Bitcoin Plateaued in 2022 and has been Running on Momentum Since

The “curling stone” model of log Bitcoin prices suggests reversion to around $40k USD (inflation-adjusted to Jan 2025) within the next few years

By Summer 2022, it became clear that Bitcoin returns were, in the words of economics commentator Noah Smith, “steadily slowing down as adoption gets saturated.” In the thread that followed, some back-of-the-napkin analytics painted a convincing, if not rigorous, picture of slowing returns.

Almost 3 years later, I still haven’t seen a model that makes a prediction about when the growth stops. It’s the internet and I’m sure it exists, but I was in the mood to do some forecasting and I found a fairly compelling model that suggests, in the most fundamental sense, the growth stopped back in 2022. This peak was at $38.4k USD in today’s money, and the price activity we’ve seen since then is momentum, caused by a mix of FOMO, fear of future inflation, etc.

You’ve read the conclusions in the title and subtitle. Here is my code. The rest of the article is the methodology, which describes the model itself, and a short conclusion.

Methodology

The first step in modeling Bitcoin’s price growth, for the purposes of calling a top, is finding a mathematical growth model where there is a top. A prominent candidate is logistic growth, which is offered by the time series forecasting package prophet and is motivated by biological equilibrium models where there is growth and saturation. But the log Bitcoin price series doesn’t even appoximately follow that shape.

Thinking from first principles, “clearly and steadily slowing down,” sounds like friction acting on an object in motion. Imagine that for a minute, on a log scale, Bitcoin prices are like a curling stone’s position after being launched. The physics are shown below.

To apply the curling stone model, we need to map physical to financial variables.

• Position x(t): This could represent the log price of Bitcoin at time t. It’s the “distance” Bitcoin has traveled in terms of value or adoption.
• Time t0: Time since Bitcoin’s launch, or a date of minimal adoption, like when Bitcoin becan trading on its first major exchange in July 2010.
• Initial velocity v0: This could represent the initial “push” or growth rate of Bitcoin’s value or adoption at the start — think of it as early enthusiasm, innovation, or the rate of early adopters joining.
• Acceleration a: In the curling model, a is negative (deceleration due to friction). In a financial context, this could represent forces that slow growth, like market saturation, regulatory challenges, or profit-taking by early investors.

Statistical modeling

Though not ideal for reasons to be discussed, a straightforward way to approach the curling stone model is through linear regression with ARIMA errors.

A cynic might point out that after the fancy analogy, this is basically a polynomial regression with a fancy error structure. It’s not fully linear because of t_stop and there is a parameter constraint in beta_3 that is not easily enforced in standard time series regression software. In the spirit of “make it work, then make it better,” the forecast package in R was used with a t_stop grid and no enforement of the parameter constraint. The following are points about the model fitting procedure:

• The Arima function in the forecast package was the basis of the modeling,
• y was the log Bitcoin price in USD, inflation adjusted to Jan 2025 levels,
• the external regressors were the terms shown above, where t_stop was hard-coded and shifted sequentially,
• the error structure wasspecified to be AR(1),
• the method used was maximum likelihood.

The bitcoin month-end closes came from the CryptoCompare API and spanned July 2010 to Feb 2025. The inflation index used was CPI-U, not seasonally adjusted (CPIAUCNS), and was extended one month past Jan 2025 assuming a 2.5% annual inflation rate.

An AR(1) model for the model residuals was chosen because the drift looks autoregressive but also because the autoregressive parameter, rho, has a useful interpretation of how soon the series is expected to mean revert, critically assuming that there is a mean to revert to. For a monthly time series example, when rho is .5, a 50% reversion to the mean is expected to occur in just one period. When rho is .8, that time is 3 months. When rho is .99, the time balloons to 69 months. At rho = 1.0, the model has a differencing (I) term and there is no mean or effective possible strategy. From a long term forecasting perspecting, you walk away.

The method of maximum likelihood provides likelihood-based fit measures, including AIC (smaller is better). Below are the AIC values over a grid of t_step values.

Fortunately, there is a valley in the AIC values that can be seen around t=140 (Feb 2022), but there are points that are lower than the general trend (indicating better fit). Upon inspection, these fits had substantial discontinuities at the plateau. Below is an plot of a discontinuity observed at t_stop = 143 (May 2022, AIC near 84), which can be seen in the blue trend line.

At t_stop = 146 (Aug 2022), there’s virtually no discontinuity, and this was the value used to create the forecast presented at the beginning. The autoregressive parameter was estimated to be 0.94 with a standard error of 0.025. The plateau in dollars was estimated to be $38.4K USD in Jan 2025 value (presented without uncertainty quantification). The upper end of the 95% forecast interval two years out (Feb 2027) is $260k, as with an autoregressive parameter this close to 1, the series can run. The forecast uncertainty estimation doesn’t incorporate that of the model selection (the curling analogy might be weak) or the grid-based selection of t_stop.

What does this all mean?

An asset class materializing out of thin air was not something I thought was possible, and I, with many others, missed out on the opportunity of a lifetime. Two months after he created the posts that motivated this article (Aug 2022, which happens to be t_stop), Noah followed up with the following statement:

“For many people who bought into Bitcoin early, the eye-popping returns of those early years are an experience they’ll spend the rest of their lives trying and failing to recapture, like Hunter S. Thompson and the drug parties of the mid 1960s.”

We all saw those returns and may be similarly inclined to chase them. If they are not there anymore, then denying that reality could be financially painful in the long run.