Autocorrelation, Hurst Exponent and CointegrationACF
Autocorrelation measures how strongly a series is related to its own past values. The Hurst exponent estimates whether a series tends to persist (above 0.5), revert (below 0.5) or behave randomly (about 0.5). Cointegration means two series that wander individually stay linked over the long run — the statistical basis of many pairs trades.
Three statistics help describe how a price series behaves over time: autocorrelation, the Hurst exponent and cointegration. Together they suggest whether a market trends, mean-reverts or wanders randomly.
- Category
- Indicators and Tools
- Reading time
- 3 min
- Also called
- Time-series statistics, Serial correlation, Long memory
- Updated
The three ideas in plain words
Autocorrelation compares a series with its own past values at different lags.1
The Hurst exponent runs from 0 to 1 and tells you whether a series tends to trend, wander randomly or revert to its mean.2 Above 0.5 is persistent, below 0.5 mean-reverting, about 0.5 random.
Cointegration asks whether two drifting series stay tied together over time — an idea Clive Granger introduced.3
How to apply them
Compute returns
Work with returns or log prices, not raw levels, for autocorrelation.
Plot the ACF
Scale it so the correlation at lag zero equals one.1
Estimate Hurst
Use rescaled range analysis over several window lengths.
Read H
Read a low Hurst value as strong mean reversion.2
Test cointegration
For Engle-Granger, regress one series on the others and test the leftovers (residuals) for a unit root.4
Decide the strategy
Persistent series suit trend-following; mean-reverting or cointegrated ones suit reversion.
Worked examples
Example 1 — Persistent series
TrendingA hypothetical commodity shows H = 0.68.
| H | 0.68 |
A persistent series is more likely to keep trending than a pure random walk.5
Example 2 — Mean-reverting spread
Mean-revertingA hypothetical spread between two banks’ shares shows H = 0.31.
| H | 0.31 |
The low Hurst value points to mean reversion, so reversion trades are worth testing.2
Example 3 — Random walk
RandomA hypothetical index’s daily returns give H = 0.50 and no significant ACF lags.
| H | 0.50 |
Close to 0.5 the series has no memory — it behaves like noise, so there's no edge here.5
Three time-series tools
| Aspect | Measures | Typical use |
|---|---|---|
| Autocorrelation | Link to own past at each lag | Short-term momentum or reversal |
| Hurst exponent | Long memory, 0 to 1 | Trend vs mean-reversion regime |
| Cointegration | Stable long-run link between series | Pairs trading |
| Correlation | Co-movement of returns | Diversification |
Where these statistics are used
Regime choice
Pick Trend Following or Mean Reversion.
Pairs trading
Test spreads beyond simple Correlation.
Strategy research
Check for an edge before Backtesting.
Risk
Persistent Volatility changes sizing.
Advantages and disadvantages
Advantages
- Quantify behaviour objectively.
- Help match strategy to market.
- Cointegration handles trending prices properly.
- Widely available in software.
Disadvantages
- Results change with sample and window.
- Hurst estimates are noisy.
- Relationships can break.
- Need long data histories.
Going deeper
Read as far as you need. Each level builds on the one before it.
ACF basics
Beginner
Lag zero is always one by construction; the other lags show how strongly the past echoes.1
Hurst regimes
Intermediate
At 0.5 there's no memory at all, while readings above 0.5 point to persistence.5
Cointegration testing
Advanced
The Engle-Granger test starts from the assumption of no cointegration, so rejecting it supports a stable long-run link.4
Common mistakes
Computing ACF on raw prices
Use returns.
Trusting one Hurst estimate
Check several windows.
Confusing correlation with cointegration
They answer different questions.
Assuming links last forever
Retest regularly.
Check your understanding
Test yourself before moving on. Each answer comes with an explanation.
Key points to remember
- Autocorrelation links a series to its own past.
- H above 0.5 is persistent.
- H below 0.5 is mean-reverting.
- H near 0.5 is random.
- Cointegration finds stable long-run links.
Frequently asked questions
What does H = 0.5 mean?
Is cointegration the same as correlation?
Which data should I use?
Further learning
- Mean Reversion
Strategies for low-H series.
- Trend Following
Strategies for high-H series.
- Correlation
The simpler co-movement tool.
- Backtesting
Testing what the stats suggest.
Sources
This article is written in our own words. The sources below support the specific facts, rules and figures marked with a number in the text. Links open the original publisher’s page.
- MathWorks. autocorr – Sample autocorrelation (opens in a new tab) Checked .
- CFA Institute (Enterprising Investor). Rescaled Range Analysis: A Method for Detecting Persistence, Randomness, or Mean Reversion (opens in a new tab) Checked .
- The Royal Swedish Academy of Sciences (Nobel Prize). Time-series Econometrics: Cointegration and ARCH (2003 advanced information) (opens in a new tab) Checked .
- MathWorks. egcitest – Engle-Granger cointegration test (opens in a new tab) Checked .
- PubMed Central (Entropy journal). The Hurst Exponent as an Indicator to Anticipate Agricultural Commodity Prices (opens in a new tab) Checked .
Educational content, not financial advice. Examples use hypothetical prices and simplified costs; actual fees, margin rules and regulations vary by broker, market and country. We are not affiliated with any publisher linked above. Last reviewed . Report an error or request a source credit.