Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test TORIo Andrew W. Lo; A. Craig MacKinlay The Review of Financial Studies, Vol. 1, No. 1( Spring, 1988), pp 41-66 Stable url: http://links.jstor.org/sici?sici=0893-9454%028198821%0291903a1%03c41%03asmpdnf93e2.0.co%03b2-u The Review of Financial Studies is currently published by Oxford University Press Your use of the jStOR archive indicates your acceptance of JSTOR,'s Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://wwwjstor.org/journ Each copy of any part of a STOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jStOR is an independent not-for-profit organization dedicated to creating and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support @ jstor. org http://www」]stor.org Wed Apr2615:39:002006
Stock market Prices Do Not Follow random Walks Evidence from a simple Specification Test Andrew W. lo Craig MacKinlay University of Pennsylvania In tbis article we test the random walk bypotbesis ance estimators derived from data sampled at dif ferent frequencies. Tbe random walk model is strongly rejected forthe entire sample period (1962- 1985)and for all subperiods for a variety ofaggre- gate returns indexes and size-sorted portfolios. largely to ior of small stocks, tbey cannot be attributed con pletely to tbe effects of infrequent trading or time- ying volatilities. Moreover, tbe rejection of tbe random walk for weekly returns does not support a mean-reverting model of asset prices ince Keynes's(1936)now famous pronouncement that most investors' decisions" can only be taken as a result of animal spirits--of a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of benefits multiplied by quantitative proba bilities, 'a great deal of research has been devoted xamining the efficiency of stock market price forma tion. In Fama's(1970)survey, the vast majority of those studies were unable to reject the" efficient markets This paper has benefited considerably from the suggestions of the editor Michae and the referee. We thank Cliff Ball, Don Keim, Whitney K Newe rsity, Ohio State University, Princeton University, Stanford UCLA ya havi Vinjamuri for preparing the manu Grant No. SES-8520054), and the u gratefully acknowledged. Any errors are of course our own. Address reprint quests to Andrew Lo, Department of Finance, Wharton School, University of Pennsylvania, Philadelphia, PA 19104 al Studies 1988 ber 1, pp. 41-66. c 1988 The Revi nancial Studies00219398/88/5904013$150 41
The Review of Financial Studies/Spring, 1988 hypothesis for common stocks. Although several seemingly anomalous departures from market efficiency have been well documented, many finan cial economists would agree with Jensen's(1978a)belief that"there is no other proposition in economics which has more solid empirical evidence supporting it than the Efficient Markets Hypothesis Although a precise formulation of an empirically refutable efficient mar kets hypothesis must obviously be model-specific, historically the majority of such tests have focused on the forecastability of common stock returns Within this paradigm, which has been broadly categorized as the"random walk"theory of stock prices, few studies have been able to reject the random walk model statistically. However, several recent papers have uncovered empirical evidence which suggests that stock returns contain predictable components. For example, Keim and Stambaugh(1986)find statistically significant predictability in stock prices by using forecasts based on certain predetermined variables. In addition, Fama and French(1987) show that long holding period returns are significantly negatively serially correlated, implying that 25 to 40 percent of the variation of longer-horizon returns is predictable from past returns In this article we provide further evidence that stock prices do not follow random walks by using a simple specification test based on variance esti mators. Our empirical results indicate that the random walk model is generally not consistent with the stochastic behavior of weekly returns especially for the smaller capitalization stocks. However, in contrast to the negative serial correlation that Fama and French(1987)found for longer horizon returns, we find significant positive serial correlation for weekly and monthly holding-period returns. For example, using 1216 weekly observations from September 6, 1962, to December 26, 1985, we compute the weekly first-order autocorrelation coefficient of the equal-weighted Center for Research in Security Prices(CRSP)returns index to be 30 per- cent! The statistical significance of our results is robust to heteroscedas ticity. We also develop a simple model which indicates that these large autocorrelations cannot be attributed solely to the effects of infrequent trading. This empirical puzzle becomes even more striking when we show that autocorrelations of individual securities are generally negative Of course, these results do not necessarily imply that the stock market is ineffcient or that prices are not rational assessments of"fundamental values As Leroy(1973)and Lucas(1978) have shown, rational expectations equilibrium prices need not even form a martingale sequence, of which the random walk is a special case. Therefore, without a more explicit conomic model of the price-generating mechanism, a rejection of the random walk hypothesis has few implications for the efficiency of market price formation. Although our test results may be interpreted as a rejection for example, the studies in Jensen's(1978b) volume on anomalous evidence regarding market eff
Test of tbe Random walk of some economic model of efficient price formation, there may exist other plausible models that are consistent with the empirical findings. Our more modest goal in this study is to employ a test that is capable of distinguishing among several interesting alternative stochastic price processes. Our test exploits the fact that the variance of the increments of a random walk is linear in the sampling interval. If stock prices are generated by a rando walk (possibly with drift), then, for example, the variance of monthly sampled log-price relatives must be 4 times as large as the variance of a weekly sample. Comparing the(per unit time) variance estimates obtained from weekly and monthly prices may then indicate the plausibility of the random walk theory. 2 Such a comparison is formed quantitatively along he lines of the Hausman(1978) specification test and is particularly simple In Section 1 we derive our specification test for both homoscedastic and heteroscedastic random walks. Our main results are given in Section 2 where rejections of the random walk are extensively documented for weekly returns indexes, size-sorted portfolios, and individual securities Section 3 contains a simple model which demonstrates that infrequent trading cannot fully account for the magnitude of the estimated autocorrelations of weekly stock returns. In Section 4 we discuss the consistency of our empirical rejections with a mean. reverting alternative to the random walk model We summarize briefly and conclude in Section 5 =和 o not provide any formal samplin Campbell and Mankiw(1987) and Cochrane(1987b)do derive the asymptotic more ce of der the nui Our variance ratio may, however, be related to the spectral-density estimates in the following fo) denote the spectral density of the increments AX, at frequency 0, we have the following relation x/(0)=1(0)+2∑?(k) where y(k)is the autocovariance function Dividing both sides by the variance y(o) then yields r"(0)=1+2∑p(k where. is the n ity and p(k) is the autocorrelation n Now sum on the right-hand side of the preceding equation m use this variance ratio, Huizinga (1987) does employ the Newey and West(1987) estimator of the normalized spectral densi
The Review of Financial Studies/Spring, 1988 1. The Specification Test Denote by P, the stock price at time t and define X, In P, as the log price process. Our maintained hypothesis is given by the recursive relation where u is an arbitrary drift parameter and E, is the random disturbance term. We assume throughout that for all t, Ele]=0, where E[] denotes the expectations operator. Although the traditional random walk hypoth esis restricts the e,'s to be independently and identically distributed (i i d) gaussian random variables, there is mounting evidence that financial time series often possess time-varying volatilities and deviate from normality Since it is the unforecastability, or uncorrelatedness, of price changes that is of interest, a rejection of the ii d. gaussian random walk because of heteroscedasticity or nonnormality would be of less import than a rejection that is robust to these two aspects of the data. In Section 1. 2 we develop a test statistic which is sensitive to correlated price changes but which is otherwise robust to many forms of heteroscedasticity and nonnormality Although our empirical results rely solely on this statistic, for purposes clarity we also present in Section 1. 1 the sampling theory for the more restrictive i i.d. gaussian random walk 1.1 Homoscedastic increments We begin with the null hypothesis H that the disturbances e, are indepen dently and identically distributed normal random variables with variance g2; thus, H: t, i.i.d. N(o, o2) In addition to homoscedasticity, we have made the assumption of inde pendent gaussian increments. An example of such a specification is the exact discrete-time process X, obtained by sampling the following well known continuous-time process at equally spaced intervals dX(t)=μat+aodW(t) (3) where dw(t denotes the standard Wiener differential. The solution to this stochastic differential equation corresponds to the popular lognormal diffusion price process One important property of the random walk X, is that the variance of its increments is linear in the observation interval That is, the variance of X, -X-2 is twice the variance of X, -X,-I Therefore, the plausibility of the random walk model may be checked by comparing the variance esti mate of X, one half the variance estimate of x This is the essence of our specification test; the remainder of this section is devoted to developing the sampling theory required to compare the vari ances quantitatively Suppose that we obtain 2n 1 observations Xo, XI X of X, at
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
extension in view of stock returns'well-documented empirical departure from normality. 10 Specifically, we consider the null hypothesis H*. 1, For all t, E(e )=0, and E(EE-r)=0 for any T 2.(6 )is o-mixing with coefficients p(m) of size r/(2r- 1) or is a-mixing with coefficients a(m)of size r/(r-1), where r>1, such that for all t and for any T 20, there exists some 8>0 for which E|∈-,|m<△<∞ (17) E(e2)=o2 4. For all t,E(∈1-天1-)=0 for any nonzero j and k where≠k but allows for onesis assumes that X, possesses uncorrelated increments ministic changes in the variance(due, for example, to seasonal factors) and Engle's(1982)ARCH processes(in which the conditional variance depends on past information) Since M, (g) still approaches zero under H*, we need only compute its asymptotic variance [call it 8(q)] to perform the standard inferences. We do this in two steps. First, recall that the following equality obtains asymp M(g∑2a-Dn cond. note that under H*(condition 4) the autocorrelation coefficient imators p(j) are asymptotically uncorrelated. 12 If we can obtain asymp totic variances 8()) for each of the p() under H*, we may readily calculate the asymptotic variance e(q)of M, q) as the weighted sum of the 8() do,however, allow for many other forms of leptokurtosis, such as that generated by Engle's(1982) autoregressive conditionally heteroscedastic (ARCH) process Condition 1 is the essential property of the random walk that we wish to test. Conditions 2 and 3 are the law of bers and the central lim condition 4 tions of are asymptotically uncorrelated; this condition may be weakened considerably at th of computational simplicity (see note 12) Although this restriction on the fourth cross-moments of e, may seem somewhat unintuitive, it of M(a via sing the results of Dufour (1981) and Dufour and Roy (1985). Again, this would sacrifice much of the icity of our asymptotic results
