Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test
Source: Lo, A. W. & MacKinlay, A. C. (1988). Review of Financial Studies 1(1), 41–66. (NBER WP #2168, 1987)
TL;DR
Introduces the variance-ratio test and uses it to strongly reject the random-walk hypothesis for weekly U.S. stock returns over 1962–1985. Under a random walk, return variance grows linearly with the holding horizon; Lo and MacKinlay find it grows faster than linearly at short horizons — evidence of positive serial correlation, strongest for equal-weighted (small-stock-heavy) portfolios. The rejection survives corrections for heteroskedasticity and nonsynchronous trading.
Problem it solves
Provides a clean, low-power-loss test of the random-walk (weak-form efficiency) hypothesis that does not require specifying the full return-generating process. The key insight: if returns are uncorrelated, the variance of a q-period return is q times the variance of a one-period return, so the variance ratio identifies departures from a random walk through a single interpretable statistic.
The method
Variance ratio VR(q) = Var(q-period return) / (q × Var(1-period return)), which equals 1 under a random walk; VR(q) > 1 signals positive autocorrelation, < 1 mean reversion.
VR(q) − 1 is a particular (declining) linear combination of the first q−1 autocorrelations, so the test aggregates serial dependence.
Two asymptotically standard-normal test statistics: z under homoskedasticity and z* robust to heteroskedasticity (so a rejection cannot be ascribed to changing volatility).
Assumptions & inputs
Inputs: a single return series (or index/portfolio), a base sampling interval, and aggregation values q.
Sample: 1216 weekly observations, 6 Sep 1962 – 26 Dec 1985, CRSP equal- and value-weighted indices and size-sorted portfolios; tested over the full period and sub-periods.
Asymptotic inference; finite-sample properties checked by simulation.
How to use it / findings
Equal-weighted index, q=2: VR = 1.30, implying a weekly first-order autocorrelation of about 30%; with 1216 obs (std error ≈ 0.03) this rejects the random walk at any level.
VRs exceed 1 and decline in q (z, z* statistics weaken as q grows) — consistent with positive, declining short-horizon autocorrelation, not the negative correlation a "price fads"/mean-reversion model predicts.
Value-weighted index rejects more weakly (mainly the first 304 weeks); at a monthly base interval the random walk is generally not rejected — short-horizon structure washes out at coarser sampling.
Rejections are largest for small stocks but are not driven by infrequent trading or heteroskedasticity.
Limitations & pitfalls
Short-horizon autocorrelation can reflect microstructure (nonsynchronous trading, bid-ask bounce) rather than exploitable predictability.
Results are horizon-, index-, and sample-dependent; rejecting the random walk does not by itself imply inefficiency or a mean-reverting (stationary) price.
Inference is asymptotic; overlapping observations require the bias-corrected estimators used in the paper.
Key references
Lo, A. & MacKinlay, A. C. (1988) — Stock Market Prices Do Not Follow Random Walks — Review of Financial Studies
Lo, A. & MacKinlay, A. C. (1990) — When Are Contrarian Profits Due to Stock Market Overreaction? — Review of Financial Studies
Fama, E. (1970) — Efficient Capital Markets — Journal of Finance
Provenance: verified/generated from the paper's full text.
