Mean Reversion Half-Life Calculator
Fit x(t) = α + φx(t−1) and derive a positive-persistence half-life only when the fitted φ is strictly between zero and one.
Enter an ordered equally spaced series
The selected period label changes display wording only; the numeric interval comes from how you sampled the observations.
Display label only; this does not validate timestamps or spacing.
Separate finite decimal values with line breaks, spaces, commas or semicolons. Keep the sampling interval equal.
Positive-persistence AR(1) half-life
Entered Mean-Reversion Diagnostics 1.0.0.
| Audit item | Observations or speed | Residual or half-life | Status or unit |
|---|
How the AR(1) mean-reversion half-life is calculated
Implied mean = α ÷ (1 − φ)
Decay speed = −ln(φ)
Half-life = ln(0.5) ÷ ln(φ), only when 0 < φ < 1
Version 1.0.0 creates adjacent lagged pairs from the ordered series and fits ordinary least squares with an intercept. The slope is φ and the intercept is α. When φ is strictly between zero and one, repeated deviations under that fitted positive-persistence AR(1) representation decay geometrically.
The implied long-run mean is α divided by one minus φ. The half-life solves φ raised to h equals one half, so h equals log 0.5 divided by log φ. Because h is measured in observation intervals, hourly data produces an hourly label only when every adjacent input is genuinely one hour apart.
If φ is zero, negative, one or above one, this page withholds the positive-persistence half-life instead of displaying a misleading number. A negative φ implies alternating signs rather than the monotonic positive decay summarized by this convention; φ at or above one does not produce the required positive decay.
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.
- Check fitted φ first; the positive half-life exists on this page only for zero below φ below one.
- Read the unit as the entered observation interval, not calendar time inferred by the calculator.
- Inspect the implied mean and latest deviation to understand the fitted equation’s scale.
- Treat R² as in-sample linear fit, not as a stationarity test or forecast-confidence score.
- Re-estimate across defensible windows and frequencies to expose instability instead of choosing the best-looking result.
- For a spread, establish the hedge relationship and run separate stationarity diagnostics before interpretation.
Worked example from the audited fixture
How to interpret the result
One bar in the fixture is a property of the constructed sequence and fitted AR(1) equation. On real prices or spreads, the estimate changes with sampling, window, transformation and market regime. It is not a promise that a deviation will halve in one bar or that the series is stationary.
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
- At least six and at most 2,000 finite, ordered, equally spaced observations are accepted.
- The calculator fits one AR(1) with an intercept; it does not select lag order or model nonlinear decay.
- A constant lagged predictor is rejected because the regression slope is undefined.
- Positive-persistence half-life is withheld unless fitted φ is strictly between zero and one.
- The displayed implied mean can be unstable when φ is near one and does not prove an economic equilibrium.
- No ADF, KPSS, cointegration, residual or structural-break test is performed inside this route.
- The result is not a forecast, trade duration, stop, target, entry, exit or position-sizing recommendation.
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
- The page fits x at t equals alpha plus phi times x at t minus one, then uses log 0.5 divided by log phi only when phi is strictly between zero and one.
- The intercept allows the fitted AR(1) equation to imply a nonzero long-run mean equal to alpha divided by one minus phi.
- It is withheld when fitted phi is zero, negative, one or above one because this page defines half-life only for positive geometric decay.
- It is measured in entered observation intervals; the selected bars, hours, days or weeks label does not validate timestamps.
- No. The estimate does not include stationarity, forecast error, spread, commission, slippage, liquidity or execution risk.
- No. It fits one AR(1) and reports descriptive evidence; ADF, KPSS or other tests are separate.
- You can enter one already constructed spread, but this page does not estimate its hedge ratio or establish cointegration.
- No. It is a mathematical transform of fitted phi, not an entry, exit, stop, target or holding-period recommendation.
Sources and methodology
- Ornstein–Uhlenbeck mean-reversion estimation article — Peer-reviewed discussion connecting discrete AR behavior, mean reversion and half-life estimation.
- NIST/SEMATECH e-Handbook — Linear least squares — Documents the intercept-included ordinary least-squares fit.
- NIST/SEMATECH e-Handbook — Autoregressive models — Provides autoregressive time-series model context and stationarity conditions.
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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