equally spaced intervals and consider the following estimators for the unknown parameters u and az (Xr -XR-)=6(X,n-Xo 6=1∑(x.-x-1-p (4c) The estimators A and aZ correspond to the maximum-likelihood estimators of the u and o? parameters; ab is also an estimator of of but uses subset of n 1 observations Xo, X2, X4 X and corre sponds formally to i times the variance estimator for increments of even-numbered observations Under standard asymptotic theory, all three estimators are strongly consistent; that is, holding all other parameters constant as the total number of observations 2n increases without bound the estimators converge almost surely to their population values In addi tion, it is well known that both a2 and ab possess the following gaussian limiting distributions 2n(G2-a2)gN(0,2oa) (5a) /2n(G3-a2)gN(0,4oa where a indicates that the distributional equivalence is asymptotic Of course, it is the limiting distribution of the difference of the variances that interests us. Although it may readily be shown that such a difference is so asymptotically gaussian with zero mean, the variance of the limiting distribution is not apparent since the two variance estimators are clearly not asymptotically uncorrelated. However, since the estimator a2 is asymp totically efficient under the null hypothesis H, we may apply Hausman's ( 1978)result, which shows that the asymptotic variance of the difference is simply the difference of the asymptotic variances. If we define Ja =a- 02. then we have the result 2n/a a N(o, 2o) Using any consistent estimator of the asymptotic variance of Ja, a standard any other estimator of 0. If not, then there exists a linear combination of A, and 4.-0, that is more efficient than i. contradicting the assumed efficiency of 0. The result follows directly, then, since where avar( ) denotes the asymptotic variance operator
ignificance test may then be performed. a more convenient alternative test statistic is given by the ratio of the variances, J, G J2N(0,2) Although the variance estimator a% is based on the differences of every other observation, alternative variance estimators may be obtained by using the differences of every qth observation. Suppose that we obtain nq 1 observations Xo, X1 ng, where q is any integer greater than 1. Defin the estimators (X-X-1)=(Xn-X (8a) q一q )2 J(q)≡0(q)-62J(q)≡ oi(q) (8d) The specification test may then be performed using Theorem 1.5 Theorem 1. Under the null hypothesis H, the asymptotic distributions of Jaq and(a are given by VnqJ(q)思M(0,2(q-1)) (9a) Vng ga N(o, 2(q-1)) (9b) Two further refinements of the statistics Ja and result in more desirable nite-sample properties. The first is to use overlapping qth differences of X, in estimating the variances by defining the following estimator of o 2 a2 ( q (X-X-q-)2 (10) This differs from the estimator ab(q) since this sum contains ng-q+1 terms, whereas the estimator ab(q contains only n terms. By using over lapping gth increments, we obtain a more efficient estimator and hence a Note that if(@a)2 is used to estimate o then the standard -test of /,-0 will yield inferences identical nose obtained from th esponding test of ,=0 for the ratio, since S Proofs of all the theorems are given in the Appendix
Test of the Random walk more powerful test. Using a2(q) in our variance-ratio test, we define the corresponding test statistics for the difference and the ratio as M(q=a2(q)-a2 Mgs a2(g) The second refinement involves using unbiased variance estimators in the calculation of the M-statistics. Denote the unbiased estimators as aa and where X-1-2 (12a) nq-1 G2(q)= (Xr -Xe-a- gu) q+1 (12b) and define the statistics MAq)≡0(q)-Ma=(q 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 Infer ence for the overlapping variance differences and ratios may then be per formed using Theorem 2 Theorem 2. Under the null bypothesis H, the asymptotic distributions of the statistics Mq), M ( q), M(q), and M, q) are given by nqM. (q)a. (a) a No (14a) 2(2q-1)(q-1) nqM, ()avngM, (q)aNo (14b) In practice, the statistics in Equations (14)may be standardized in the usual manner[e.g, define the(asymptotically) standard normal test statistic x(q)=VnqM(q)(2(24-1)(q-1)/3q)-aNO,1) results of Monte Carlo experiments in Lo and MacKinlay (1987b), the be tatistics(which we denote as M,()and M, ( q)] does not depart significantly from that of limits even for small sample sizes. Therefore, all our empirical results are based on th M, (q)statistic. 47
Tbe Review of Financial Studies/ Spring 1988 To develop some intuition for these variance ratios, observe that for an aggregation value q of 2, the M, (q statistic may be reexpressed as M(2)=p(1) X-X-a)2+(Xn-X2n-1-a)]=(1)(15) Hence, for q=2 the M, (q) statistic is approximately the first-order auto correlation coefficient estimator p(1) of the differences. More generally, it may be shown that 9(1)+2(q-2)(2)+…+q-1)(16) M(q)≈2(q-1) here p(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 q- 1 autocorrelation coefficient estimators of the first differences with arithmetically declining weights, B 1.2 Heteroscedastic increments Since there is already a growing consensus among financial economists that volatilities do change over time, 9 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 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, q) 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 7 See Equation(A2-2) in the Appendix. s Note the similarity berween these variance ratios and the Box-Pierce Q-statistic, which is a linear combi xpect the finite-sample behavior of the variance ratios to be comparable to tha hey can have very different power properties under various MacKinlay(1987b)for further details. ee, for example, Merton (1980), Poterba and Summers(1986), and French, Schwert, and Stambaugh