🔍Common Objections to the Power Law

Welcome to the next edition of Bitcoin Insights. This week we’re addressing one of the more controversial topics in Bitcoin at the moment, the Power Law and whether it can live up to the hype.

Thoughtful, data driven debate and constructive criticism are essential for progress, and for a better understanding of the power and limitations of the tools at our disposal.

This article aims to engage with different perspectives of the power law to refine our approaches, clarify assumptions, and enhance our understanding of what this model can and cannot do. We cover both common criticisms of the power law and the (often misunderstood) concept of autocorrelation.

Let’s get started!

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Table of Contents

1. Criticism #1 “Its Too Simple”

2. Criticism #2 “All Models Are Useless”

3. Criticism #3 “Forecasting the Future is Futile”

4. Autocorrelation: Why is it so Misunderstood?

5. Key Takeaways

Criticism #1 “Its Too Simple”

The most prevalent criticism of the Power Law model is that it oversimplifies the world, suggesting that all events in Bitcoin's future cannot be modeled based on just a single line on a chart.

This is an interesting comment that requires discussion on what’s known as “multi-scale analysis”. When analyzing complex systems, we often observe that although a system may have complex behavior at the micro level, we often find rather simple patterns at the macro level.

For example, when observing a flock of birds flying together, each bird follows a complex set of rules for how to fly. However, the flock often creates a consistent “V” pattern. Although billions of variables are at play at the micro level, they aggregate into a simple macro pattern.

This example illustrates how, despite the micro chaos and volatility in Bitcoin, we might still observe a predictable adoption rate at the macro level that follows Metcalfe's Law.

Even though the V shape is a likely scenario, it is not a certain outcome, and it will vary slightly each time. These patterns repeat over time but not exactly in the same way.

We therefore need to ensure we mix in a dose of humility when attempting to model out future probabilities, but when done correctly it can provide useful and actionable insights.

Criticism #2 “All Models Are Useless”

Many Bitcoiners often reason that because the future is unknowable, there is no point in trying to model out these various probabilities. As such, all models including those such as the Power Law are useless.

While I sympathize with the caution regarding models predicting the future with certainty, this criticism is nuanced. The reasons for this are:

#1 Everyone has a model (whether you’d like to admit it or not)

If you ask those same Bitcoiners who criticize models, whether they believe Bitcoin will potentially reach $1M in the future, many are likely to say “absolutely”. This belief is a mental model — a model built on their past observations of fundamental growth of the Bitcoin ecosystem.

In reality though, how do they know Bitcoin’s price will rise in the future? Well, they’ve observed past events, analyzed Bitcoin’s fundamentals, and developed a thesis that certain properties should result in a future increased price—a model!

Everyone has a model of how they believe the future will play out, whether they explicitly recognize it or not!

The problems usually arise only when models are abused, causing subsequent misinterpretation, proliferated by social media apps such as Twitter which promote shorter and often imperfect communication of data.

#2 Models help improve our decision making

It is important to note that even the best models are imperfect. Models are, by definition, abstractions of reality. Done right however, they can capture many important parts.

Take Albert Einstein, for example who developed his general theory of relativity between 1905 and 1915. His theory has been one of the most important ideas in modern physics, predicting phenomena like gravitational waves, confirmed a century later. However, even Einstein's theory breaks down in certain extreme situations such as at the center of the black hole (known as the “singularity”).

As George Box, a British statistician, famously said in 1976, “All models are wrong, some are useful”. The quote emphasizes the importance of practicality over perfection. No theory or model is 100% correct, but the real measure of its value lies in its usefulness.

Einstein's work, though not flawless, has been incredibly useful. For example, the Global Positioning System (GPS) relies on relativity to provide accurate directions. Without it, our navigation systems would fail.

Criticism #3 “Forecasting the Future is Futile”

Another common criticism of Bitcoin price models is the assertion that the past cannot predict the future, and thus, the forecasting the future using models is a futile endeavor.

Before discussing the technical workings of any model, I first teach students that if anyone claims to have a model, their first question should always be “what are your assumptions?”. This is because a model is only ever as useful as the quality of the assumptions. If you put garbage assumptions in, you’ll get garbage results out.

The core assumption behind a regression projection of price into the future is that the forces that shaped the price in the past will very likely continue to shape the future in the same pattern. This is a critical assumption that any good data analyst should explicitly state it in every model.

So, can we guarantee that Bitcoin’s price will continue to evolve as it did in the past? No, but the most likely scenario is based on historical data. Deviations are possible and should not be dismissed.

The power of statistical modeling lies in its ability to capture the resilient and persistent parts of the system.

For example, the Power Law curve captures the Bitcoin’s adoption rate, which is based on Metcalfe’s Law. For now, our best guess is that this rate of adoption will continue into the future. This pattern is likely to hold in the future.

Note that most broad macro trends typically play out gradually over the long term. As long as your model captures the changes in these long-term fundamental forces of an asset, you can build a model that accurately assesses future predictions based on that.

Unfortunately, many “modelers” fail to incorporate the long term nature of these models, misleading audiences who lack statistical training. However, just because analysts are not explicit about the limitations of their own models, does not necessarily mean modeling data is a useless endeavor.

Autocorrelation: Why is it so Misunderstood?

Let’s move forward from conceptual concerns to a very technical one – autocorrelation. A common response on Bitcoin twitter is that “the presence of autocorrelation in Bitcoin's price data invalidates the model's predictions”. Given the prevalence of this misunderstanding, we believed it warranted a more thorough response.

Introduction

Many have argued that the log-log regression of price over time is invalid because of autocorrelation. However, this argument often stems from a misunderstanding of autocorrelation and the problem it causes.

What is Autocorrelation?

Autocorrelation (also known as ‘serial correlation’) refers to the correlation of a time series with its own past values. In the context of financial data, autocorrelation suggests that past price movements can influence future prices.

What Problem Does It Cause?

Ordinary Least Squares (OLS) regression is a widely used method for estimating relationships between variables. However, OLS assumes that the error terms (the difference between the observed and predicted values) are independent. When autocorrelation is present, this assumption is violated.

Although the predictions remain unbiased—meaning the average prediction is still correct—the standard errors, confidence intervals, and p-values may become unreliable.

While an econometrician needs to address autocorrelation, if the purpose is only to obtain an average prediction and not to estimate the standard errors, it does not invalidate the results.

In fact, it has been mathematically shown that autocorrelation only matters for standard errors and significance tests, not the average value.

In 'Introductory Econometrics,' the de facto undergraduate econometrics textbook, Jeff Wooldridge explains:

“What will happen if we violate the assumption that the errors are not serially correlated or autocorrelated? We demonstrated that the OLS estimators are unbiased, even in the presence of autocorrelated errors [… this] alone does not cause bias nor inconsistency in the OLS point estimates.”

OLS Estimation

To explore the impact of autocorrelation, we performed a log-log regression on the price of Bitcoin to derive a Power Law model.

Here is the regression equation derived from the OLS model:

Log(Price) = −16.5653 + 5.7067 × Log(Days)

R²: 0.9552
Intercept: -16.5653 (SE: 0.0594)
Slope: 5.7067 (SE: 0.0173)
95% CI:
Intercept: (-16.6816, -16.4489)
Slope: (5.6729, 5.7405)
p-values: (0.0000, 0.0000)

Durbin-Watson Test Results

Despite the effectiveness of the OLS model, we noted that Bitcoin prices do in fact exhibit autocorrelation.

This was confirmed by the Durbin-Watson test, which returned a test statistic of 0.0045 with a p-value from the Breusch-Godfrey test for autocorrelation of 0.0000. These results indicate that the data is autocorrelated, meaning the OLS assumption of independent errors is violated.

However, it is essential to understand that autocorrelation only affects the standard errors, not the overall predictions made by the model.

Addressing Autocorrelation with Newey-West Estimation

To address the issue of autocorrelation, we can also apply the Newey-West estimator.

This method adjusts the standard errors to account for both autocorrelation and heteroscedasticity (non-constant variance in the errors).

The regression equation derived from the Newey-West estimator is:

Log(Price) = −16.5653 + 5.7067 × Log(Days)

R²: 0.9552
Intercept: -16.5653 (SE: 0.0974)
Slope: 5.7067 (SE: 0.0276)
95% CI:
Intercept: (-16.7561, -16.3745)
Slope: (5.6526, 5.7607)
p-values: (0.0000, 0.0000)

OLS vs. Newey-West Estimation

Below are the predictions from both models plotted side by side, with the confidence intervals shaded.

As you can see, the mean predictions are identical, but the Newey-West model shows wider confidence intervals. This widening occurs because the Newey-West estimator accounts for the autocorrelation, leading to larger standard errors.

In the context of the Power Law model, our primary interest is in the mean prediction of Bitcoin prices. While autocorrelation affects the precision of our confidence intervals and p-values, it does not invalidate the model's mean prediction. The Newey-West adjustment shows that even after correcting for autocorrelation, the predicted prices remain consistent with the OLS model.

Therefore, autocorrelation, while an important concept to recognize, does NOT invalidate the Power Law model.

Key Takeaways

As with any framework that attempts to provide insight into the future, there is a lot of associated criticism. These range from the model being “overly simplistic” to rendering statistical modelling itself a useless practice.

While some criticisms of the model can partly be addressed by being explicit about the limitations of data modeling, others stem from misunderstandings or a failure to appreciate the nuances of statistical analysis.

Finally, a significant portion of the discussion focused on autocorrelation. We conclude that autocorrelation only affects the standard errors, and not the overall predictions made by the model, rendering the common argument “log-log regression of price over time is invalid because of autocorrelation” null and moot.

Like any model, the power law model certainly has its limitations, but when utilized correctly, it provides valuable insights. Statistical modelling is an art and not a science, and hopefully by addressing these concerns, we can have a more informed discussion about the strengths and limitations of the Power Law model in predicting Bitcoin prices.