The Review of Financial Studies Spring,1988 1.The Specification Test Denote by P,the stock price at time t and define X,In P,as the log- price process.Our maintained hypothesis is given by the recursive relation X,=4+X,-1+e, (1) where u is an arbitrary drift parameter and e,is the random disturbance term.We assume throughout that for all t,Efe]=0,where E[]denotes the expectations operator.Although the traditional random walk hypoth- esis restricts the e,'s to be independently and identically distributed (i.i.d.) gaussian random variables,there is mounting evidence that financial time series often possess time-varying volatilities and deviate from normality. Since it is the unforecastability,or uncorrelatedness,of price changes that is of interest,a rejection of the i.i.d.gaussian random walk because of heteroscedasticity or nonnormality would be of less import than a rejection that is robust to these two aspects of the data.In Section 1.2 we develop a test statistic which is sensitive to correlated price changes but which is otherwise robust to many forms of heteroscedasticity and nonnormality. Although our empirical results rely solely on this statistic,for purposes of clarity we also present in Section 1.1 the sampling theory for the more restrictive i.i.d.gaussian random walk 1.1 Homoscedastic increments We begin with the null hypothesis H that the disturbances e,are indepen- dently and identically distributed normal random variables with variance a;thus, H:ei.i.d.N(0,2) (2) In addition to homoscedasticity,we have made the assumption of inde pendent gaussian increments.An example of such a specification is the exact discrete-time process X,obtained by sampling the following well- known continuous-time process at equally spaced intervals: dX(t)-udt+o。dw(t) (3) where dw(t)denotes the standard Wiener differential.The solution to this stochastic differential equation corresponds to the popular lognormal diffusion price process. One important property of the random walk X,is that the variance of its increments is linear in the observation interval.That is,the variance of X,-X,-2 is twice the variance of X,-X-1.Therefore,the plausibility of the random walk model may be checked by comparing the variance esti- mate of X,-X,-1 to,say,one-half the variance estimate of X,-X,-2.This is the essence of our specification test;the remainder of this section is devoted to developing the sampling theory required to compare the vari- ances quantitatively. Suppose that we obtain 2n +1 observations X,Xi,...,Xam of X,at 44