Serial Correlation: When Return Observations Depend on Time

Specify the lag and estimator

Lag one compares observations separated by one interval; lag five compares observations five intervals apart. The intervals should have a clear meaning, such as consecutive trading-day returns. Comparing a daily series with irregularly spaced transaction outcomes without a stated convention can make a lag difficult to interpret.

NIST defines an autocorrelation estimator using the overall sample mean and a denominator based on the full series. At lag k, it sums products of deviations k observations apart and divides by the sum of squared deviations. Other implementations may center or normalize differently, especially in short samples.

Calculate two four-observation examples

Use percentage-point returns of −1, −1, +1 and +1. Their mean is zero. For lag one, the adjacent deviation products are +1, −1 and +1, summing to +1. The denominator is 1 + 1 + 1 + 1 = 4. Under the stated estimator, lag-one autocorrelation is +0.25.

Now reorder the same values as −1, +1, −1 and +1. The mean and variance are unchanged, but all three adjacent products are −1. The numerator becomes −3 and the lag-one value is −0.75. The examples show why a histogram or standard deviation alone cannot describe the temporal ordering. Four observations are far too few to establish a stable trading pattern.

Separate outcome dependence from measurement artifacts

Overlapping holding-period returns share underlying price changes by construction. Smoothed valuations and stale quotes can also create apparent persistence. Repeated follower accounts may share the same source exposure, so treating every account-day as an unrelated draw understates dependence.

Check the data pipeline before attaching an economic story. Preserve timestamps, identify missing sessions and verify that returns are not accidentally duplicated. Do not fill absent returns with zeros unless that value represents the intended valuation rule; a convenient fill policy can change the estimated pattern.

Use the finding to question assumptions

Confidence intervals based on independent trials and simple square-root volatility annualization can be misleading when relevant dependence is present. The appropriate adjustment depends on the question and model; there is no universal correction obtained by multiplying the trade count by one minus an observed correlation.

Inspect several lags with a predeclared purpose, sample length and uncertainty context. Searching hundreds of lags and publishing only the most extreme one introduces selection. A zero linear autocorrelation at one lag also does not prove complete independence or rule out nonlinear dependence.

For copied-account research, retain a chronological record and compare source and follower series at compatible frequencies. An observed lag relationship could reflect exposure timing, conversion, missing marks or data alignment. It is evidence to investigate, not proof that a delay caused a return difference or that the next outcome can be profitably predicted.