This study examined the Efficiency Market hypothesis in its weak form using run tests, unit root tests and the Ljung-Box Q-statistics. The motivation was to determine whether foreign exchange rate returns follow a random walk. The data covered the period starting January 1994 to June 2007 for the daily closing spot price of the Kenya shillings per US dollar exchange rate. The main finding of this study is that the foreign exchange rate market is not efficient. The results showed that most of the rejections are due to significant patterns, trend stationarity and autocorrelation in foreign exchange returns. This is attributed to both
exchange rate undershooting and overshooting phenomena.