The Review of Financial Studies/Spring,1988 To develop some intuition for these variance ratios,observe that for an aggregation value g of 2,the M,()statistic may be reexpressed as 1 M2）=1）-4iaX-X-2+(x.-X-1-2月≥(10(15) Hence,for g=2 the M(g)statistic is approximately the first-order auto- correlation coefficient estimator p(1)of the differences.More generally,it may be shown that M(g)-29-卫1)+2g-262)+…+2(g-1） 2 (16) q q where (k)denotes the kth-order autocorrelation coefficient estimator of the first differences of X.7 Equation (16)provides a simple interpretation for the variance ratios computed with an aggregation value g:They are (approximately)linear combinations of the first g-1 autocorrelation coefficient estimators of the first differences with arithmetically declining weights.8 1.2 Heteroscedastic increments Since there is already a growing consensus among financial economists that volatilities do change over time,?a rejection of the random walk hypothesis because of heteroscedasticity would not be of much interest. We therefore wish to derive a version of our specification test of the random walk model that is robust to changing variances.As long as the increments are uncorrelated,even in the presence of heteroscedasticity the variance ratio must still approach unity as the number of observations increase without bound,for the variance of the sum of uncorrelated increments must still equal the sum of the variances.However,the asymptotic variance of the variance ratios will clearly depend on the type and degree of het- eroscedasticity present.One possible approach is to assume some specific form of heteroscedasticity and then to calculate the asymptotic variance of M,(g)under this null hypothesis.However,to allow for more general forms of heteroscedasticity,we employ an approach developed by White (1980)and by White and Domowitz (1984).This approach also allows us to relax the requirement of gaussian increments,an especially important See Equation (A2-2)in the Appendix. Note the similarity between these variance ratios and the Box-Pierce O-statistic,which is a linear combi- nation of squared autocorrelations with all the weights set identically equal to unity.Although we may expect the finite-sample behavior of the variance ratios to be comparable to that of the o-statistic under the null hypothesis,they can have very different power properties under various alternatives.See Lo and MacKinlay (1987b)for further details. See,for example,Merton (1980),Poterba and Summers (1986),and French,Schwert,and Stambaugh (1987). 48