The Review of Financial Studies/Spring,1988 where the weights are simply the weights in relation(18)squared. More formally, we have Tbeorem 3 Denote byd(i) ande(q) the asymptotic variances ofp()and M,( Q), respectively. Tben under the null hypothesis H* 1. Tbe statistics (@),J,q), M(q), M,( q), Ma(q), and M, a)all con verge almost surely to zero for all q as n increases without bound. 2. T'be following is a beteroscedasticity-consistent estimatorof d(j) ∑(X-X1-)2(X-x1-1)2 6(j)= (19) (Xr -Xr 3. The following is a beteroscedasticity-consistent estimator ofe(q) 2(q-j) 6(j) (20) Despite the presence of general heteroscedasticity, the standardized test statistic*(q=VnqM, q)/ve is still asymptotically standard normal. In Section 2 we use the z*(q) statistic to test empirically for random walks in weekly stock returns data 2. The Random Walk Hy pothesis for Weekly Returns To test for random walks in stock market prices, we focus on the 1216. week time span from September 6, 1962, to December 26, 1985. Our choice of a weekly observation interval was determined by several considerations Since our sampling theory is based wholly on asymptotic approximations a large number of observations is appropriate. While daily sampling yields many observations, the biases associated with nontrading, the bid-ask spread ynchronous prices, etc, are troublesome. Weekly sampling is the ideal compromise, yielding a large number of observations while minimizing the biases inherent in daily data The weekly stock returns are derived from the CrSP daily returns file The weekly return of each security is computed as the return from Wednes day s closing price to the following Wednesday's close If the following Wednesday's price is missing, then Thursday's price(or Tuesday's if Thurs day' s is missin d. If both Tuesday's and Thursday s prices are missing, the return for that week is reported as missing. 13 The average fraction(over all securities)of the entire sample where this occurs is less than 0.5 percent