Autocorrelation and Partial 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#

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.

Returns vs squared returns#

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.

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 and 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.

Testing for autocorrelation#

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.

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.