Bitcoin Power-Law Corridor Methodology

Property	Meaning
Independent variable	Positive network age in declared time units
Dependent variable	Positive Bitcoin price in a fixed currency
Basic fit	Linear regression in logarithmic coordinates
Distinct outputs	Fitted trend, residual corridor, parameter uncertainty, future scenario

Let d be elapsed days from a stated origin, with d greater than zero, and P be a positive price in currency per BTC. A day-zero observation cannot enter a logarithm. Changing the time unit changes the intercept; changing the origin can change more than the intercept.

The sample has a first date, last date, observation frequency, close convention, and treatment of gaps. Daily sampling gives a different weighting of market history from annual observations. The model's meaning depends on those choices as well as the fitted coefficients.

For the base-ten ordinary least-squares specification:

• x = log10(d), y = log10(P)
• y_hat = a + b × x
• P_hat = 10^a × d^b
• Residual e_t = y_t - y_hat_t
• Residual standard error s = sqrt(sum(e_t^2) / (n - 2))
• Descriptive k-sigma corridor = 10^(a + b × log10(d) ± k × s)

The residual standard-error formula requires more than two observations and an identifiable fit. A corridor in logarithmic space becomes multiplicative in price space. Exponentiating a fitted log price does not automatically estimate the arithmetic mean future price.

Price-to-trend is P / P_hat. When b is nonzero, the model age corresponding to a price is d_star = (P / 10^a)^(1/b). A time-shift interpretation compares d_star with current d. Converting that difference to years requires a stated day-count convention and does not predict when the market will revisit a price.

Take a hypothetical coefficient pair a = 0 and b = 1. At d = 100 days the fitted price is 100 currency units per BTC. A log10 residual scale of 0.1 gives a one-scale corridor from 79.432823 to 125.892541. These are transformed reference bands, not an empirically certified probability of future containment.

Ordinary least squares minimizes squared errors in log space for this specification. NIST's regression reference explains least-squares estimation and its extrapolation limits. Robust regression, weighted regression, and fitting price levels answer different estimation questions. Selecting an early market start after examining the outcome can introduce substantial discretion.

Residual corridors, confidence intervals for the fitted relationship, and prediction intervals for future observations are different objects. Resampling individual residuals assumes a structure that may be inappropriate when errors cluster in time. A block bootstrap preserves some local dependence, but its results still depend on block length, resampling scheme, and sample history. The corridor tool on this site displays only the descriptive sigma-based corridor from the published fit; it does not offer a separate resampled or bootstrapped band, so none of these alternative-method caveats apply to anything shown on the page.

A statistical price curve contains no purchases or disposals. Fees, tax, spreads, custody charges, and financing are absent from the fit. A future wealth calculation based on the curve needs a separate transaction and cash-flow model; the fitted line alone does not supply investor returns.

Nonpositive prices and invalid ages have no logarithmic value. Excluding them changes the sample, while replacing them invents observations unless an estimation rule is disclosed. Extending the curve past its last observed price is extrapolation even when the date is today. Updated data and refitting can move the historical fitted line as well as its forward extension. On this page, the displayed trend line uses a fixed set of coefficients calculated once from price history through a stated date; they are updated by hand occasionally rather than recalculated automatically each time someone visits.

Network age is not a mechanism that purchases Bitcoin or guarantees demand. A visually close fit can reflect long-term trends without establishing causal or predictive power. Structural change and persistent residual errors weaken simple sigma interpretations. The fit cannot represent an economic regime absent from its sample merely by drawing wider bands.

Treat a fitted curve as a conditional relationship with a reproducible sample. Its most revealing test is whether reasonable changes to the sample or estimator change the conclusion, not whether the line looks convincing on one chart.