Sensitivity analysis: how stable are the Power Law parameters?

Fairly stable, but less precise than the three decimal places suggest. We reran the regression with every data end from 2014 to 2026: since 2016 the exponent has stayed between 5.62 and 5.89, and the line for 1 January 2030 between $330’000 and $537’000. The published exponent of 5.688 sits in the middle of that range. How certain it is, though, depends heavily on how honestly you calculate the uncertainty.

Why the parameters can move at all

The Bitcoin Power Law is a straight line on a log-log chart: days since the genesis block on one axis, price on the other. Two numbers define this line. The slope, the exponent b, says how fast the price grows with time. The intercept a moves the whole line up or down.

These two numbers are not a law of nature. They come from a regression over the daily prices so far. When new prices arrive, a crash enters the data, or the first years are left out, a and b change. The question of this article is: by how much? And what does that mean for the values you plan with?

The fundamental criticism of the model, such as whether a straight line on log-log axes proves anything at all, is in Criticism of the Bitcoin Power Law. Here we look only at the numbers themselves.

Test 1: how do a and b change with the end of the data?

The most obvious test: pretend today is 31 December of an earlier year and run the regression only on the prices up to then. That shows what someone would have calculated at the time.

Three things stand out.

The first years were not enough. With data up to 2014 the regression would have put 2030 at almost $3.9 million, ten times today's value. Four and a half years of data with two huge bubbles (2011 and 2013) are not enough for a stable slope.

Since 2016, b has moved within a band. All values lie between 5.62 and 5.89. A high at the end of the data makes the line steeper: from the end of 2020 to the end of 2021, b rose from 5.74 to 5.89. A bear market at the end makes it flatter: with the end of 2022, b fell to 5.82, with the end of 2023 to 5.71. The prices at the edge pull the line along, because on the time axis they lie farthest from the centre of the data.

2026 pushed the line down noticeably. At the end of 2025 the regression matched the published formula almost exactly (5.700 against 5.688). Ten months of prices at around half the line lowered b to 5.636 and took 9 % off the value for 2030.

Test 2: how precise is the slope? The standard error

Statistics software gives a standard error for every slope. For all 5’957 days it is 0.015. The 95 % interval would then be about 5.61 to 5.66, and the line for 2030 would lie between $354’000 and $373’000. That sounds very precise.

But this number is far too optimistic. The formula assumes that every day is independent new information. With Bitcoin that is not the case: if the price is 50 % above the line today, it will almost certainly be there tomorrow too. The deviations from the line hang together for months. So 5’957 days contain much less information than 5’957 independent measurements.

Only after about a year is today's deviation forgotten. Cycles last about four years, and the data contain only four of them. That trending series moving together produce far too small errors and too high an R² in a regression has been known in statistics since Granger and Newbold (1974).

Taking this dependence into account makes the standard error about ten times as large:

So the honest answer is: b very probably lies somewhere between about 5.2 and 6.1. The three decimal places in 5.688 are a choice, not measurement precision.

Test 3: which start date?

The site's data start on 18 July 2010. Some Power Law calculations start later, because the first prices come from a few thinly traded exchanges. Leaving out the early years makes the line flatter:

Starting in 2013 instead of 2010 lowers b by 0.4 and the line for 2030 by a fifth. The reason: on a log time axis the first years are long. From day 561 (July 2010) to day 1’459 (January 2013) the x-axis has two thirds as much room as from 2013 to November 2026. Leaving these years out drops the points that pin down the slope most. What remains is a shorter stretch with large cycles on it, and its tilt depends heavily on where the data end.

Test 4: which weighting?

Weighting every day equally means: 2025 counts with 365 points, 2010 with only 167. On the log time axis, though, recent years lie close together. So you can also weight differently:

Weighting the early years more makes the line steeper, weighting the latest more makes it flatter. That fits what the site's research paper describes: recent years lay below the line, above all the highs have become smaller. Between the flattest and the steepest variant lie 0.33 in the exponent and 37 % in the value for 2030.

Test 5: what does a change of 0.1 in the exponent do?

There is a trap here. If you change only b by 0.1 and leave a as it is, the value for 2030 changes by the factor 10^(0.1 × log10(7’668)) = 2.45. $401’000 would become $981’000 or $164’000. But that calculation is meaningless, because such a line would lie far away from every price so far.

a and b are linked. If the line gets steeper, it must start lower to still fit the data. It turns around the centre of the data, which on the log time axis lies in April 2017. The farther a date is from it, the more a different slope matters.

So a tenth in the exponent moves the value for 2030 by about 9 to 10 %, and for 2040 by 12 to 14 %. That is noticeable, but not a factor of 2.5. The last row is interesting too: with the published exponent, the best intercept for all the data is −16.514, only 0.021 away from the published −16.493. That is a 5 % difference in price.

What the site does with this: fixed parameters

bitcoin-powerlaw.com calculates the floor and the power law line with the published parameters a = −16.493 and b = 5.688. Other Power Law pages use the same values. They are fixed: a day's floor is the same on all these pages and never changes afterwards.

Until November 2026 the site reran the regression with every new daily close. You can see the result in Test 1: a weak 2026 lowered the floor by 9 % within ten months. A floor that gives way whenever the price falls is of little use as a lower bound. So today:

• The formula is the model. It does not change with every price.
• The regression is the check. It keeps running with every new daily close, and the methodology page shows it beside the formula. Up to the end of 2025 it confirmed the formula to within about 1 %; with all prices up to November 2026 it is 8.6 % below.
• The formula stays within the uncertainty. b = 5.688 lies in the middle of the band the regression has moved in since 2016, and well inside the honest 95 % range of about 5.2 to 6.1.

Switching to new parameters would only make sense if the data showed a clearly different slope, not as soon as a bear market pulls the regression down for a year. The clear test comes by itself: according to the formula, the floor passes the all-time high of October 2025 in December 2028. If Bitcoin is still below it then, the price is below the floor and the model no longer holds.

What this means for your planning

The sensitivity mainly affects long horizons. For the next one or two years, other parameters move the lines by only a few percent, because today is close to the centre of the data. For 2030 it is already about 10 % per tenth in the exponent, for 2040 more.

So calculate with ranges, not a single number. If your plan only works if the line in 2030 is exactly at $401’000, it is too tight. Check it with a line at $330’000 and a floor at about $140’000 as well. What the values for 2030, 2035 and 2040 look like in the site's model is in the Bitcoin forecast 2030.