It is much weaker than it sounds: series that rise over several orders of magnitude generally look straight on those axes, and the fitted exponent here changes — 5.73, 5.89, 5.65 — depending only on where the data is cut off.
Is a high R² proof that the model works?
No, and this is measurable. We ran 2,000 random walks with Bitcoin’s own drift and volatility but no power law built in; their median R² in the same regression was 0.917, against 0.909 for the real price. A fit that pure noise reproduces slightly better is not evidence of a law. The test that matters is out-of-sample, and there the 2017 fit missed the 2021 peak by a factor of 2.7.
What about the network-growth argument?
That version is worth taking seriously — the claim is that adoption, users and hash rate compound together and drive value, in the spirit of Metcalfe’s law. But that mechanism has to be measured to do any work. A regression whose only explanatory variable is elapsed time contains no adoption term at all; the network story is narration alongside the arithmetic, not inside it.
Sources & method
Every figure on this page comes from one public price series and arithmetic anyone can repeat. Where a claim would need a source we could not verify, it is not on this page.
- Coin Metrics Community Network Data — BTC reference rate and hash rate, daily — the price and hash-rate series every figure on this page is computed from
- Moonkelp’s own analysis — the calculation itself — method described above, reproducible from the series
The other charts
Perspectives, not investment advice. This page criticises a method, not the people who publish it — and it says so where a construction does something real. How Moonkelp works
This study is reviewed quarterly; corrections normally ship within days. Corrections:contact@moonkelp.com