# Autocorrelation and Partial Autocorrelation

> Autocorrelation measures how a series relates to its own past values. Learn the formula, the ACF, what positive and negative autocorrelation mean and why it matters.

Source: https://learn.tradelabsai.com/math/autocorrelation/  
Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Autocorrelation and Partial Autocorrelation", https://learn.tradelabsai.com/math/autocorrelation/

Autocorrelation, also called serial correlation, measures how strongly a time series is related to its own past values. If today's return tends to be positive after a positive return yesterday, returns have positive autocorrelation; if today tends to reverse yesterday's move, they have negative autocorrelation. Autocorrelation is the statistical fingerprint of momentum and mean reversion, and it also affects how reliable statistical tests on trading results are.

## The formula

The autocorrelation at lag k:

```
ρ(k) = Cov(r_t, r_(t-k)) / Var(r_t)
```

A lag 1 autocorrelation compares each value with the previous one; lag 5 compares with the value five periods earlier.

## Interpreting autocorrelation

| Autocorrelation | Meaning | Trading implication |
|---|---|---|
| Positive | Moves tend to continue | Momentum or trend following. See [Momentum Trading](https://learn.tradelabsai.com/strategies/momentum-trading/) |
| Near zero | Little linear relationship with the past | Close to a random walk. See [White Noise and Random Walks](https://learn.tradelabsai.com/math/white-noise-and-random-walks/) |
| Negative | Moves tend to reverse | Mean reversion. See [Mean Reversion](https://learn.tradelabsai.com/strategies/mean-reversion/) |

## The autocorrelation function (ACF)

The ACF plots autocorrelation at many lags. With a sample of n observations, autocorrelations within about ±2 / √n of zero are generally not statistically significant. For 1,000 daily returns, that band is about ±0.063.

**Example: Reading an ACF**
A trader calculates the ACF of 2,500 daily returns for a stock index. Lag 1 autocorrelation is minus 0.04, inside the ±0.04 significance band (2 / √2,500 = 0.04), and other lags are near zero. Returns show little linear predictability. But the ACF of squared returns shows large positive values for many lags, revealing volatility clustering: big moves follow big moves, even though their direction is unpredictable. See [GARCH](https://learn.tradelabsai.com/math/garch/).

## Returns vs squared returns

| Series | Typical autocorrelation in markets |
|---|---|
| Daily returns of liquid indices | Close to zero; sometimes slightly negative at lag 1 |
| Squared or absolute returns | Strongly positive for many lags (volatility clustering) |
| Returns of illiquid assets | Positive at short lags, partly due to stale prices |
| Prices (levels) | Very high, close to 1 (non stationary). See [Stationarity, Differencing and Unit Roots](https://learn.tradelabsai.com/math/stationarity/) |

## Why autocorrelation matters

### For strategy design

Positive autocorrelation supports trend strategies; negative supports reversal strategies. Many documented effects, such as short term reversal in individual stocks and medium term momentum, are forms of autocorrelation at different horizons. See [Short and Long-Term Reversal](https://learn.tradelabsai.com/research/short-and-long-term-reversal/).

### For statistical tests

Standard errors assume independent observations. Positive autocorrelation in strategy returns makes standard errors too small and results look more significant than they are. Adjusted methods include Newey West standard errors and block bootstrapping. See [Sampling and Standard Error](https://learn.tradelabsai.com/math/sampling-and-standard-error/) and [Bootstrap and Permutation Tests](https://learn.tradelabsai.com/math/bootstrap-and-permutation-tests/).

### For performance measurement

Smoothed returns, common in illiquid assets such as private equity, real estate funds and some hedge funds, show positive autocorrelation and understate true volatility, inflating Sharpe ratios. Andrew Lo and others have shown how to adjust Sharpe ratios for autocorrelation. See [Sharpe Ratio](https://learn.tradelabsai.com/portfolio/sharpe-ratio/).

## Testing for autocorrelation

| Test | Checks |
|---|---|
| ACF and PACF plots | Pattern of correlations by lag |
| Ljung Box test | Whether a group of autocorrelations is jointly zero |
| Durbin Watson test | Autocorrelation in regression residuals. See [Regression Analysis](https://learn.tradelabsai.com/math/regression-analysis/) |

The partial autocorrelation function (PACF) shows the correlation at each lag after removing the effect of shorter lags, and helps choose ARIMA models. See [ARIMA](https://learn.tradelabsai.com/math/arima/).

## Autocorrelation and holding periods

Autocorrelation can differ by horizon. Daily returns may show slight reversal while monthly returns show momentum. Checking autocorrelation at the holding period your strategy actually uses avoids building a trend system on a market that tends to reverse at that horizon.

## Frequently asked questions

### What is autocorrelation?

The correlation of a time series with its own past values, showing whether movements tend to continue or reverse.

### Do stock returns have autocorrelation?

Daily returns of liquid indices show little autocorrelation, but squared returns show strong autocorrelation because volatility clusters.

### Why does autocorrelation matter for backtests?

Because autocorrelated returns make standard errors too small, so results can look more statistically significant than they really are.

Next, learn why stable statistical properties matter in [Stationarity, Differencing and Unit Roots](https://learn.tradelabsai.com/math/stationarity/).

## Continue learning

- Next lesson: [Stationarity, Differencing and Unit Roots](https://learn.tradelabsai.com/math/stationarity/)
- Previous lesson: [White Noise and Random Walks](https://learn.tradelabsai.com/math/white-noise-and-random-walks/)
- Related: [White Noise and Random Walks](https://learn.tradelabsai.com/math/white-noise-and-random-walks/): A random walk is a path built from random steps; white noise is pure randomness. Learn how they model prices, the random walk hypothesis and the evidence against it.
- Related: [Stationarity, Differencing and Unit Roots](https://learn.tradelabsai.com/math/stationarity/): A stationary series has stable statistical properties over time. Learn why prices are non stationary, how to test with ADF and KPSS and how to make data stationary.
- Related: [ARIMA](https://learn.tradelabsai.com/math/arima/): ARIMA models forecast a time series from its own past values and errors. Learn the AR, I and MA terms, how to choose orders and why returns are hard to predict.
- Related: [GARCH](https://learn.tradelabsai.com/math/garch/): GARCH models capture volatility clustering, where big moves follow big moves. Learn the GARCH(1,1) formula, persistence, forecasting and uses in risk and options.
- Related: [Momentum Trading](https://learn.tradelabsai.com/strategies/momentum-trading/): Momentum trading buys assets that are rising fastest and sells those falling fastest. Learn the research, intraday and multi month methods and momentum crashes.