How does the variance ratio test detect mean reversion or momentum?

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AcadiFi · Accepted Answer

Under a random walk, the variance of k-period returns equals k times the variance of one-period returns. The variance ratio is VR(k) = Var(r_t(k)) / (k * Var(r_t))...

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Lo and MacKinlay (1988) gave asymptotic standard errors under both homoskedastic and heteroskedastic nulls. The test statistic z = (VR(k) - 1) / sqrt(phi(k)) is asymptotically standard normal. For robustness, they derived a heteroskedasticity-consistent version widely used in practice.

At Falconhurst, suppose you compute VR(4) = 1.18 with standard error 0.06 using weekly returns over 10 years. z = 3.0, rejecting random walk at 1%. The 18% excess variance suggests short-term momentum — actionable for a trend-following overlay but you'd still have to beat transaction costs.

Beware subsampling bias with overlapping observations: Lo and MacKinlay's adjusted estimator is the correct one to use. Also run the test at multiple horizons k = 2, 4, 8, 16 and look for a pattern — persistent VR > 1 across horizons is stronger evidence than an isolated rejection. An AcadiFi Level III tip: don't infer tradability just from statistical rejection; costs, slippage, and regime stability matter more than a z-score.

How does the variance ratio test detect mean reversion or momentum?

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