Bitcoin’s power-law really debunked?
A more detailed explanation of the time-based power-law
Harold Christopher Burger and Giovanni Santostasi
In this article we’re going to review an article entitled “Bitcoin’s power-law corridor debunked” written by Tim Stolte of Amdax Asset Management, claiming that bitcoin’s power-law, as originally described by Giovanni on reddit in 2018 and revisited by Chris on medium in 2019, has been “debunked” and shows nothing more than a spurious correlation, i.e. it looks correct because of sheer luck.
“Flawed”, “highly inadequate”, “ridiculous”, “irrational”, “no logic nor wisdom, just pure guesswork”, “logically and statistically invalid”, “We might as well draw price predictions by hand while we’re at it”, “useless”: Tim does not mince words when it comes to his critique. Yet, the out-of-sample forecasts made by the power-law model have held up incredibly well since first publication in 2018. Surprising for a spurious model! What gives? Is Tim right, or is his article as flawed as he claims the power-law is? Let us review.
Tim’s criticism can be summarized into two main points.
- Tim spends roughly half the article criticizing the use of a log transformation on time, claiming it does not make sense.
- Tim claims that the model does not display cointegration, and is therefore invalid, showing a mere spurious correlation.
We were most surprised by Tim’s first claim, as we thought that an aspiring econometrician should easily and intuitively understand the approach. We will take this opportunity to explain the log-log transformation behind the time-based power-law in more detail than we have so far. It should then become obvious why Tim is wrong.
Regarding cointegration, Chris has co-authored an article with Peter Vijn demonstrating that Tim’s (and others’) claims about cointegration are bunk. We will therefore refer the reader mostly to that article. However, Tim makes a number of claims not completely related to cointegration that are patently false and we will refute them here.
For clarity we note that Tim actually refers to Chris’ 2019 article and therefore refers to Chris when he says things like “he uses”. For the purposes of this article there is no meaningful distinction between Giovanni’s and Chris’ description of the time-based power-law. Therefore a criticism of one article applies equally to the other.
Log transformation of time: The log-log plot
The time-based power-law is best displayed in a log-log plot — a plot on which both the x-axis and y-axis are scaled logarithmically. The main reason to plot the log of price vs the log of time is to identify a possible scaling relationship between these two quantities. Data often has a simple relationship with observable quantities like time. The simplest relationship possible is a linear relationship where one quantity changes proportionally with another.
When the data is not linear, it often follows exponential growth, in particular when there is a process with a constant doubling time (for example the growth of a population). Exponentials are easily identified even without sophisticated regression or fitting methods by visual inspection, namely by checking if the data looks linear when plotting the data in a log-linear chart, where we take the log of the y-axis but leave the x-axis linear. The log of the y-axis in a sense “cancels” the exponential and this is why the exponential looks linear in such a chart.
This type of graph is often used in finance, in particular when showing the performance of an asset over a long period of time where the asset may have grown by several orders of magnitude. Populations can also grow exponentially for a limited amount of time.
Bitcoin’s USD price is not exponential because it doesn’t look like a straight line in a log-linear chart. It is curved downwards, indicating that the growth of BTC is not accelerating as fast as an exponential process but is in a sense “slowing down” (diminishing returns) over time.
It is quite obvious, though, that there is a regular long-term pattern in the data and that one could fit a curve through it and find out if there is a simple equation that describes the general growth of the price.
Physicists are trained, after failing to find simple relationships like linear or exponential, to plot data in log-log charts. So differently from what Tim claims, it is a standard mathematical transformation — definitely in physics, but also in financial time series, as in Sornette [1].
A log-log chart means plotting the logs of both variables. The x variable doesn’t have to be time. It can be the size of an animal (in terms of its body weight) for example vs the size of the brain of that animal. In the following graph one can notice that the values of the x and y axis are given in terms of “scale” or changes in size or order of magnitude, in other words 0.1s, 10s, 100s, 1000s of kg for the weight of the brain and similar changes in the body weight. When we see a chart that has these equally spaced values in terms of scale then we are dealing with a log-log chart. The idea is to emphasize how the scaling of a quantity changes with respect to the scaling of another. There are important reasons to compare how the scaling of a quantity changes with the scaling of another because if we see a simple linear relationship in such a chart then we are dealing with some general process that regulates the change of the size of a quantity with respect to the size of the other.
Such a relationship is quite likely not due by chance but rather indicates that something important is organizing this scaling proportionality, a physical or biological organizing “law”.
In physics we call these simple relationships between the scaling of physical quantities “power laws” because it can be shown with some simple math that a straight line in a log of x vs the log of y implies a relationship of the kind y=A*x^n where n is called the “power” of the law and it is a number specific to the process being studied. In fact, measuring n can tell us a lot of things about the nature of the process itself, almost like a fingerprint.
Power laws are ubiquitous in nature and human made phenomena and they have many interesting properties and consequences in terms of the processes that created them, their stability and universality. Here is a very informative video that discusses the relevance and importance of power laws in nature and human made phenomena. And here is a famous book by Geoffrey West on why studying how systems scale up (often via power laws) is a very interesting and a valid scientific endeavor.
A log log plot is also how Kepler discovered his famous laws by plotting the planets’ log of the distance from the Sun against the log of the time it takes a planet to orbit around our star. He was amazed to find a perfect straight line — a power law!
Giovanni, being a physicist and passionate about BTC and having worked on BTC models for several years, decided to plot the log of the price vs the log of days since the Genesis Block. He was also astonished that the data looked like a straight line!
It is evident from simple visual inspection that in a log-log plot, the price evolution does not curve any more — it is “straight”. It is true that there are large deviations from this linear trend and the data doesn’t look as nice as in the case of the planets (it is after all a highly random or stochastic process) but interestingly enough, the general path of the growth of the price is a clear straight line. One doesn’t even need very sophisticated tools to see that we are dealing with a straight line (with large oscillations around the general path).
Notice that this straight line behavior covers 6 orders of magnitudes (from about $0.05 to almost $70,000), indicating that the behavior of BTC followed this particular growth when it was only a few cents, through a few dollars, to thousands and tens of thousands of dollars. That is the true usefulness of the log-log graph that is completely missed by some of the critics of this approach.
So far, we are not talking about finding a particular statistical model that passes all possible tests but showing in very general terms that BTC has very regular scaling properties, that it seems that its growth in price by orders of magnitude requires equivalent (though not identical) changes in orders of magnitude of time, so for the price to increase from 1 to 10 dollars requires equivalent changes in time from a few days to weeks, changes from 10s of dollars to 100s requires months (100s of days) and so on. You do not really need sophisticated math to show this — visual inspection of the log-log plot is sufficient. It is obvious. Intuition and understanding should come before sophisticated mathematical analysis. Without understanding and intuition using mathematics is a dangerous affair. And what is missing on Tim’s side is specifically this understanding.
The beauty of finding a simple relationship in time for the price evolution is that it allows us to make predictions and extrapolations for the future because we can simply plug any future time (expressed as days from the Genesis Block) into the formula and obtain a “fair” value for the price (or the mean or trendline price). If the real price of BTC is below such a fair value it means the price is undervalued and if it is above the price it is overvalued. Identifying a general trend is also useful so it can be used to express prices as a deviation from this trend and identify different phases of the BTC cycle, as shown in the graph below. So besides being a valid scientific hypothesis our power law model is also useful from a practical point of view. Chris has written an article about this.
As any scientific hypothesis, it is falsifiable and future data will show if BTC continues to scale up in this very predictable fashion or if something changes in how it grows. This is another useful contribution in identifying BTC price evolution as a power law because any changes in the future of the currently observed law will indicate a change of regime and it would be an equivalently interesting and useful observation.
In general, data can always be transformed using any chosen mathematical expression or process if the transformation allows us to see interesting patterns that cannot be seen easily in the usual ways of displaying the data like in linear graphs (one can use polar coordinates, transform the data into frequency space via Fourier transform and so on). The important thing is to then correctly interpret what the data tells us through the transformation.
Now, back to debunking the debunker
It is clear that Tim struggles with the concept of logarithmically scaling time, and using a critical point in time (the date of bitcoin’s Genesis block in this case):
“Logarithmically scaling time is possibly the weirdest thing I have ever seen in time series analysis.”
“A scaled time axis, on the other hand, makes no sense at all”
“If we were to log-scale time, we are effectively modelling the real-world time to pass increasingly faster. That is ridiculous.”
“He uses an absurdly arbitrary starting integer that equals the number of days since 1 January 2009, which equals 577 in my case (it’s 563 in the original model). There’s no logic or wisdom there, just pure guesswork and picking whatever looks nice…”
The statements reveal a deep misunderstanding of our method, which, as explained, is quite standard in disciplines like physics. In addition, non-linear transformations of time are not limited to physics — using such transformations on the time component is a common operation in financial time series analysis [1]. Tim being weirded out by something so trivial is a shortcoming on his side, not ours.
Use of a non-linear transformation on time
Let us take apart one specific statement:
“A scaled time axis, on the other hand, makes no sense at all, because we want the change from 2022 to 2023 to be exactly equal to the change from 2011 to 2012. The amount of time in a year is constant. If we were to log-scale time, we are effectively modelling the real-world time to pass increasingly faster. That is ridiculous.”
This is nonsense. The correct interpretation is that the bunching-up of time seen in the log-log chart reveals that it takes more and more time for price to go up by factors of 10. That is exactly what happens and it doesn’t require changing the nature of time in real life.
Let us make this even more clear by using an analogy: let's consider the trajectory of an object in free fall. The distance traveled by the object is given by a function of the square of time: s = ½ * a * t² (a power-law with a power or 2). Here t refers to the time elapsed since the object was released, very similarly to how we use the amount of time elapsed since the Genesis block. We can plot the distance traveled on a log-log plot, with log time since release of the object on the x-axis and the log of the distance traveled on the y-axis. We discover a straight line with a slope of 2, correctly identifying the power factor of 2.
We see that it can be quite natural to represent time in a different than linear scale. The trajectory is described very neatly using a log-log plot.
Choice of the genesis block as a starting time
Our time-based power-law uses a starting point in time which is equal to the date of Bitcoin’s Genesis block. Tim finds this choice arbitrary, even suggesting it was chosen in such a way as to artificially produce a seemingly linear relationship between time and price which should not be there:
“He uses an absurdly arbitrary starting integer that equals the number of days since 1 January 2009, which equals 577 in my case (it’s 563 in the original model). There’s no logic or wisdom there, just pure guesswork and picking whatever looks nice…”
Yet the Genesis block seems like a very logical choice for the starting point of the model: The Genesis block is the beginning of the process we are studying. Again using the analogy of an object in free fall (as above), you would use the instant you let go of the object as the starting point: time should be counted from then on.
Mathematically, consider the expression of the model:
log(price) = a*log(days since Genesis) + bWhen we go back in time and approach the Genesis block (the number of days since the Genesis block approaches 0), the model price approaches 0. When we move forward in time (the number of days since the Genesis block increases), the model price increases. Hence, the built-in assumption of our model is that bitcoin has had (an implicit at first) price since the Genesis block (January 3rd 2009). This does not seem like an unreasonable assumption because bitcoin’s hashrate has been non-zero since the Genesis block, meaning that people assigned value to bitcoin (it had a price to them). Thus the Genesis block can really be seen as the beginning of the process we are studying: bitcoin’s price history.
If this a priori choice still seems too arbitrary: it is possible to slightly modify the model so as to contain a third free parameter (let’s call it c) which controls for the date of Bitcoin’s price Genesis:
log(price) = a*log(days since c) + bThis is exactly what Peter Vijn has done. Fitting c to Bitcoin’s price data yields a date that hovers around the Genesis block, with very similar price forecasts as the version of the power-law advanced by Giovanni originally and later by Chris.
Yet another statement betrays Tim’s confusion:
“Again, note the ambiguity of this interpretation. How can time move 1%? This time movement is fully dependent on the arbitrary starting date of 1 January 2009. Moreover, the further we move away from 2009, the more days fit into that 1% time interval. So we are not able to convert that 5.88% to average daily return or anything similar, giving us basically no understanding of the estimation results.”
It is not “time” that moves 1%, but the time elapsed since the Genesis block that moves 1%. We don’t understand what Tim means with “giving us basically no understanding of the estimation results”, as the model is easily interpretable. Maybe power-laws are not quite as easy to understand as exponentials, but it should still not be very difficult to understand. Let’s give it a try: We’re going to investigate how the price behaves when we increase the amount of time to the Genesis block by a scalar amount (e.g. 1% more time has passed). Let p1 be the initial price and p2 be the price after a scalar amount s more time has passed. We have:
log(p1) = a + b*log(d),
log(p2) = a + b*log(s*d) = a + b*(log(s) + log(d))We’re interested in the ratio of the current price p1 to the new price p2 (the price after 1% more time has passed). We’re going to take the log of that ratio for mathematical convenience:
log(p2/p1) = log(p2)-log(p1) = a + b*log(s*d) - a - b*log(d) = b*log(s) We note that this expression depends only on the power coefficient b, and not on a. As an example, for the price to increase ten-fold, log10(p2/p1) = log10(10) = 1. Solving for s we get s = 10^(1/b). By fitting the time-based power-law on bitcoin’s price history until now we obtain b=5.723, giving us s = 1.495. In other words, for the price to increase 10-fold, the amount of time elapsed to the Genesis block must increase by about 50%. Quite an intuitive result.
Part 2: cointegration
Use of regressions
For some reason, Tim is opposed to the use of regressions:
“He makes his prediction based on the linear approximation, which he obtains by means of a linear regression. I am strongly opposed to this particular method for this particular purpose. If you want to make price predictions, don’t use regressions, but just calculate some growth rate and extrapolate it into the future.”
Maybe he is confused as to what a regression is. Indeed what we are doing is estimating growth rates and extrapolating them into the future.
“Why does he still use a regression for his predictions? One reason is that it sounds sophisticated even though it’s very easy to implement.”
It is ridiculous to say that we use regression because it “sounds sophisticated”. We are disappointed in what appears to be an ad hominem attack.
Trends are usually described with the help of regressions, e.g. Moore’s law, or estimating the stock market’s long-term returns.
It is also important to note that regression is the standard method to extract the power n of a power relationship because as explained above, in a log-log graph the data looks linear and the slope of the regression fit turns out to be the power of the power law and this number is very important because it often reveals the nature of the process underlying the power law. In the case of the object in free fall above, it is regression that allows us to correctly identify the power of 2 (so n=2) inherent in the law of gravity. We therefore find Tim's opposition to regression bizarre and unfounded.
As a side note, it turns out that many biological processes have a value of n that is less than 1 (also called sublinear power laws) while human made phenomena like social networks (cities, nations, companies) show values of n above 1 (or superlinear). It is an interesting observation that n in the case of BTC is around 5.82 — larger than 1, which is what is expected for a human-made network-like entity.
Problems with R²
The coefficient of determination R² of the time-based power-law increases over time:
This is normally interpreted as being a good sign: A high R² is indicative of a good model fit. Yet Tim claims that the increasing R² of the power-law fit is actually expected:
“His conclusion is that his model performs exceptionally well and even increasingly better over time. This is, as it appears, a false conclusion. In statistics, there is a well-known case in which the R-squared of a linear regression approaches its maximum value of 1 as we add more and more observations, regardless of how good the model fit is. This happens if both variables (log-price and log-time) are 1) non-stationary and 2) not cointegrated. [3]”
The reasoning seems to be as follows. If the two variables (log time and log price) are both 1) non-stationary and 2) not cointegrated, then in the long run, R² MUST approach 1, and supposedly this is what is happening here. This is a bizarre statement on multiple counts:
- Tim himself seems to acknowledge that at least the log price variable might be stationary, hence these two conditions are not met even according to his own criteria.
- He cites a source [3] to support his claim that under these two conditions, R² must tend to 1. The problem is that his source does not make any such claim (though it does claim that “moderate” values of R² are expected under some assumptions).
- It should be obvious that Tim is confused, and that under the two named conditions, R² need not tend to 1. It is trivial to construct non-stationary, non-cointegrated time series, and show that the R² does not tend to 1: We can generate two independent random walks. Those are non-stationary by construction and will also not be cointegrated. The coefficient of determination will not tend to 1 over time. As more observations are added, it becomes LESS likely that the coefficient of determination approaches 1.
What Tim probably means to say is that it is POSSIBLE for two time series which are non-stationary and not cointegrated to have a high coefficient of determination even though there is no relation between the two. Indeed in that case the regression is spurious. However, as more observations are made (more data is collected), this problem gets better, not worse, precisely because it is spurious.
So Tim’s statement is completely wrong. The increasing R² value of the time-based model is actually evidence of a fit that continues to be good.
We further note that the coefficient of determination is just one method by which the quality of the model fit can be estimated. For example, the mean squared error (MSE) of the model fit decreases with time, which also points towards increasing evidence of a good fit:
Finally, the power-law parameters themselves stabilize over time, with the power-law being virtually unchanged since roughly 2017 or so:
The parameters can of course never totally stabilize due to the fact that the price of bitcoin has strong variations (it’s a random variable). The parameters can only stabilize if new observations are in line with the forecasts of previously estimated parameters. Hence stabilizing parameter values are a certain sign that the fit continues to be good.
Supposed lack of cointegration
We highly encourage our readers to read the article by Chris and Peter Vijn on the time-based power-law and cointegration, which will give the reader a much deeper understanding of the subject matter. The article also clearly demonstrates that the time-based power-law is strong from a statistical point of view. We will here give a short overview of the problems with Tim’s statements on cointegration.
Cointegration is a property that exists between two or more non-stationary random variables if a linear combination between them can be found which is stationary. More intuitively, two non-stationary random variables (time series) are cointegrated if they have a common stochastic drift in the long term (an intuitive summary we owe to Tu et al [2]). The existence of cointegration is seen as a positive for a model that describes one stochastic non-stationary random variable in terms of another because it means that the error term is not expected to drift over time — the fit is expected to stay good over time. This is why Tim sets out to look for the existence or non-existence of cointegration between log time and log price.
Now, the problems:
Both variables must be stochastic for cointegration to be possible. Time is emphatically not a random variable, but a deterministic variable. Tim throwing log time into a cointegration test does not make sense because it violates the assumptions of what cointegration is. In fact, the very first sentence in the paper Tim cites is very clear about this: “If there exists a stationary linear combination of non-stationary random variables, the variables combined are said to be cointegrated”. Tim writes “log-time is non-stationary by construction” but this is wrong, it is deterministic and does not belong in a cointegration test.
Next: Both variables must be non-stationary for cointegration to be possible. We will not investigate log time, as it is already out of the picture because it is deterministic. What about log price? Tim is ambiguous about whether log price is stationary or not:
“Hence, we are not able to firmly reject non-stationary and conclude that there are signs of non-stationarity in the log-price.”
In that case, he should also have been ambiguous about whether a cointegration test is reasonable or not (disregarding his mistake regarding log time). But the bigger problem is that log price is actually trend-stationary! Hence looking for cointegration makes no sense. Both variables violate the necessary assumptions.
So the existence of cointegration between log time and log price is impossible because the assumptions are violated. Is Tim correct in claiming that therefore, “we are dealing with a so-called spurious regression”? Even though it is not possible for cointegration to exist in this model, it is possible to determine if the model error (the residuals) are stationary or not. A stationary model error achieves the same effect as the presence of cointegration: No drift in the error, hence the model can be reliable over time. Had he eye-balled the model residuals or done a stationarity test on the model error correctly, he would have determined the error to be stationary and hence fine from a statistical point of view — as is actually the case.
Tim’s section about the coefficient of determination (R²) and cointegration is wrong from start to finish. The premise that R² necessarily increases under his stated conditions is wrong (and the conditions under which this would supposedly happen are also not present). His claim that log time is random and non-stationary is wrong. His claim that log price is ambiguously stationary or non-stationary is wrong. His attempt to look for cointegration is wrong. Therefore, it is no surprise that his conclusion that the time-based power-law is statistically invalid is also wrong.
Conclusion
We were initially inclined to politely ignore Tim’s unfortunate write-up, but have changed our minds due to the fact that a lack of a reaction on our side might be interpreted as an implicit acknowledgement. With this write-up we hope to not only have clearly demonstrated that all of Tim’s article is hopelessly misguided, but also to have provided some more insights into the motivations behind the time-based power-law model.
We remain highly confident that bitcoin’s long term price history is going to be adequately described by the power-law model.
References
- “Why Stock Markets Crash. Critical Events in Complex Financial Systems” Didier Sornette, 2002.
- “Universal Cointegration and Its Applications” Tu et al., including supplemental information
- Phillips, P. C. (1986). Understanding spurious regressions in econometrics. Journal of econometrics, 33(3), 311–340.
