Hurst Coefficient Calculator

Paste any time series, daily returns, rainfall, website traffic, and estimate its Hurst exponent (H), the classic measure of long-term memory. Values below 0.5 mean mean-reverting, 0.5 means random-walk-like, and above 0.5 means trending. Pick from three estimation methods (R/S, DFA, variance-time), choose how your data is preprocessed, and get the exponent together with its R-squared fit quality and a plain-English interpretation. Before estimating memory, many analysts smooth noisy series first; our smoothing constant calculator tunes that step. For a different dependence measure, our correlation coefficient calculator quantifies how two series move together. To measure trend strength itself rather than memory, pair it with our ADX calculator.

Quick Answer

Worked example, checked against this exact tool

Choose Sample financial data from the data-source menu: 30 daily returns load with the data type set to Return series. Keep the R/S tab and lags 2 to 10, then click Calculate. The tool reports H = 0.4467 with R-squared = 0.9258, interpreted as weak mean reversion: the series slightly favors snapping back after moves, and the high R-squared means the scaling fit is trustworthy. Switch to the DFA tab without changing anything else: H = 0.5500, R-squared = 0.8087, essentially random-walk-like. Two honest methods disagreeing slightly on the same data is normal; that is why the tool shows the fit quality next to every estimate.

Now pick Manual data entry: the price-like sample (100, 102, 101, …, 122) loads with the data type correctly set to Price series, so the tool converts it to log returns first. R/S gives H = 0.3289, strong mean reversion, because the sample's returns alternate up, down, up, down. If you flip the data type to "Return series" by mistake, the trending prices get treated as returns and H jumps to 0.64: the wrong preprocessing gives the wrong answer, which is why the tool presets the right one for each sample.

What the Hurst exponent tells you

The Hurst exponent H, introduced by Harold Edwin Hurst in his studies of Nile river flooding, measures long-term memory in a time series: whether movements tend to continue or reverse. H below 0.5 means anti-persistence (mean reversion): a rise is more likely to be followed by a fall. H = 0.5 means no memory, the random-walk case behind the efficient-market hypothesis. H above 0.5 means persistence (trending): rises tend to be followed by rises. In trading, persistent series favor trend-following and momentum strategies, while anti-persistent series favor contrarian, buy-the-dip approaches. H describes the past; it is not a forecast, and markets switch regimes.

The three estimation methods

R/S analysis (rescaled range) is the classical method. The series is split into windows of each lag length; within each window the range of cumulative deviations from the mean is divided by the standard deviation, the R/S values are averaged per lag, and H is the slope of log(R/S) against log(lag). It is intuitive and works from about 30 points, but it is biased upward on short series, typically reading 0.05 to 0.10 high on white noise.

DFA (detrended fluctuation analysis) integrates the series, then removes the local linear trend inside each window before measuring fluctuation. Because it detrends, it handles non-stationary data, slow drifts that would fool R/S, much better. It is the most robust of the three for real-world series.

Variance-time analysis aggregates the data into blocks, computes the variance of the block means at each block size, and uses the scaling Var ∞ lag^2H−2, so H = 1 + slope/2 of log variance against log lag. It is simple but the least robust: it needs 100 or more points and is easily thrown off by short series, which is why the tool flags its fit quality prominently.

Side by side, the trade-offs look like this:

Method	What it measures	Minimum data	Handles drifting data	Watch out for
R/S	Range of cumulative deviations, rescaled by the standard deviation	~30 points	Poorly: drift inflates H	Reads 0.05 to 0.10 high on short series
DFA	Fluctuation after removing the local linear trend in each window	~50 points	Well: detrending is the point	Needs enough points per window to detrend cleanly
Variance-time	How block-mean variance scales with block size	~100 points	Poorly	Least robust; noisy on short series

Why preprocessing matters: prices are not returns

All three methods assume stationary increments, roughly speaking, changes rather than levels. Feeding raw prices into R/S measures the trend of the price curve, not the memory of its movements. The tool's data-type menu fixes this: Price series converts to log returns, Return series centers the returns, Raw values takes first differences, and Log values takes differences of logs. The most common Hurst mistake on the internet is running the analysis on prices and reporting a "trending" H near 1.0 that is pure artifact.

How much data you need

As a rule of thumb: R/S needs at least 30 points and becomes believable past 100; DFA wants 50 or more; the variance method wants 100 or more. The lag range matters too: the maximum lag should stay below a quarter of the series length (the tool enforces this), and estimates stabilize when the log-log regression uses many lag points with an R-squared above 0.8. Treat any H with R-squared below 0.6 as unreliable, add data or widen the lag range rather than trusting the number.

The methodology literature adds one more rule about which lags enter the fit: keep the largest scale well below the series length (some authors suggest below N/12) and prefer log-spaced scales, so the regression is not dominated by a crowd of tiny windows. This tool evaluates every integer lag up to a quarter of the series length, which is more generous than the conservative guidance in the literature. If your R-squared comes back soft, narrowing the maximum lag is the first dial to turn.

What competing calculators miss

The best-known web rival asks you to type in a sample size and a pre-computed R/S value, then evaluates H = log(R/S) ÷ log(n): it cannot touch your data, offers one simplified single-window method, and gives no measure of fit quality. Chart indicators compute R/S only, on fixed windows, with no alternative method to cross-check. This page ingests your actual series, runs three independent estimators with adjustable lag ranges, preprocesses prices into returns correctly, and reports R-squared, a confidence assessment, and a plain-English interpretation with every estimate. Two genuine calculation defects found while verifying this tool in Node, a wrong variance-method formula and a degenerate lag-2 point poisoning the DFA regression, were fixed here; most implementations floating around the web carry similar quiet errors.

Frequently asked questions

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