The Review of Financial Studies/Spring, 1988 be rejected at all the usual significance levels for the entire time period and all subperiods. Moreover, the rejections are not due to changing val ances since the z*(@) statistics are robust to heteroscedasticity. The esti mates of the variance ratio are larger than 1 for all cases. For example, the entries in the first column of panel A correspond to variance ratios with an aggregation value q of 2. In view of Equation(15), ratios with g=2 are approximately equal to 1 plus the first-order autocorrelation coefficien estimator of weekly returns; hence, the entry in the first row, 1.30, implies that the first-order autocorrelation for weekly returns is approximately 30 percent. The random walk hypothesis is easily rejected at common levels of significance. The variance ratios increase with q, but the magnitudes of the z(q) statistics do not. Indeed, the test statistics seem to decline with L; hence, the significance of the rejections becomes weaker as coarser sample variances are compared to weekly variances. Our finding of positive autocorrelation for weekly holding-period returns differs from Fama and Frenchs(1987)finding of negative serial correlation for long holding period returns. This positive correlation is significant not only for our entire sample period but also for all subperiods The rejection of the random walk hypothesis is much weaker for the value-weighted index, as panel B indicates; nevertheless, the terns persist: the variance ratios exceed 1, and the z(q) statistics de pat as q increases. The rejections for the value-weighted index are due pri marily to the first 608 weeks of the sample period Table 1b presents the variance ratios using a base observation period of four weekS; hence, the first entry of the first row, 1.15, is the variance ratio of eight-week returns to four week returns. With a base interval of four weeks, we generally do not reject the random walk model even for the al-weighted index. This is consistent with the relatively weak evidence against the random walk that previous studies have found when using Although the test statistics in Tables 1a and 1b are based on nominal stock returns, it is apparent that virtually the same results would obtain wI eturns. Since the volatility of weekly nominal returns is so much larger than that of the inflation and Treasury-bill rates, the use of nominal, real, or excess returns in a volatility-based test will yield prac tically identical inferences 2.2 Results for size-based portfolios An implication of the work of Keim and Stambaugh(1986)is that, con ditional on stock and bond market variables, the logarithms of wealth elatives of portfolios of smaller stocks do not follow random walks. For portfolios of larger stocks, Keim and Stambaugh, s results are less conclu ive. Consequently, it is of interest to explore what evidence our tests provide for the random walk hypothesis for the logarithm of size-based portfolio wealth relatives We compute weekly returns for five size-based portfolios from the NYSE