Plan C@TheRealPlanC
🚨 Very Important: Bitcoin Modeling
The original Bitcoin power law was fit with OLS regression.
It was invalid from day one.
OLS cannot produce valid results on data with Bitcoin's characteristics:
• Non-stationary
• Autocorrelated
• Right-skewed
• Fat-tailed
It was not until I introduced the idea of using quantile regression to the Bitcoin power law that we had a statistically valid regression for Bitcoin's unique Time-series dataset.
This is not a matter of preference.
OLS requires four assumptions to produce valid results: constant variance, independent errors, normally distributed residuals, and no outlier dominance.
Bitcoin violates all four. Simultaneously. Always.
Bitcoin is a non-stationary time series.
Constant variance. OLS assumes the spread of residuals is the same everywhere.
Bitcoin's volatility has declined by an order of magnitude.
In 2011, price could move 100x in months. In 2024, a 3x move is a major cycle.
OLS cannot tell the difference. It treats both eras as if they have the same uncertainty.
Independent errors. OLS assumes each observation is an independent draw. Bitcoin prices are serially correlated.
Today's price is highly predictive of tomorrow's.
OLS dramatically underestimates the true standard errors.
The confidence intervals it reports are far too narrow. They look precise. They are not.
Normal residuals. OLS is the maximum likelihood estimator only if the errors are Gaussian.
Bitcoin's residuals are not Gaussian.
They are right-skewed because bull market overshoots are larger than bear market drawdowns.
They are fat-tailed because extreme moves happen far more often than a normal distribution predicts.
These are two separate problems.
Skew means the distribution is asymmetric. Fat tails mean extremes are too frequent. Bitcoin has both.
No outlier dominance.
OLS minimizes squared errors.
A data point 5 times away from the line has 25 times the pull.
Bitcoin's bubble peaks are exactly this kind of extreme observation.
A handful of bubble tops move the entire fitted line more than thousands of normal observations.
The mean gets dragged toward the bubbles.
The fit represents the outliers, not the data.
Bitcoin does not partially violate these assumptions.
It violates all four, all the time, across the entire dataset.
There is no subset of Bitcoin data where OLS assumptions hold.
The confidence intervals are wrong.
The standard errors are wrong.
The point estimate itself, the mean, is a misleading summary of a skewed distribution.
The left chart shows OLS fitting the mean.
The right chart shows quantile regression fitting the median.
Same data. The correct tool.
