The Runs Test for Randomness

Take a sequence of up (+) and down (−) daily returns and ask: is this genuinely random, or is there hidden structure — streakiness (momentum) or excessive flip-flopping (mean reversion)? Counting the actual number of "runs" — maximal streaks of the same sign — and comparing it to how many runs a truly random coin-flip sequence would typically produce answers this without needing to model the returns at all.

A sequence of length $n$ with $n_1$ pluses and $n_2$ minuses has an expected number of runs under randomness of $\mu_R = \frac{2 n_1 n_2}{n_1+n_2} + 1$, with a known variance formula alongside it, so a z-score comparing the observed run count to $\mu_R$ gives a standard hypothesis test.

A worked example

Twenty days of sign returns: + + + − − − + + + − − − + + + − − − + +. That's only 7 runs in 20 observations with roughly 12 pluses and 8 minuses — far fewer than the roughly 10-11 runs expected under randomness. Too few runs means the signs cluster into long streaks: a momentum pattern, not noise. Conversely, a sequence that alternates almost every single day (+ − + − + − ...) would show far more runs than expected, which is the signature of mean reversion, not randomness either.