Test of the Random Walk more powerful test.Using 2()in our variance-ratio test,we define the corresponding test statistics for the difference and the ratio as M(q)=2(q)-2M,(q)= (①-1 (11) 62 The second refinement involves using unbiased variance estimators in the calculation of the M-statistics.Denote the unbiased estimators as and (g),where ng-1 觉(X-X-,- (12a) g=1(x-X-,- m m=q(ng- - (12b) and define the statistics: MAq)=(q)- a.()= (①-1 (13) G品 Although this does not yield an unbiased variance ratio,simulation exper- iments show that the finite-sample properties of the test statistics are closer to their asymptotic counterparts when this bias adjustment is made.s Infer- ence for the overlapping variance differences and ratios may then be per- formed using Theorem 2. Tbeorem 2.Under the null bypotbesis H,the asymptotic distributions of the statistics Ma(q),M,(q),Ma(q),and M.(q)are given by VngMa(q)a VngMd(g)4 N0, (29-1)(9-1) (14a 3q VnaM.o Vngn(No,22-D(g-1 (14b) 3q In practice,the statistics in Equations (14)may be standardized in the usual manner [e.g.,define the (asymptotically)standard normal test statistic z(q)=VngM,(q(2(2q-1)(q-1)/3q-N(0,1]: .According to the results of Monte Carlo experiments in Lo and Mackinlay (1987b),the behavior of the bias-adjusted M-statistics [which we denote as M()and M,()]does not depart significantly from that of their asymptotic limits even for small sample sizes.Therefore,all our empirical results are based on the M(g)statistic. 47