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This paper examines the predictability of equity index returns for 18 developed countries. Based on the variance ratio test, the random walk hypothesis can be rejected at conventional significance levels for 11 countries with daily data and for 15 countries with weekly data. Monthly indices may well be characterized as a random walk for the majority of countries. The excess returns from buying past winners and selling past losers are positive and particularly striking for daily data, where they are not only statistically significant but also economically important in the absence of transaction costs. Imposing a reasonable transaction cost substantially reduces the profitability.