Bitcoin heading to $1 million sooner, FGLS vs. Ordinary Least Squares

A 5.9 power law slope for price vs. age with generalized least squares

Figure 1. Log-linear graph of a Feasible Generalized Least Squares model that mitigates against autocorrelation in Bitcoin residuals. It has a higher power law slope, about 5.9, and thus projects higher future prices.

This is not investment advice. Bitcoin is highly volatile. Long-term holding makes volatility your friend.

When we perform Ordinary Least Squares regression on Bitcoin’s price history we find that residuals are auto-correlated, with a high correlation between adjacent residuals of value ρ = 0.98, and thus R^2 may be spuriously high as is often found in time-series analysis (Granger and Newbold 1974, “Spurious Regressions in Econometrics”,  Journal of Econometrics 2 (1974) 111-120; Nelson and Kang, “Pitfalls in the Use of Time as an Explanatory Variable in Regression”,  Journal of Business and Economic Statistics,  Jan. 1984, Vol. 2, No. 1, pp. 73-82). This article is not meant to obviate the power law relationship  to which Bitcoin adheres, but to temper the analysis and interpretation of it, and in fact indicate the possibility of a steeper power law relationship.

With the power law theory as applied to Bitcoin we have certain advantages with respect to typical econometric and financial time series. The first one is, we do not start from an arbitrary date, unlike most financial and economic time series. We start from the beginning of a new revolutionary monetary technology, the Genesis block of January 3, 2009. This is the only proper starting point to use. The second advantage is that we have an underlying theory of price behavior, as developed by Plan G (@Giovann35084111) that relates value to the square of adoption, in accordance with Metcalfe’s Law. The third advantage is that there is underlying price support from the economics of mining which is strongly influenced by the difficulty and halving adjustments of the Nakamoto proof-of-work protocol. Hashrate and thus difficulty are extremely steep power law functions, rising roughly as the 12th power of Bitcoin’s age.

When a rocket is launched on a roughly parabolic trajectory its next position is highly correlated to its present position because there is underlying physics of gravity and the dependence on the rocket equation for fuel consumption and vehicle weight and it has a velocity vector toward its next position it will reach. Bitcoin, we believe, is adhering to underlying physics of networks, with an emergent power law behavior and despite some wobbles (volatility and exponential bubbles) is on a steep upward trajectory.

The Power Law has a decade of success under its belt, it has worked for that long in general, and for the past eight years the power law relationship with time has been quite stable. Typically OLS regressions, such as the one I performed for this article, yield a slope or power law index of around 5.7. Price goes as Bitcoin Age^5.7, or nearly the square of a cubic law. This reflects the cubic growth of adoption with the number of nonzero balance addresses as the proxy, and price growing as the square of the adoption, in accordance with Metcalfe’s law, resulting in the steep growth of value.

Nevertheless we have evaluated both OLS and two Feasible Generalized Least Squares models against weekly Bitcoin data from 2010 to mid-October 2024, consisting of 744 prices, in order to examine the autocorrelation issue. There are several Generalized Least Squares methods, we have evaluated two such methods that are designed to mitigate against auto-correlation. Both assume AR(1) auto-regression of lag 1. In addition to the parameter ρ, another measure of auto-correlation is the Durbin-Watson (D-W) statistic, which always ranges from 0 to 4. For the OLS we find a very small 0.03, also indicative of a high degree of positive auto-correlation. A value near 2 for D-W would indicate minimal auto-correlation (over 2 indicates negative correlation).

Table 1 has the results of applying the three techniques: OLS, FGLS ‘manual’, and FGLS ‘Prais-Winsten’ to the weekly price history (we call these FGLS and FGLS PW in the table). With autocorrelated series it is expected that R^2 and the F-test statistic both continue to rise as more data is added (R^2 rises toward one, and the F-statistic rises in an unbounded fashion), and we are seeing evidence of that here for the OLS model. The D-W statistic is very low, which is consistent with high positive autocorrelation.

Table 1. Statistical parameters for the OLS model and two FGLS model. The first of the FGLS models is the preferred one with better D-W, AIC, and log likelihood statistics. It also has the steepest power law slope of the three models.

By contrast, both FGLS models have good D-W statistics, approaching 2.0 which is the value where autocorrelation is repaired fully. The manual FGLS has the best D-W statistic, the best AIC (lower is better in this case), and the best Log Likelihood (higher is better). It also has the steepest slope and a much smaller error in the slope as compared to the FGLS model.

We can thus select the first FGLS method as the best performing model since it eliminates most of the autocorrelation issue and has excellent AIC and log likelihood measures. This suggests a higher power law slope and higher values for long-term projections than are found with an OLS regression.

Figure 2. Log-log plot showing both the OLS (red dashed line) and FGLS (solid green line) fits to weekly price series. I have also plotted the projection of the FGLS model out 15 years, to 2039 (purple dots).

In Figure 2 I plot both the OLS regression and the FGLS model of the middle row of Table 1 against the data. Furthermore the FLGS model trend of a 5.893 power law is projected out 15 years into the future. Note the model is steeper and also better fits the earliest data of the series. It crosses the OLS model (of power law index 5.68) in price around the year 2020

The volatility of Bitcoin has been falling over time. For the final third of the data set, about 5 years’ duration, the standard deviation is 0.237 in the log10. This corresponds to a multiplicative factor of 1.73. Thus the one standard deviation downside from trend is 42%, the upside is 72%, and the two standard deviation upside is a factor of 3. Generally Bitcoin prices have been in the range Z of [-1, +2] standard deviations. You should consider that substantial volatility as you examine Table 2, the trend of future price.

Table 2. The projection of future trend prices with the FGLS model with power law slope of 5.893. Given the high volatility the price at a given time could vary by a factor of about 2 on the downside and about 3 on the upside. These values of course assume a continuation of the trend and no flattening of the power law.

In Table 2 we have price projections for the beginning of the year from 2025 through 2039, based on the FGLS model with a 5.89 power law. You should consider the substantial volatility as you exam the table, downside is around a multiplicative factor of 2, and upside around a factor of 3.

The table indicates we could reach $250K in 2028 even without an exponential bubble upward, and by the end of 2025 with a decent sized bubble (factor of 2 or more). The $1 million level is reached on trend by 2033, and the $2.5 million level by 2037, just over a dozen years from now.

If the power law persists with this same high index close to 6, Bitcoin could be well above $1 million in a decade from now.

I thank Professor Emeritus Prem Jain of Georgetown University for pointers to the classic papers on the autocorrelation impacts on OLS R^2 values and regression results.