The weak form of the efficient markets hypothesis is tested for eight African stock markets using three finite‐sample variance ratio tests. A rolling window captures short‐horizon predictability, tracks changes in predictability and is used to rank markets by relative predictability. These stock markets experience successive periods when they are predictable and then not predictable; this is consistent with the adaptive markets hypothesis. The degree of predictability varies widely: the least predictable African stock markets are those located in Egypt, South Africa and Tunisia, while the most predictable are in Kenya, Zambia and Nigeria.