This paper investigates the distribution of the least squares estimator of the coefficient @a in the model @c"t = @a@c"t -"1 + @?"t where the @?"t where the @?"t are independently distributed N (O, @s^2). The exact finite sample and limiting distributions are calculated when @a @> 1 and finite sample distributions when @? < 1. These distributions are used to compute the power functions of tests of the random walk hypothesis @a = 1 as well as the hypotheses.