Serial correlation, also called autocorrelation, measures the linear relationship between observations of the same variable at different times. For daily returns, lag one compares each day with the preceding trading day. It helps explain why a long list of returns may contain less independent evidence than its row count suggests.
Positive autocorrelation means above-average returns tend to be followed by above-average returns, and below-average returns by below-average returns. Negative autocorrelation means those deviations tend to alternate. These are tendencies, not rules about the next day. An above-average return can still be a loss when the average is negative.
Ordinary cross-asset correlation compares two different investments at the same times. Autocorrelation compares one series with its own past. A lag is the number of observation intervals between the values being compared. Daily and monthly lag-one correlations describe different horizons. Persistent price levels are a separate issue: a price carries forward its previous level even when successive returns are independent.
Using the NIST sample convention, r₁ = Σ(t=2 to n)[(Rₜ − R̄)(Rₜ₋₁ − R̄)] ÷ Σ(t=1 to n)(Rₜ − R̄)². Rₜ is the return at observation t, R̄ is the mean of all n returns, and Σ means add the terms. Observations must use a consistent interval. The numerator uses n − 1 adjacent pairs; the denominator uses all n deviations. A constant series has a zero denominator, so the statistic is undefined.
Illustrative example: four successive daily returns are +1%, +1%, −1%, −1%. Their mean is 0%. In percentage-point units, adjacent products sum to 1 − 1 + 1 = 1; squared deviations sum to 1 + 1 + 1 + 1 = 4. Thus r₁ = 1 ÷ 4 = +0.25. Reorder the same returns as +1%, −1%, +1%, −1% and the numerator becomes −3, so r₁ = −0.75. The mean and dispersion are unchanged; the order changed.
Software can instead compute Pearson correlation between two shifted, separately centred lists. That is a different sample convention and need not give the same answer in a short sample. State the convention before comparing results.
Four observations illustrate arithmetic, not statistical evidence of persistence. A near-zero lag-one value does not prove independence: relationships may occur at other lags or be nonlinear. For example, the size of returns can depend on past sizes even when their signed linear correlation is small.
Lo and MacKinlay distinguish rejecting a random-walk model of prices from establishing market inefficiency. A random walk is a model in which successive price changes have specified unpredictable increments; rejecting one specification does not identify a dependable alternative. A historical pattern also needs separate evidence about stability, trading costs and future performance before any trading claim could follow.
No. It concerns deviations from the average return, including runs of below-average observations. It is a statistical tendency, not a direction forecast.
No. Zero linear correlation at a specified lag rules out only that linear relationship. Other lags and nonlinear dependence can remain.
No. Uncertainty in an average depends on dependence across all relevant lags and on the assumptions about the process. The Academy lesson shows an explicitly assumed example.
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