This paper tests the random walk hypothesis for weekly stock market returns using variance estimators derived from data sampled at different frequencies. The authors find strong evidence against the random walk model for the entire sample period (1962-1985) and all sub-periods, across various aggregate returns indexes and size-sorted portfolios. The rejections are primarily due to the behavior of small stocks, but cannot be attributed to infrequent trading or time-varying volatilities. The rejection of the random walk hypothesis is more consistent with a specific nonstationary alternative hypothesis rather than a mean-reverting stationary model. The authors also demonstrate that infrequent trading cannot account for the significant positive serial correlation in weekly and monthly holding-period returns, which is inconsistent with the random walk model. The results suggest that stock prices do not follow random walks and highlight the importance of considering nonstationary models in financial markets.This paper tests the random walk hypothesis for weekly stock market returns using variance estimators derived from data sampled at different frequencies. The authors find strong evidence against the random walk model for the entire sample period (1962-1985) and all sub-periods, across various aggregate returns indexes and size-sorted portfolios. The rejections are primarily due to the behavior of small stocks, but cannot be attributed to infrequent trading or time-varying volatilities. The rejection of the random walk hypothesis is more consistent with a specific nonstationary alternative hypothesis rather than a mean-reverting stationary model. The authors also demonstrate that infrequent trading cannot account for the significant positive serial correlation in weekly and monthly holding-period returns, which is inconsistent with the random walk model. The results suggest that stock prices do not follow random walks and highlight the importance of considering nonstationary models in financial markets.