Hurst Exponent Calculator
Estimate a classical rescaled-range Hurst slope across three to six declared block sizes, with the scale-level evidence shown.
Enter a series and block sizes
Use one equally spaced ordered series. Each block size must provide at least two complete non-overlapping blocks.
Three through six distinct whole numbers; each must yield at least two blocks.
Separate finite decimal values with line breaks, spaces, commas or semicolons. A maximum of 2,000 is accepted.
Entered-series Hurst estimate
Entered Mean-Reversion Diagnostics 1.0.0.
| Block size | Complete blocks | Mean R/S | Dropped tail |
|---|
How the classical rescaled-range Hurst estimate is calculated
For each scale: average the block R/S values
H = OLS slope of log(mean R/S) against log(block size)
Version 1.0.0 implements a transparent classical non-overlapping rescaled-range estimator. For every block, it subtracts that block’s mean, cumulatively sums the deviations, measures the range of that cumulative path, and divides by the block population standard deviation. It then averages R/S across all complete blocks of the same size.
The final H estimate is the ordinary least-squares slope, with an intercept, of natural log mean R/S against natural log block size. At least three distinct sizes are required. The result panel also reports the log-log intercept, R-squared, complete-block counts and trailing observations omitted at each scale.
The familiar 0.5 value is presented only as a numerical reference. Small samples, deterministic trends, short scale ranges, structural breaks and estimator choice can all influence the slope. The model deliberately does not clamp the estimate to zero through one, because clamping would conceal what the actual fitted line produced.
A careful interpretation workflow
Start with the estimator and visible audit fields, then check whether the data and assumptions support the question you want to ask. A clean number is not a substitute for a valid series.
- Inspect the scale table before the headline slope; each regression point comes from one block size.
- Use the displayed complete-block count to spot scales supported by very little data.
- Treat R² as fit to the chosen log-log points, not as proof the underlying market model is correct.
- Compare estimates only when sampling, preprocessing, block sizes and estimator convention match.
- Test sensitivity by changing reasonable scale sets rather than relying on one convenient selection.
- Use independent stationarity, structural-break, execution-cost and out-of-sample analysis before trading.
Worked example from the audited fixture
How to interpret the result
The near-one fixture value reflects the deliberately smooth arithmetic path and this exact classical estimator. It is not evidence that a traded market will continue trending. On real data, treat H as estimator-, scale- and sample-dependent descriptive evidence that must be checked for robustness.
Which mean-reversion diagnostic answers which question?
These three pages are complementary rather than interchangeable. They describe different properties of the same entered numbers, and none establishes a complete trading strategy by itself.
| Diagnostic | Question answered | Main dependency | What it does not prove |
|---|---|---|---|
| Latest z-score | How far is the latest value from its selected sample mean? | Window and sample standard deviation | Normality or future reversal |
| Classical Hurst R/S | What log-log rescaled-range slope appears across selected scales? | Estimator, scale set and sample path | A stable persistent or mean-reverting regime |
| AR(1) half-life | What positive geometric decay time follows from the fitted φ? | Sampling interval and AR(1) specification | Stationarity, cointegration or forecast accuracy |
Assumptions and limits
- The input needs at least 16 and at most 2,000 finite ordered observations.
- Three through six distinct integer block sizes are required; each is at least eight and must yield two complete blocks.
- Trailing observations that do not fill a complete block are excluded separately at each scale and disclosed.
- A complete constant block is rejected because its standard deviation and R/S denominator are zero.
- Classical R/S can be affected by short-range dependence, deterministic trends, nonstationarity and the chosen scale range.
- R-squared measures the selected log-log line fit only; it is not a regime-confidence probability.
- The page does not emit persistent, mean-reverting, efficient, trend, entry or exit labels.
Prepare the entered series before calculating
Choose the economic object first: a price level, log price, return, spread, residual or indicator value is not interchangeable with the others. Export completed observations in chronological order, keep one feed and one transformation, and remove headers before pasting.
Check timestamps outside this calculator. A numeric list cannot reveal a missing weekend rule, duplicated bar, daylight-saving shift or gap in broker history. If observations are unevenly spaced, a bar-labelled decay or time-series interpretation can be false even though the arithmetic runs.
Record the symbol, timeframe, time zone, sample dates, preprocessing, window and model version with any saved result. That audit trail makes later comparisons meaningful and reduces the risk of selecting only the most attractive statistic.
Frequently asked questions
- Version 1.0.0 uses classical non-overlapping rescaled range with population standard deviation inside each block and intercept-included log-log OLS.
- Enter three through six distinct whole-number sizes; each must be at least eight and yield at least two complete blocks.
- Trailing observations that do not fill a complete block are dropped independently at each scale and their counts are displayed.
- Estimator convention, sampling, preprocessing, scale selection, block overlap and finite-sample corrections can all change the result.
- No. The page reports the difference from 0.5 descriptively and emits no persistence, trend, efficiency or trading verdict.
- It measures in-sample fit of the selected log mean-R/S points to one line; it does not prove the market model or regime is correct.
- A constant complete block has zero standard deviation, so its rescaled-range quotient is undefined.
- No. Unit-root, stationarity and structural-break questions require separate diagnostics.
Sources and methodology
- NIST Journal of Research — Rescaled Range Analysis — Methodological discussion of rescaled-range analysis and Hurst estimation.
- H. E. Hurst — Methods of using long-term storage — Historical source for the rescaled-range relationship.
- NIST/SEMATECH e-Handbook — Linear least squares — Documents the intercept-included least-squares line used for the log-log slope.
The operational contract is version 1.0.0. Its formulas, fixture outputs, maximum sample and withheld-output rules are tested locally before staging release.
Continue the statistical diagnostic workflow
Compare the chart feed and trading terms
Before transferring an entered-series result to execution, confirm the broker’s symbol specification, chart history, time zone, spreads, commissions and financing. Calculations based on one feed need not reproduce on another.
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