Stock Market Prices Do Not
Follow Random Walks:
Evidence from a Simple
S p e c i f i c a t i o n T e s t
Andrew W. Lo
A. Craig MacKinlay
University of Pennsylvania
In this article we test the random walk hypothesis
for weekly stock market returns by comparing variance
estimators derived from data sampled at different
frequencies. The random walk model is
strongly rejected for the entire sample period (1962-
1985) and for all subperiod for a variety of aggregate
returns indexes and size-sorted portofolios.
Although the rejections are due largely to the behavior
of small stocks, they cannot be attributed completely
to the effects of infrequent trading or timevarying
volatilities. Moreover, the rejection of the
random walk for weekly returns does not support a
mean-reverting model of asset prices.
Since 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 probabilities,"
a great deal of research has been devoted to
examining the efficiency of stock market price formation.
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 Michael
Gibbons and the referee. We thank Cliff Ball, Don Keim, Whitney K. Newey,
Peter Phillips, Jim Poterba, Krishna Ramaswamy. Bill Schwert, and seminar
participants at MIT, the NBER-FMME Program Meeting (November 1986).
Northwestern University, Ohio State University, Princeton University, Stanford
University, UCLA, University of Chicago, University of Michigan, University of
Pennsylvania, University of Western Ontario, and Yale University for helpful
comments. We are grateful to Stephanie Hogue, Elizabeth Schmidt, and Madhavi
Vinjamuri for preparing the manuscript. Research support from the Geewax-Terker
Research Program in Investments, the National Science Foundation
(Grant No. SES-8520054), and the University of Pennsylvania Research Fund
is gratefully acknowledged. Any errors are of course our own. Address reprint
requests to Andrew Lo, Department of Finance, Wharton School, University of
Pennsylvania, Philadelphia, PA 19104.
The Review of Financial Studies 1988, Volume 1, number 1, pp. 41-66.
 1988 The Review of Financial Studies 0021-9398/88/5904-013 $1.50
41
hypothesis for common stocks. Although several seemingly anomalous
departures from market efficiency have been well documented,1
many financia1
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 markets
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 estimators.
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 longerhorizon
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 percent!
The statistical significance of our results is robust to heteroscedasticity.
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 inefficient 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
economic 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
1
See, for example, the studies in Jensen's (1978b) volume on anomalous evidence regarding market effi-
ciency.
42
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 random
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
the lines of the Hausman (1978) specification test and is particularly simple
to implement.
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.
2
The use of variance ratios is, of course, not new. Most recently, Campbell and Mankiw (1987), Cochrane
(1987a. 1987b), Fama and French (1987). French and Roll (1986). and Huizinga (1987) have all computed
variance ratios in a variety of contexts; however, these studies do not provide any formal sampling theory
for our statistics. Specifically, Cochrane (1987a), Fama and French (1987). and French and Roll (1986)
all rely on Monte Carlo simulations to obtain standard errors for their variance ratios under the null.
Campbell and Mankiw (1987) and Cochrane (1987b) do derive the asymptotic variance of the vat-lance
ratio but only under the assumption that the aggregation value q grows with (but more slowly than) the
sample size T. Specifically, they use Priestley's (1981, page 463) expression for the asymptotic variance
of the estimator of the spectral density of AX, at, frequency 0 (with a Bartlett window) as the appropriate
asymptotic variance of the variance ratio. But Priestley's result requires (among other things) that q  ,
T'  , and q/T  0. In this article we develop the formal sampling theory of the variance-ratio statistics
for the more general case.
Our variance ratio may, however, be related to the spectraldensity estimates in the following way. Letting
f(0) denote the spectral density of the increments  Xt, at frequency 0, we have the following relation:
where g (k) is the autocovariance function. Dividing both sides by the variance  (0) then yields
where f* is the normalized spectral density and p(k) is the autocorrelation function. Now in order to
estimate the quantity  f*(O). the infnite sum on the right-hand side of the preceding equation must
obviously be truncated. If, in addition to truncation, the autocorrelations are weighted using Newey and
West's (1987) procedure, then the resulting estimator is formally equivalent to our Mr(q) statistic. Although
he does not explicitly use this variance ratio, Huizinga (1987) does employ the Newey and West (1987)
estimator of the normalized spectral density.
43
1. The Specification Test
Denote by P, the stock price at time t and define Xt  ln Pt, as the logprice
process. Our maintained hypothesis is given by the recursive relation
(1)
where  is an arbitrary drift parameter and , t is the random disturbance
term. We assume throughout that for all t, E[ , t] = 0, where E[ˇ] denotes
the expectations operator. Although the traditional random walk hypothesis
restricts the , t'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 i.i.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 of
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 , t, are independently
and identically distributed normal random variables with variance
 0
2
; thus,
( 2 )
In addition to homoscedasticity, we have made the assumption of independent
gaussian increments. An example of such a specification is the
exact discrete-time process Xt; obtained by sampling the following wellknown
continuous-time process at equally spaced intervals:
( 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 Xt is that the variance of its
increments is linear in the observation interval. That is, the variance of
Xt- Xt-2 is twice the variance of Xt - Xt-1. Therefore, the plausibility of
the random walk model may be checked by comparing the variance estimate
of Xt - Xt-1 to, say, one-half the variance estimate of Xt - Xt-2. This
is the essence of our specification test; the remainder of this section is
devoted to developing the sampling theory required to compare the variances
quantitatively.
Suppose that we obtain 2n + 1 observations X0 X1, . . . , X2n of Xt at
44
equally spaced intervals and consider the following estimators for the
unknown parameters  and  0
2
:
The estimators and correspond to the maximum-likelihood estimators
of the  and  0
2
parameters; is also an estimator of  0
2
but uses only the
subset of n + 1 observations X0, X2, X4, . . . , X2n and corresponds
formally to 1/2 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 addition,
it is well known that both and possess the following gaussian
limiting distributions:
where 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
also 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 is asymptotically
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.3
If we define
then we have the result
Using any consistent estimator of the asymptotic variance of Jd, a standard
3
Briefly, Hausman (1978) exploits the fact that any asymptotically efficient estimator of a parameter , say
must possess the property that it is asymptotically uncorrelated with the difference where is
any other estimator of . If not, then there exists a linear combination of that is more efficient
than , contradicting the assumed efficiency of The result follows directly, then, since
where avar (ˇ) denotes the asymptotic variance operator.
45
significance test may then be performed. A more convenient alternative
test statistic is given by the ratio of the variances, Jr: 4
Although the variance estimator 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 X0, X1, . . . , Xnq, where q is any integer greater than 1. Define
the estimators:
The specification test may then be performed using Theorem l.5
Theorem 1. Under the null hypothesis H, the asymptotic distributions of
Jd(q) and Jr(q) are given by
Two further refinements of the statistics Jd and Jr result in more desirable
finite-sample properties. The first is to use overlapping qth differences of
Xt, in estimating the variances by defining the following estimator of
This differs from the estimator since this sum contains nq - q + 1
terms, whereas the estimator contains only n terms. By using overlapping
qth increments, we obtain a more efficient estimator and hence `a
4
Note that if is used to estimate  0
4
, then the standard t-test of Jd = 0 will yield inferences identical to
those obtained from the corresponding test of Jr = 0 for the ratio, since
5
Proofs of all the theorems are given in the Appendix.
46
more powerful test. Using in our variance-ratio test, we define the
corresponding test statistics for the difference and the ratio as
The second refinement involves using unbiased variance estimators in the
calculation of the M -statistics. Denote the unbiased estimators as and
w h e r e
and define the statistics:
Although this does not yield an unbiased variance ratio, simulation experiments
show that the finite-sample properties of the test statistics are closer
to their asymptotic counterparts when this bias adjustment is made.6
Inference
for the overlapping variance differences and ratios may then be performed
using Theorem 2.
Theorema 2. Under the null hypothesis H, the asymptotic distribution of
the statistics Md(q), Mr(q), and are given by
In practice, the statistics in Equations (14) may be standardized in the
usual manner [e.g., define the (asymptotically) standard normal test statistic
6
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 and does not depart significantly from that of
their asymptotic limits even for small sample sizes. Therefore, all our empirical results are based on the
statistic.
47
To develop some intuition for these variance ratios, observe that for an
aggregation value q of 2, the Mr(q) statistic may be reexpressed as
Hence, for q= 2 the M,(q) statistic is approximately the first-order autocorrelation
coefficient estimator of the differences. More generally, it
may be shown that
where denotes the kth-order autocorrelation coefficient estimator of
the first differences of Xt.7
Equation (16) provides a simple interpretation
for the variance ratios computed with an aggregation value q: They are
(approximately) linear combinations of the first q - 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,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
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 heteroscedasticity
present. One possible approach is to assume some specific
form of heteroscedasticity and then to calculate the asymptotic variance
of 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.
8
Note the similarity between these variance ratios and the Box-Pierce Q-statistic, which is a linear combination
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 Q-statistic under
the null hypothesis, they an have very different power properties under various alternatives. See Lo and
MacKinlay (1987b) for further details.
9
See, for example, Merton (1980). Poterba and Summers (1986). and French, Schwert and Stambaugh
(1987).
48
extension in view of stock returns' well-documented empirical departures
from normality.10
Specifically, we consider the null hypothesis H*:11
l.For all t, E ( , t)=0, and E( , t , t - ) =0 for any 0.
2. { , t} is &mixing with coefficients  (m) of size r/(2r - 1) or is a-mixing
with coefficients  (m) of size r/(r - l), where r > 1, such that for all t
and for any   0, there exists some  > 0 for which
4. For all t, for any nonzero j and k where j  k.
This null hypothesis assumes that Xt possesses uncorrelated increments
but allows for quite general forms of heteroscedasticity, including deterministic
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 still approaches zero under H*, we need only compute its
asymptotic variance [call it  (q)] to perform the standard inferences. We
do this in two steps. First, recall that the following equality obtains asymp-
totically:
Second, note that under H* (condition 4) the autocorrelation coefficient
estimators are asymptotically uncorrelated.12
If we can obtain asymptotic
variances  (j) for each of the under H*, we may readily calculate
the asymptotic variance  (q) of as the weighted sum of the  (j),
10
Of course, second moments are still assumed to be finite; otherwise, the variance ratio is no longer well
defined. This rules out distributions with infinite variance, such as those in the stable Pareto-Levy family
(with characteristic exponents that are less than 2) proposed by Mandelbrot (l963) and Fama (1965). We
do, however, allow for many other forms of leptokurtosis, such as that generated by Engle's (1982)
autoregressive conditionally heteroscedastic (ARCH) process.
11
Condition 1 is the essential property of the random walk that we wish to test. Conditions 2 and 3 are
restrictions on the maximum degree of dependence and heterogeneity allowable while still permitting
some form of the law of large numbers and the central limit theorem to obtain. See White (1954) for the
precise definitions of . and  -mixing random sequences. Condition 4 implies that the sample autocorrelations
of , t are asymptotically uncorrelated; this condition may be weakened considerably at the expense
of computational simplicity (see note 12).
12
Although this restriction on the fourth crass-moments of , t may seem somewhat unintuitive, it is satisfied
for any process with independent increments (regardless of heterogeneity) and also for linear gaussian
ARCH processes. This assumption may be relaxed entirely, requiring the estimation of the asymptotic
covariances of the autocorrelation estimators in order to estimate the limiting variance  of via
relation (18). Although the resulting estimator of  would be more complicated than Equation (20), it is
conceptually straightforward and may readily be formed along the lines of Newey and West (1987). An
even more general (and possibly more exact) sampling theory for the variance ratios may be obtained
using the results of Dufour (1981) and Dufour and Roy (1985). Again, this would sacrifice much of the
simplicity of our asymptotic rezults.
49
where the weights are simply the weights in relation (18) squared. More
formally, we have:
Theorem 3. Denote by d (j) and q (q) the asymptotic variances of and
, respectively. Then under the null hypothesis H*:
verge almost surely to zero for all q as n increases without bound.
2. The following is a heteroscedasticity-consistent estimator of dd (j):
3. The following is a heteroscedasticity-consistent estimator of qq (q):
Despite the presence of general heteroscedasticity, the standardized test
statistic is still asymptotically standard normal. In
Section 2 we use the z*(q) statistic to test empirically for random walks
in weekly stock returns data.
2. The Random Walk Hypothesis for Weekly Returns
To test for random walks in stock market prices, we focus on the 1216week
time span from September 6, 1962, to December 26, 1985. Our choice
of a weekly observation interval was determined by several considerations.
Since our sampling theory is based wholly on asymptotic approximations,
a large number of observations is appropriate. While daily sampling yields
many observations, the biases associated with nontrading, the bid-ask spread,
asynchronous prices, etc., are troublesome. Weekly sampling is the ideal
compromise, yielding a large number of observations while minimizing
the biases inherent in daily data.
The weekly stock returns are derived from the CRSP daily returns file.
The weekly return of each security is computed as the return from Wednesday's
closing price to the following Wednesday's close. If the following
Wednesday's price is missing, then Thursday's price (or Tuesdays if Thursday's
is missing) is used. If both Tuesday's and Thursday's prices are
missing, the return for that week is reported as missing.13
13
The average fraction (over all securities) of the entire sample where this occurs is less than 0.5 percent
of the time for the 1216-week sample period.
Table 1a
Market index results for a one-week base observation period
Variance-ratio test of the random walk hypothesis for CRSP equal- and value-weighted indexes, for the
sample period from September 6,1962, to December 26, 1985, and subperiods. The variance ratios 1 +
are repotted in the main rows, with the heteroscedasticity-robust test statistics z*(q) given in
parentheses immediately below each main row. Under the random walk null hypothesis, the value of the
variance ratio is 1 and the test statistics have a standard normal distribution (asymptotically). Test statistics
marked with asterisks indicate that the corresponding variance ratios are statistically different from 1 at
the 5 percent level of significance.
In section 2.1 we perform our test on both equal- and value-weighted
CRSP indexes for the entire 1216-week period, as well as for 608-week
subperiods, using aggregation values q ranging from 2 to 16.14
Section 2.2
reports corresponding test results for size-sorted portfolios, and Section
2.3 presents results for individual securities.
2.1 Results for market indexes
Tables 1 a and 1 b report the variance ratios and the test statistics z*(q) for
CRSP NYSE-AMEX market-returns indexes. Table la presents the results
for a one-week base observation period, and Table 1 b reports similar results
for a four-week base observation period. The values reported in the main
rows are the actual variance ratios and the entries enclosed
in parentheses are the z*(q) statistics.15
Panel A of Table 1a displays the results for the CRSP equal-weighted
index. The first row presents the variance ratios and test statistics for the
entire 1216-week sample period, and the next two rows give. the results
for the two 608-week subperiods. The random walk null hypothesis may
14
Additional empirical results (304-week subperiods, larger q values, etc.) arc reported In Lo and MacKinlay
(1987a).
15
Since the values of z*(q) are always smaller than the values of z(q) in our empirical results, to conserve
space we report only the more conservativc statistics. Both statistics are reported in Lo and MacKinlay
(1987a).
be rejected at all the usual significance levels for the entire time period
and all subperiods. Moreover, the rejections are not due to changing variances
since the z*(q) statistics are robust to heteroscedasticity. The estimates
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 q = 2
are approximately equal to 1 plus the first-order autocorrelation coefficient
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
q; hence, the significance of the rejections becomes weaker as coarsersample
variances are compared to weekly variances. Our finding of positive
autocorrelation for weekly holding-period returns differs from Fama and
French's (1987) finding of negative serial correlation for long holdingperiod
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 general patterns
persist: the variance ratios exceed 1, and the z*(q) statistics decline
as q increases. The rejections for the value-weighted index are due primarily
to the first 608 weeks of the sample period.
Table 1 b 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
equal-weighted index. This is consistent with the relatively weak evidence
against the random walk that previous studies have found when using
monthly data.
Although the test statistics in Tables l a and 1 b are based on nominal
stock returns, it is apparent that virtually the same results would obtain
with real or excess returns. 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 practically
identical inferences.
2.2 Results for size-based portfolios
An implication of the work of Keim and Stambaugh (1986) is that, conditional
on stock and bond market variables, the logarithms of wealth
relatives of portfolios of smaller stocks do not follow random walks. For
portfolios of larger stocks, Keim and Stambaugh's results are less conclusive.
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-
52
Table 1 b
Market index results for a four-week base observation period
Variance-ratio test of the random walk hypothesis for CRSP equal- and value-weighted Indexes, for
the sample period from September 6, 1962, to December 26, 1985, and subperiods. The variance ratios
are reported in the main rows, with the heteroscedasticity-robust test statistics z*(q) given in
parentheses immediately below each main row. Under the random walk null hypothesis, the value of the
variance ratio is 1 and the test statistics have a standard normal distribution (asymptotically). Test statistics
marked with asterisks indicate that the corresponding variance ratios are statistically different from 1 at
the 5 percent level of significance.
AMEX universe on the CRSP daily returns file. Stocks with returns for any
given week are assigned to portfolios based on which quintile their market
value of equity is in. The portfolios are equal-weighted and have a continually
changing composition. 16
The number of stocks included in the
portfolios varies from 2036 to 2720.
Table 2 reports the test results for the size-based portfolios, using
a base observation period of one week. Panel A reports the results for the
portfolio of small firms (first quintile), panel B reports the results for the
portfolio of medium-size firms (third quintile), and panel C reports the
results for the portfolio of large firms (fifth quintile). Evidence against the
random walk hypothesis for small firms is strong for all time periods considered;
in panel A all the z*(q) statistics are well above 2.0, ranging from
6.12 to 11.92. As we proceed through the panels to the results for the
portfolio of large firms, the r*(q) statistics become smaller, but even for
the large-firms portfolio the evidence against the null hypothesis is strong.
As in the case of the returns indexes, we may obtain estimates of the firstorder
autocorrelation coefficient for returns on these size-sorted portfolios
simply by subtracting 1 from the entries in the q = 2 column. The values
16
We also performed our tests using value-weighted portfolios and obtained essentially the same results.
The only difference appeared in the largest quintile of the value-weighted portfolio, for which the random
walk hypothesis was generally not rejected. This, of course, is not surprising, given that the largest valueweighted
quintile is quite similar to the value-weighted market index.
53
Table 2
Size-sorted portfolio results for a one-week base observation period
in Table 2 indicate that the portfolio returns for the smallest quintile have
a 42 percent weekly autocorrelation over the entire sample period! Moreover,
this autocorrelation reaches 49 percent in subperiod 2 (May 2,1974,
to December 26, 1985). Although the serial correlation for the portfolio
returns of the largest quintile is much smaller (14 percent for the entire
sample period), it is statistically significant.
Using a base observation interval of four weeks, much of the evidence
against the random walk for size-sorted portfolios disappears. Although
the smallest-quintile portfolio still exhibits a serial correlation of 23 percent
with a z*(2) statistic of 3.09, none of the variance ratios for the largestquintile
portfolio is significantly different from 1. In the interest of brevity,
we do not report those results here but refer interested readers to Lo and
MacKinlay (1987a).
The results for size-based portfolios are generally consistent with those
for the market indexes. The patterns of (1) the variance ratios increasing
in q and (2) the significance of rejections decreasing in q that we observed
for the indexes also obtain for these portfolios. The evidence against the
T a b l e 3
Individual securities results for a one-week base observation period
Means of variance ratios over all individual securities with complete return histories from September 2,
1062, to December 26, 1985 (625 stocks). Means of variance ratios for the smallest 100 stocks, the Intermediate
100 stocks, and the largest 100 stocks are also reported. For purposes of comparison, panel B
reports the variance ratios for equal- and value-weighted portfolios, respectively, of the 625 stocks. Parenthetical
entries for averages of individual securities (panel A) are standard deviations of the cross section
of variance ratios. Because the variance ratios are not cross-sectionally independent, the standard deviations
cannot be used to perform the usual significance tests; they are reported only to provide an indication of
the variance ratios' cross-sectional dispersion. Parenthetical entries for portfolio variance ratios (panel B)
are the heteroscedasticity -robust z*(q) statistics. Asterisks indicate variance ratios that are statistically
different from 1 at the 5 percent level of significance.
random walk hypothesis for the logarithm of wealth relatives of small-firms
portfolios is strong in all cases considered. For larger firms and a one-week
base observation interval, the evidence is also inconsistent with the random
walk; however, as the base observation interval is increased to four weeks,
our test does not reject the random walk model for larger firms.
2.3 Results for individual securities
For completeness, we performed the variance-ratio test on all individual
stocks that have complete return histories in the CRSP database for our
entire 1216-week sample period, yielding a sample of 625 securities. Owing
to space limitations, we report only a brief summary of these results in
Table 3. Panel A contains the cross-sectional means of variance ratios for
the entire sample as well as for the 100 smallest, 100 intermediate, and
100 largest stocks. Cross-sectional standard deviations are given in parentheses
below the main rows. Since the variance ratios are clearly not crosssectionally
independent, these standard deviations cannot be used to form
the usual tests of significance; they are reported only to provide some
indication of the cross-sectional dispersion of the variance ratios.
The average variance ratio for individual securities is less than unity
when q = 2, implying that there is negative serial correlation on average.
55
For all stocks, the average serial correlation is -3 percent, and -6 percent
for the smallest 100 stocks. However, the serial correlation is both statistically
and economically insignificant and provides little evidence against
the random walk hypothesis. For example, the largest average z*(q) statistic
over all stocks occurs for q = 4 and is -0.90 (with a cross-sectional
standard deviation of 1.19); the largest average z*(q) for the 100 smallest
stocks is -1.67 (for q = 2, with a cross-sectional standard deviation of
1.75). These results complement French and Roll's (1986) finding that
daily returns of individual securities are slightly negatively autocorrelated.
For comparison, panel B reports the variance ratios of equal- and valueweighted
portfolios of the 625 securities. The results are consistent with
those in Tables 1 and 2; significant positive autocorrelation for the equalweighted
portfolio, and less significant positive autocorrelation for the
value-weighted portfolio.
That the returns of individual securities have statistically insignificant
autocorrelation is not surprising. Individual returns contain much company-specific,
or "idiosyncratic," noise that makes it difficult to detect the
presence of predictable components. Since the idiosyncratic noise is largely
attenuated by forming portfolios, we would expect to uncover the predictable
"systematic" component more readily when securities are combined.
Nevertheless, the negativity of the individual securities' autocorrelations
is an interesting contrast to the positive autocorrelation of the
portfolio returns. Since this is a well-known symptom of infrequent trading,
we consider such an explanation in Section 3.
3. Spurious Autocorrelation Induced by Nontrading
Although we have based our empirical results on weekly data to minimize
the biases associated with market microstructure issues, this alone does
not ensure against the biases' possibly substantial influences. In this section
we explicitly consider the conjecture that infrequent or nonsynchronous
trading may induce significant spurious correlation in stock retums.17
The
common intuition for the source of such artificial serial correlation is that
small capitalization stocks trade less frequently than larger stocks. Therefore,
new information is impounded first into large-capitalization stock
prices and then into smaller-stock prices with a lag. This lag induces a
positive serial correlation in, for example, an equal-weighted index of stock
returns. Of course, this induced positive serial correlation would be less
pronounced in a value-weighted index. Since our rejections of the random
walk hypothesis are most resounding for the equal-weighted index, they
may very well be the result of this nontrading phenomenon. To investigate
this possibility, we consider the following simple model of nontrading.18
17
See, for example, Scholes and Williams (1977) and Cohen et al. (1983).
18
Although our model is formulated in discrete time for simplicity, it is in fact slightly more general than
the Scholes and Williams (1977) continuous-time model of nontrading. Specifically, Scholes and Williams
56
Suppose that our universe of stocks consists of N securities indexed by
i, each with the return-generating process
where RMt represents a factor common to all returns (e.g., the market) and
is assumed to be an independently and identically distributed (i.i.d.) random
variable with mean M and variance  M
2
. The , it term represents the
idiosyncratic component of security i's return and is also assumed to be
i.i.d. (over both i and t), with mean 0 and variance  M
2
. The return-generating
process may thus be identified with N securities each with a unit beta such
that the theoretical R2
of a market-model regression for each security is
0.50.
Suppose that in each period t there is some chance that security i does
not trade. One simple approach to modeling this phenomenon is to distinguish
between the observed returns process and the virtual returns
process. For example, suppose that security i has traded in period t - 1;
consider its behavior in period t. If security i does not trade in period t,
we define its virtual return as Rit [which is given by Equation (21)], whereas
its observed return Ro
it is zero. If security i then trades at t + 1, its observed
return R0
it+1 is defined to be the sum of its virtual returns Rit and Rit+1 hence,
nontrading is assumed to cause returns to cumulate. The cumulation of
returns over periods of nontrading captures the essence of spuriously
induced correlations due to the nontrading lag.
To calculate the magnitude of the positive serial correlation induced by
nontrading, we must specify the probability law governing the nontrading
event. For simplicity, we assume that whether or not a security trades may
be modeled by a Bernoulli trial, so that in each period and for each security
there is a probability p that it trades and a probability 1 - p that it does
not. It is assumed that these Bernoulli trials are i.i.d. across securities and,
for each security, are i.i.d. over time. Now consider the observed return
R0
t at time t of an equal-weighted portfolio:
The observed return R0
it for security i may be expressed as
where X,,(j), j = 1, 2, 3, . . . are random variables defined as
(23)
implicitly assume that each security trades at least once within a given time interval by "ignoring periods
over which no trades occur" (page 311), whereas our model requires no such restriction. As a consequence,
it may be shown that, ceteris paribus, the magnitude of spuriously induced autocorrelation is lower in
Scholes and Williams (1977) than in our framework. However, the qualitative predictions of the two models
of nontrading are essentially the same. For example, both models imply that returns for individual securities
will exhibit negative serial correlation but that portfolio returns will be positively autocorrelated.
57
The Xit(j) variables are merely indicators of the number of consecutive
periods before t in which security j has not traded. Using this relation, we
have
For large N, it may readily be shown that because the , it, component of
each security's return is idiosyncratic and has zero expectation, the following
approximation obtains:
It is also apparent that the averages become arbitrarily close,
again for large N, to the probability of j consecutive no-trades followed by
a trade; that is,
The observed equal-weighted return is then given by the approximation
Using this expression, the general jth-order autocorrelation coefficient q(j)
may be readily computed as
Assuming that the implicit time interval corresponding to our single period
is one trading day, we may also compute the weekly (five-day) first-order
autocorrelation coefficient of R°t as
By specifying reasonable values for the probability of nontrading, we may
calculate the induced autocorrelation using Equation (30). To develop
58
Table 4
Magnitudes of nontrading-induced autocorrelation in returns
Spuriously induced autocorrelations are reported for nontrading probabilities 1 - p of 10 to 50 percent.
In the absence of the nontrading phenomenon, the theoretical values of daily jth-order autocorrelations
q(j) and the weekly first-order autocorrelation qw
(1) are all zero.
some intuition for the parameter p, observe that the total number of securities
that trade in any given period t is given by the sum Under
our assumptions, this random variable has a binomial distribution with
parameters (N, p); hence, its expected value and variance are given by Np
and Np(1 - p), respectively. Therefore, the probability p may be interpreted
as the fraction of the total number of N securities that trades on
average in any given period. A value of .90 implies that, on average, 10
percent of the securities do not trade in a single period.
Table 4 presents the theoretical daily and weekly autocorrelations induced
by nontrading for nontrading probabilities of 10 to 50 percent. The first
row shows that when (on average) 10 percent of the stocks do not trade
each day, this induces a weekly autocorrelation of only 2.1 percent! Even
when the probability of nontrading is increased to 50 percent (which is
quite unrealistic), the induced weekly autocorrelation is 17 percent.19
We
conclude that our rejection of the random walk hypothesis cannot be
attributed solely to infrequent trading.
19
Severa1 other factors imply that the actual sizes of the spurious autocorrelations induced by infrequent
trading are lower than those given in Table 4. For example, in calculating the induced correlations using
Equation (29). we have ignored the idiosyncratic components in returns because dversification makes
these components trivial in the limit; in practice, perfect diversification is never achieved. But any residual
risk increases the denominator of Equation (29) and does not necessarily increase the numerator (since
the , it's are cross-sectionally uncorrelated). To see this explicitly, we simulated the returns for 1000 stocks
over 5120 days, calculated the weekly autcorrelations for the virtual returns and for the observed returns,
computed the difference of those autocorrelations, repeated this procedure 20 times, and then averaged
the differences. With a (daily) nontrading probability of 10 percent, the simulations yield a difference in
weekly autocorrelations of 2.1 percent, of 4.3 percent for a nontrading probability of 20 percent, and of
7.6 percent for a nontrading probability of 30 percent.
Another factor that may reduce the spurious positive autocorrelation empirically is that, within the CRSP
files, if a security does not trade, its price is reported as the average of the bid-ask spread. As long as the
specialist adjusts the spread to reflect the new information, even if no trade occurs the reported CRSP
price will reflect the new information. Although there may still be some delay before the bid-ask spread
is adjusted, it is presumably less than the lag between trades.
Also, if it is assumed that the probability of no-trades depends upon whether or not the security has
traded recently, it is natural to suppose that the likelihood of a no-trade tomorrow is lower if there is a
no-trade today. In this case, it may readily be shown that the induced autocorrelation is even lower than
that computed in our i.i.d. framework.
59
The positive autocorrelation of portfolio returns and the negative autocorrelation
of individual securities is puzzling. Although our stylized model
suggests that infrequent trading cannot fully account for the 30 percent
autocorrelation of the equal-weighted index, the combination of infrequent
trading and Roll's (1984) bid-ask effect may explain a large part of
the small negative autocorrelation in individual returns.
One possible stochastic model that is loosely consistent with these observations
is to let returns be the sum of a positively autocorrelated common
component and an idiosyncratic white-noise component. The common
component induces significant positive autocorrelation in portfolios since
the idiosyncratic component is trivialized by diversification. The whitenoise
component reduces the positive autocorrelation of individual stock
returns, and the combination of infrequent trading and the bid-ask spread
effects drives the autocorrelation negative. Of course, explicit statistical
estimation is required in order to formalize such heuristics and, ultimately,
what we seek is an economic model of asset prices that might give rise to
such empirical findings. This is beyond the scope of our present article,
but it is the focus of current investigation.
The Mean-Reverting Alternative to the Random Walk
Although the variance-ratio test has shown weekly stock returns to be
incompatible with the random walk model, the rejections do not offer any
explicit guidance toward a more plausible model for the data. However,
the patterns of the test's rejections over different base observation intervals
and aggregation values q do shed considerable light on the relative merits
of competing alternatives to the random walk. For example, one currently
popular hypothesis is that the stock-returns process may be described by
the sum of a random walk and a stationary mean-reverting component, as
in Summers (1986) and in Fama and French (1987).20
One implication of
this alternative is that returns are negatively serially correlated for all holding
periods. Another implication is that, up to a certain holding period,
the serial correlation becomes more negative as the holding period
increases.2l
If returns are in fact generated by such a process, then their
variance ratios should be less than unity when q = 2 (since negative serial
20
Shiller and Perron (1985) propose only a mean-reverting process (the Ornstein-Uhlenbeck process),
whereas Poterba and Summers (1987) propose the sum of a random walk and a stationary mean-reverting
process. Although neither study offers any theoretical justification for its proposal, both studies motivate
their alternatives as models of investors' fads.
21
If returns are generated by the sum of a random walk and a stationary mean-reverting process, their serial
correlation will be a U-shaped function of the holding period; the first-order autocorrelation becomes
more negative as shorter holding periods lengthen, but it gradually returns to zero for longer holding
periods becausc the random walk component domnates. The curvature of thls U-shaped function depends
on the relative variability of the random walk and mean-reverting components. Fama and French's (1987)
parameter estimates imply that the autocorrelation coefficient is monotonically decreasing for holding
periods up to three years; that is, the minimum of the U-shaped curve occurs at a holding period greater
than or equal to three years.
60
5.
correlation is implied by this process). Also, the rejection of the random
walk should be stronger as q increases [larger z*(q) values for larger q].22
But Tables 1 and 2 and those in Lo and MacKinlay (1987a) show that both
these implications are contradicted by the empirical evidence.23
Weekly
returns do not follow a random walk, but they do not fit a stationary meanreverting
alternative any better.
Of course, the negative serial correlation in Fama and French's (1987)
study for long (three- to five-year) holding-period returns is, on purely
theoretical grounds, not necessarily inconsistent with positive serial correlation
for shorter holding-period returns. However, our results do indicate
that the sum of a random walk and a mean-reverting process cannot
be a complete description of stock-price behavior.
Conclusion
We have rejected the random walk hypothesis for weekly stock market
returns by using a simple volatility-based specification test. These rejections
cannot be explained completely by infrequent trading or time-varying
volatilities. The patterns of rejections indicate that the stationary meanreverting
models of Shiller and Perron (1985), Summers (1986), Poterba
and Summers (1987)) and Fama and French (1987) cannot account for the
departures of weekly returns from the random walk.
As we stated in the introduction, the rejection of the random walk model
does not necessarily imply the inefficiency of stock-price formation. Our
results do, however, impose restrictions upon the set of plausible economic
models for asset pricing; any structural paradigm of rational price formation
must now be able to explain this pattern of serial correlation present in
weekly data. As a purely descriptive, tool for examining the stochastic
evolution of prices through time, our specification test also serves a useful
purpose, especially when an empirically plausible statistical model of the
price process is more important than a detailed economic paradigm of
equilibrium. For example, the pricing of complex financial claims often
depends critically upon the specific stochastic process driving underlying
asset returns. Since such models are usually based on arbitrage considerations,
the particular economic equilibrium that generates prices is of less
consequence. One specific implication of our empirical findings is that
the standard Black-Scholes pricing formula for stock index options is mis-
specified.
Although our variance-based test may be used as a diagnostic check for
the random walk specification, it is a more difficult task to determine
22
This pattern of stronger rejections with larger q is also only true up to a certain value of q. In view of Fama
and French's (1987) results, this upper limit for q is much greater than 16 when the base observation
interval is one week. See note 21.
23
See Lo and MacKinlay (1987b) for explicit power calculations against this alternative and against a more
empirically relevant model of stock prices.
61
precisely which stochastic process best fits the data. The results of French
and Roll (1986) for return variances when markets are open versus when
they are closed add yet another dimension to this challenge. The construction
of a single stochastic process that fits both short and long holdingperiod
returns data is one important direction for further investigation.
However, perhaps the more pressing problem is to specify an economic
model that might give rise to such a process for asset prices, and this will
be pursued in subsequent research.
Appendix
Proof of Theorem 1
Under the i.i.d. gaussian distributional assumption of the null hypothesis
are maximum-likelihood estimators of  0
2
with respect to data
sets consisting of every observation and of every qth observation, respectively
(the dependence of on q is suppressed for notational simplicity).
Therefore, it is well known that
Since, under the null hypothesis H, is the maximum-likelihood estimator
of  0
2
using every observation, it is asymptotically efficient. Therefore, following
Hausman's (1978) approach, we conclude that the asymptotic variance
of is simply the difference of the asymptotic variances
Thus, we have
The asymptotic distribution of the ratio then follows by applying the "delta
method" to the quantity where the bivariate
function g is defined as g(u, v) " v/u; hence,
Proof of Theorem 2
To derive the limiting distributions of we require the
asymptotic distribution of (the dependence of on q is
suppressed for notational convenience). Our approach is to reexpress this
variance estimator as a function of the autocovariances of the
(Xk - Xk-q) terms and then employ well-known limit theorems for autocovariances.
Consider the quantity
62
where denotes a quantity that is
of an order smaller than, n-1/2
in probability. Now define the (q ×1) vector
A standard limit theorem for sample autocovariances
of a stationary time series with independent gaussian increments
is [see, for example, Fuller (1976, chap. 6.3)]
where e1 is the (q × 1) vector [l 0 . . . 0]' and Iq is the identity matrix of
order q. Returning to the quantity we have
63
Combining Equations (A2-3) and (A2-4) then yields the following result:
where
Given the asymptotic distributions (Al-l) and (A2-5), Hausman's (1978)
method may be applied in precisely the same manner as in Theorem 1 to
yield the desired result:
The distributional results for follow immediately since
asymptotically these statistics are equivalent to Md(q) and Mr(q), respec-
tively.
Proof of Theorem 3
1. We prove the result for the proofs for the other statistics follow
almost immediately from this case. Define the increments process as
Yt  Xt - Xt-1, and define as
Consider first the numerator A (t) of
Since converges almost surely (a.s.) to , the first term of Equation (A3-
2b) converges a.s. to zero as nq  . Moreover, under assumption (A2)
it is apparent that { , t} satisfies the conditions of White's (1984) corollary
64
3.48; hence, H*'s condition 1 implies that the second and third terms of
(A3-2b) also vanish a.s. Finally, because , t , t - is clearly a measurable function
of the , t`s, { , t , t -} is also mixing with coefficients of the same size as
{ , t}. Therefore, under condition 2, corollary 3.48 of White (1984) may also
be applied to { , t , t -}, for which condition 1 implies that the fourth term
of Equation (A3-2b) converges a.s. to zero as well. By similar arguments,
it may also be shown that
Therefore, we have for all   0; hence, we conclude that
2. By considering the regression of increments AX, on a constant term
and lagged increments  Xt-j, this follows directly from White and Domowitz
(1984). Taylor (1984) also obtains this result under the assumption
that the multivariate distribution of the sequence of disturbancesis sym-
metric.
3. This result follows trivially from Equation (14) and condition 4.
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