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Unformatted text preview: THE JOURNAL OF FINANCE • VOL. LV, NO. 2 • APRIL 2000 Trading Volume and CrossAutocorrelations
in Stock Returns
TARUN CHORDIA and BHASKARAN SWAMINATHAN*
ABSTRACT
This paper finds that trading volume is a significant determinant of the leadlag
patterns observed in stock returns. Daily and weekly returns on high volume portfolios lead returns on low volume portfolios, controlling for firm size. Nonsynchronous trading or low volume portfolio autocorrelations cannot explain these findings.
These patterns arise because returns on low volume portfolios respond more slowly
to information in market returns. The speed of adjustment of individual stocks
confirms these findings. Overall, the results indicate that differential speed of
adjustment to information is a significant source of the crossautocorrelation patterns in shorthorizon stock returns. BOTH ACADEMICS AND PRACTITIONERS HAVE LONG BEEN interested in the role played
by trading volume in predicting future stock returns.1 In this paper, we examine the interaction between trading volume and the predictability of short
horizon stock returns, specifically that due to leadlag crossautocorrelations
in stock returns. Our investigation indicates that trading volume is a significant determinant of the crossautocorrelation patterns in stock returns.2
We find that daily or weekly returns of stocks with high trading volume lead
daily or weekly returns of stocks with low trading volume. Additional tests
indicate that this effect is related to the tendency of high volume stocks to
respond rapidly and low volume stocks to respond slowly to marketwide
information. * Chordia is from Vanderbilt University and Swaminathan is from Cornell University. We
thank Clifford Ball, Doug Foster, Roger Huang, Charles Lee, Craig Lewis, Ron Masulis, Matt
Spiegel, Hans Stoll, Avanidhar Subrahmanyam, two anonymous referees, the editor René Stulz,
and seminar participants at the American Finance Association meetings, Eastern Finance Association meetings, Southern Finance Association meetings, Southwestern Finance Association
meetings, Utah Winter Finance Conference, Chicago Quantitative Alliance, and Vanderbilt University for helpful comments. We are especially indebted to Michael Brennan for stimulating
our interest in this area of research. The first author acknowledges support from the Dean’s
Fund for Research and the Financial Markets Research Center at Vanderbilt University. The
authors gratefully acknowledge the contribution of I0B0E0S International Inc. for providing
analyst data. All errors are solely ours.
1
For the literature on volume and volatility see Karpoff ~1987! and Gallant, Rossi, and
Tauchen ~1992!.
2
To be specific, we use the average daily stock turnover as a proxy for trading volume. 913 914 The Journal of Finance This paper is closely related to the literature on crossautocorrelations
initiated by Lo and MacKinlay ~1990!. Lo and MacKinlay find that positive
autocorrelations in portfolio returns are due to positive crossautocorrelations
among individual security returns. Specifically, they find that the correlation between lagged large firm stock returns and current small firm returns
is higher than the correlation between lagged small firm returns and current large firm returns. Our results show that trading volume has important information about crossautocorrelation patterns beyond that contained
in firm size.
The explanations that have been proposed for these crossautocorrelation
patterns ~see Mech ~1993!! can be classified into three groups. The first
group of explanations claims that crossautocorrelations are the result of
timevarying expected returns ~see Conrad and Kaul ~1988!!. A variant of
this explanation suggests that crossautocorrelations are simply a restatement of portfolio autocorrelations and contemporaneous correlations ~see
Hameed ~1997! and Boudoukh, Richardson, and Whitelaw ~1994!!. Once account is taken of portfolio autocorrelations, according to this explanation,
portfolio crossautocorrelations should disappear. The second group of explanations ~see Boudoukh et al.! suggests that portfolio autocorrelations and
crossautocorrelations are the result of market microstructure biases such as
thin trading.
The final explanation for the leadlag crossautocorrelations claims that
these leadlag effects are due to the tendency of some stocks to adjust more
slowly ~underreact! to economywide information than others ~see Lo and
MacKinlay ~1990! and Brennan, Jegadeesh, and Swaminathan ~1993!!.3 We
refer to this explanation as the speed of adjustment hypothesis. Why do
these leadlag patterns not get arbitraged away? Most likely because of the
high transaction costs that any trading strategy designed to exploit these
shorthorizon patterns would face ~see Mech ~1993!!.
Our empirical tests are designed to take into account the issues raised by
the first two explanations. First, we conduct vector autoregressions involving pairs of high and low volume portfolio returns. Holding firm size constant, we examine whether lagged high volume portfolio returns can predict
current low volume portfolio returns controlling for the predictive power of
lagged low volume portfolio returns. We use both daily and weekly returns
in our empirical tests and take other precautions to minimize the impact of
nonsynchronous trading on our results. We find that high volume portfolio
returns significantly predict low volume portfolio returns even in the largest
size quartile. We also find that these results are robust in the post1980
time period. These results show that own autocorrelations and nonsynchronous trading cannot fully explain the observed leadlag patterns in stock
returns.
Others who have provided similar explanations include Badrinath, Kale, and Noe ~1995!,
McQueen, Pinegar, and Thorley ~1996!, and Connolly and Stivers ~1997!.
3 Trading Volume and CrossAutocorrelations 915 Next, in order to examine the source of these crossautocorrelations, we
conduct Dimson market model regressions ~see Dimson ~1979!! using returns on zero investment portfolios that are long in high volume portfolios
and short in low volume portfolios of approximately the same size. The results indicate that the leadlag effects are related to the tendency of low
volume stocks to respond more slowly to marketwide information than high
volume stocks. Finally, we use a speed of adjustment measure based on lagged
betas from Dimson regressions to examine the ex ante firm characteristics
of a subset of stocks that contribute the most ~or the least! to portfolio autocorrelations and crossautocorrelations. The evidence indicates that there
are striking differences in trading volume across stocks that contribute the
most and the least to portfolio autocorrelations and crossautocorrelations.
Specifically, stocks that contribute the most have 30 percent to 50 percent
lower trading volume.
The key conclusions are as follows. Returns of stocks with high trading
volume lead returns of stocks with low trading volume primarily because
the high volume stocks adjust faster to marketwide information. This is
consistent with the speed of adjustment hypothesis. Thus, trading volume
plays a significant role in the dissemination of marketwide information.
Thin trading can explain some of the leadlag effects, but it cannot explain all of them. The leadlag effects are also not explained by own
autocorrelations.
The rest of the paper is organized as follows. Section I discusses the
data and the empirical tests and Section II discusses the empirical results.
Section III provides additional evidence using individual stock data and Section IV concludes. I. Data and Empirical Tests
A. Data
Since Lo and MacKinlay ~1990! document that large firm returns lead
small f irm returns, we control for size effects in examining the crossautocorrelation patterns between high volume and low volume stocks. We do
this by forming a set of 16 portfolios based on size and trading volume, using
turnover as our measure of trading volume. Most previous studies ~see Jain
and Joh ~1988! and Campbell, Grossman, and Wang ~1993!! have used turnover, defined as the ratio of the number of shares traded in a day to the
number of shares outstanding at the end of the day, as a measure of the
trading volume in a stock. Moreover, using turnover disentangles the effect
of firm size from trading volume. Raw trading volume and dollar trading
volume are both highly correlated with firm size. In our sample, the crosssectional correlations between firm size and raw trading volume and firm
size and stock price are 0.78 and 0.72 respectively; the correlation between
size and turnover is 0.15 and the correlation between turnover and raw vol 916 The Journal of Finance ume is 0.60. Thus, turnover is highly correlated with raw volume but more
or less uncorrelated with firm size, which is exactly what we seek from this
variable.4
For the period from 1963 to 1996, four size quartiles are formed at the
beginning of each year by ranking all firms in the CRSP NYSE0AMEX stock
file by their market value of equity as of the December of the previous year,
and then dividing them into four equal groups. Only firms with ordinary
common shares are included in these portfolios. Additionally, all closedend
funds, real estate investment trusts, American Depositary Receipts, and Americus trust components are excluded from these portfolios. Firms in each size
quartile are further divided into four equal groups based on their average
daily trading volume over the previous year. To be included in one of these
16 portfolios, a firm must have at least 90 daily observations of trading
volume available in the previous year.
Once portfolios are formed in this manner at the beginning of each year,
their composition is kept the same for the remainder of the year. Daily and
weekly equalweighted portfolio returns are computed for each portfolio by
averaging the nonmissing daily or weekly returns of the stocks in the portfolio. Foerster and Keim ~1998! report that the likelihood of a NYSE0AMEX
stock going without trading for two consecutive days is 2.24 percent and for
five consecutive days it is only 0.42 percent. Therefore, in order to minimize
the effect of nonsynchronous trading on crossautocorrelations, returns of
stocks that did not trade at date t or t 1 are excluded from the computation
of portfolio returns for date t. This ensures that the daily returns of any
stock that did not trade for two consecutive days are excluded from the
computation of portfolio returns for those two days and for the following day.
As is common in the literature, we measure weekly returns from Wednesday close to the following Wednesday close.5 The use of weekly returns should
further alleviate concerns of nontrading. Daily and weekly stock returns,
average trading volume, and annual firm size are all obtained from CRSP
from January 1963 through December 1996.
Table I presents descriptive statistics on the 16 sizevolume portfolios.
The mean portfolio returns suggest a negative crosssectional relationship
between trading volume and average stock returns.6 The daily means for 4 Henceforth, unless otherwise stated, trading volume refers to this specific definition of
trading volume.
5
Seasonal patterns in weekly autocorrelations have been examined in detail by Keim and
Stambaugh ~1984!, Bessembinder and Hertzel ~1993!, and Boudoukh et al. ~1994!. Bessembinder and Hertzel find, for example, that the patterns in autocorrelations across weekdays are
related to the importance of weekend returns versus nonweekend returns in autocorrelation
patterns and are robust to alternative market microstructures. Though this is an interesting
issue, as far as our paper is concerned we simply want to show that our results are robust to
these patterns. In order to check the robustness of the weekly results, we repeat all of our
analysis using weekly returns computed from Friday close to the following Friday close and
Tuesday close to the following Tuesday close. The results are similar.
6
See Brennan, Chordia, and Subrahmanyam ~1998! and Datar, Naik, and Radcliffe ~1998!. Table I Summary Statistics for SizeVolume Portfolios Statistics for Weekly Returns Statistics for Daily Returns
Mean
~%! Std. Dev.
~%! r1 S10 N Mean
~%! Std. Dev.
~%! P11
P12
P13
P14 0.32
0.24
0.19
0.13 1.09
1.03
1.06
1.15 0.22
0.28
0.28
0.30 1.37
1.29
1.08
0.99 72
95
107
116 0.58
0.54
0.45
0.30 P21
P22
P23
P24 0.11
0.09
0.07
0.05 0.64
0.80
0.96
1.19 0.36
0.34
0.31
0.26 1.27
1.00
0.79
0.61 104
124
129
130 P31
P32
P33
P34 0.07
0.07
0.06
0.05 0.56
0.70
0.90
1.20 0.37
0.35
0.32
0.22 1.03
0.81
0.63
0.42 P41
P42
P43
P44 0.05
0.05
0.06
0.05 0.65
0.73
0.83
1.10 0.25
0.25
0.24
0.19 EW 0.09 0.80 0.34 Wednesday Tuesday Volume
Friday Size S4 r1 S4 r1 S4 N Med. Mean Med.
~%! Mean
~%! 2.33
2.56
2.74
3.15 0.39
0.37
0.34
0.29 0.95
0.88
0.75
0.62 0.36
0.33
0.27
0.24 0.87
0.77
0.60
0.53 0.46
0.41
0.39
0.34 1.06
0.97
0.84
0.71 123
130
131
129 0.010
0.010
0.012
0.014 0.011
0.012
0.013
0.014 0.045
0.088
0.146
0.275 0.043
0.089
0.149
0.343 0.34
0.35
0.30
0.23 1.72
2.25
2.66
3.14 0.33
0.28
0.23
0.22 0.67
0.56
0.46
0.40 0.29
0.23
0.18
0.15 0.59
0.46
0.35
0.29 0.37
0.33
0.28
0.24 0.77
0.67
0.54
0.47 132
133
133
131 0.055
0.056
0.058
0.063 0.060
0.061
0.063
0.066 0.051
0.111
0.192
0.366 0.049
0.113
0.195
0.433 122
133
134
131 0.31
0.32
0.29
0.24 1.59
1.99
2.47
3.09 0.27
0.23
0.19
0.17 0.49
0.40
0.34
0.28 0.23
0.17
0.13
0.10 0.40
0.30
0.24
0.19 0.31
0.26
0.22
0.18 0.59
0.47
0.41
0.32 134
136
135
131 0.229
0.246
0.235
0.239 0.252
0.266
0.258
0.259 0.057
0.116
0.196
0.379 0.052
0.117
0.200
0.449 0.36
0.28
0.27
0.25 134
138
137
135 0.24
0.26
0.28
0.23 1.66
1.88
2.12
2.75 0.13
0.10
0.09
0.10 0.19
0.14
0.13
0.14 0.07
0.05
0.05
0.06 0.10
0.08
0.06
0.08 0.11
0.08
0.08
0.11 0.22
0.17
0.16
0.18 138
138
138
135 1.321
1.420
1.312
1.076 3.516
2.632
2.185
1.595 0.074
0.120
0.172
0.294 0.067
0.120
0.174
0.363 0.85 — 0.33 2.19 0.26 0.51 0.20 0.40 0.29 0.59 — — — — — 917 r1 Trading Volume and CrossAutocorrelations Summary statistics for sizevolume portfolios are computed over 1963–1996. Pij refers to a portfolio of size i and volume j. i
1 refers to the
smallest size portfolio and i 4 refers to the largest size portfolio. Similarly j 1 refers to the lowest volume and j 4 refers to the highest
volume portfolio. EW is an equalweighted market index of NYSE0AMEX firms. Summary statistics for returns are computed using both daily
returns and nonoverlapping weekly returns. Each week ends on a Wednesday. For comparison, we also report autocorrelations computed using
weekly returns with weeks ending on Tuesday and Friday. The columns titled Wednesday, Tuesday, and Friday refer to weeks defined with those
ending days. Statistics of portfolio size and volume are obtained as follows: First, the crosssectional statistics ~median and mean! of size and
volume are computed for each portfolio for each year. Then the yearly crosssectional statistics of each portfolio are averaged over time and
reported below. N refers to the average number of firms in each portfolio each day or each week over 1963–1996. The number of daily returns
for all portfolios from 1963 to 1996 is 8,560. The number of nonoverlapping weekly returns over the same time period is 1,774. rk refers to the
kth order autocorrelation. Sk refers to the sum of first k autocorrelations. The size figures are in billions of dollars. The volume numbers
represent average daily percentage turnover. 918 The Journal of Finance small size stocks are higher than usual because we drop daily returns on
days a stock does not trade. The firstorder autocorrelation in daily portfolio
returns, r1 , decreases with volume in each size quartile except in the smallest size quartile ~ r1 is 0.22 for portfolio P11 and 0.30 for portfolio P14!.7 On
the other hand, the sum of the first 10 autocorrelations of the daily portfolio
returns is positive and declines monotonically with trading volume in each
size portfolio.
Table I also reports autocorrelations for weekly portfolio returns with weeks
ending on Wednesday, Tuesday, and Friday. Consistent with the findings of
Boudoukh et al. ~1994!, we find that autocorrelations based on a Tuesday
close are too low and those based on a Friday close are too high. The autocorrelations based on Wednesday close are not at either extreme and justify
the use of Wednesday close weekly returns. Therefore, all of our empirical
results from Table II onward are based on Wednesday close weekly returns.
The weekly autocorrelations, both at lag one and the sum of the first four
lags, decline monotonically with trading volume in each size portfolio.8
Not surprisingly, both daily and weekly autocorrelations also decline
with firm size. However, the autocorrelations remain fairly large even in
the largest size quartile, especially at the daily frequency. The firstorder
autocorrelations for P41 at the daily and weekly frequencies are 0.25 and
0.13 respectively. Predictably, the autocorrelations are lower using weekly
returns.
If security prices adjust slowly to information, then price increases ~decreases! will be followed by increases ~decreases!. This would give rise to
positive autocorrelation in stock returns.9 The portfolio autocorrelation evidence in Table I ~except for four portfolios of size 1 involving daily returns! is, therefore, consistent with the hypothesis that returns of stocks
with high trading volume adjust faster to common information. On the
other hand, positive portfolio autocorrelations are also symptomatic of nontrading problems. However, as Boudoukh et al. ~1994! point out, even heterogeneity in nontrading cannot explain all of the autocorrelations reported
in Table I. They estimate, for instance, that with extreme heterogeneity in
nontrading and betas, the firstorder weekly autocorrelation implied by
nontrading can be as high as 0.18. This is still less than half of the first7
One reason this happens is because of the way we compute portfolio returns. Note that we
drop firms that do not trade at day t or t
1 from the portfolio at day t. This throws away
valuable information about delayed reaction to private information and reduces the autocorrelations for the low turnover portfolio.
8
For P11, P12, and P13, the firstorder daily autocorrelations are somewhat lower than
firstorder weekly autocorrelations. This is the result of persistence in daily autocorrelations.
The sum of the first 10 daily autocorrelations are, however, uniformly higher than the sum of
the first four weekly autocorrelations.
9
Contrary to this hypothesis, most individual stocks exhibit a small negative autocorrelation
in daily and weekly returns ~see Lo and MacKinlay ~1990!! but portfolio returns exhibit positive
autocorrelations. Trading Volume and CrossAutocorrelations 919 order weekly autocorrelation of 0.39 estimated for P11 ~see Table I!. For
larger size portfolios, where nontrading problems are minimal, the nontradingimplied autocorrelations are much smaller ~see Figure 2, p. 559 in Boudoukh et al. ~1994!!. This suggests that nontrading issues cannot be the
sole explanation for the autocorrelations in Table I and other evidence to
be presented in this paper.
Table I also reports the median and average size and the median and
average trading volume for each portfolio. These are obtained by averaging
the annual crosssectional statistics. As expected, the median and mean trading volume increase within each size quartile. The median and mean size,
however, increase with trading volume only in the first three size quartiles.
In the largest size quartile ~size quartile 4!, the median and mean size decrease with trading volume. This provides an opportunity to test whether
trading volume has an independent inf luence on the crossautocorrelations
patterns. If trading volume has an independent effect then returns on high
volume stocks should continue to lead returns on low volume stocks even in
the largest size quartile. If, on the other hand, trading volume is simply a
proxy of firm size then, in the largest size quartile, low volume portfolio
returns should lead high volume portfolio returns. The autocorrelation evidence in Table I suggests that trading volume has an independent effect on
portfolio autocorrelations. Additional evidence in support of this is provided
later using tests based on crossautocorrelations.
Finally, Table I reports the average number of firms in each portfolio
each day or week during 1963 to 1996. The daily averages are significantly lower for portfolios P11 and P12 ~small size, low trading volume
portfolios!, indicating that many small firms had to be dropped from daily
portfolios due to nontrading problems ~recall that when computing portfolio returns we drop returns of firms that did not trade today or yesterday!. However, as Table I shows, nontrading problems are minimal in the
larger size quartiles. Moreover, the weekly averages suggest that at the
weekly frequency, nontrading problems are minimal even in the smallest
size quartile.
Although the autocorrelation evidence is consistent with the hypothesis
that the prices of high volume stocks adjust more rapidly to information,
it is important to point out that autocorrelations are not likely to provide
unambiguous inferences on the differences in speed of adjustment. To
see this clearly, consider two stocks A and B. Suppose that the return on
stock A responds to both today’s market information and yesterday’s market
information and the return on stock B responds only to yesterday’s market
information. Stock A, which adjusts faster to information, would exhibit
positive autocorrelation in daily returns. On the other hand, stock B,
which adjusts more slowly to information, would exhibit zero autocorrelation. Crossautocorrelations, on the other hand, do not suffer from this problem. Therefore, in the rest of the paper, we focus our attention on differences
in crossautocorrelations. 920 The Journal of Finance B. Empirical Tests
B.1. Vector Autoregressions
Following Brennan et al. ~1993!, we consider two types of time series tests:
~1! vector autoregressions ~VARs!, and ~2! Dimson beta regressions. The VAR
tests are designed to address two questions: ~a! Do crossautocorrelations
have information independent from own autocorrelations? ~b! Is the ability
of returns on high volume stocks to predict returns on low volume stocks
better than the ability of returns on low volume stocks to predict returns on
high volume stocks?
To understand the VAR tests, let us suppose that we want to test whether
returns of portfolio B lead returns of portfolio A. The leadlag effects between the returns of these two portfolios can be tested using a bivariate
vector autoregression: 10
K r A, t a0 ( K a k r A, t k k1 K r B, t c0 ( c k r A, t
k1 ( b k r B, t k ut , ~1! k1 K k ( d k r B, t
k1 k vt . ~2! In regression ~1!, if lagged returns of portfolio B can predict current returns
of portfolio A, controlling for the predictive power of lagged returns of portfolio A, returns of portfolio B are said to granger cause returns of portfolio A.
In our analysis, we use a modified version of the granger causality test by
examining whether the sum of the slope coefficients corresponding to return
B in equation ~1! is greater than zero.11 The granger causality test allows
us to determine if crossautocorrelations are independent of portfolio
autocorrelations.
Next, we are interested in testing formally whether the ability of lagged
returns of B to predict current returns of A is better than the ability of
lagged returns of A to predict current returns of B. We test this hypothesis
K
K
by examining if ( k 1 bk in equation ~1! is greater than ( k 1 ck in equation
~2!. We refer to this test as the crossequation test. This test is crucial to
establishing that returns of portfolio B lead returns of portfolio A and is a
formal test of any asymmetry in crossautocorrelations between high trading
volume and low trading volume stocks.
10
Since the regressors are the same for both regressions, the VAR can be efficiently estimated by running ordinary least squares ~OLS! on each equation individually.
11
The usual version is to jointly test whether the slope coefficients corresponding to the
lagged returns of the portfolio B are equal to zero. Our version tests not only for predictability
but also for the sign of predictability. Therefore, it is a more stringent test. Trading Volume and CrossAutocorrelations 921 B.2. Dimson Beta Regressions
In the VAR tests, we control for sizerelated differences in speed of adjustment by forming four size portfolios and estimating the VAR within each
size quartile. We control for other systematic effects in our tests of speed of
adjustment by running a market model regression suggested by Dimson ~1979!
which includes leads and lags of market returns as additional independent
variables. The Dimson beta regressions allow us to analyze the pattern of
under or overreaction of portfolio returns to market returns. They also allow
us to measure the speed of adjustment of each stock or portfolio relative to
a single common benchmark, which is helpful in comparing the speed of
adjustment across individual stocks or portfolios. In contrast, the VAR tests
measure speed of adjustment of two portfolios relative to one another.
However, both VAR and Dimson beta regressions do capture similar leadlag
effects.
In order to understand the Dimson beta regressions, consider a zero net
investment portfolio O that is long in portfolio B and short in portfolio A.
Now consider a regression of the return on the zero net investment portfolio
on leads and lags of the return on the market portfolio:
K rO, t aO
k ( K b O, k r m, t k u O, t , ~3! where bO, k
b B, k
bA, k . It is easy to show that portfolio B adjusts more
rapidly to common information than portfolio A if and only if the contemporaneous beta of portfolio B, bB, 0 , is greater than the contemporaneous
beta of portfolio A, bA, 0 , and the sum of the lagged betas of portfolio B,
K
K
( k 1 bB, k , is less than the sum of the lagged betas of portfolio A, ( k 1 bA, k .
In terms of the regression in equation ~3!, this translates into examining
K
0 and ( k 1 bO, k , 0. The basic intuition behind this result
whether bO, 0
is that if portfolio B responds more rapidly to marketwide information
than portfolio A, its sensitivity to today’s common information ~market return! should be greater than that of portfolio A. In the same vein, since
portfolio A responds sluggishly to contemporaneous information, it should
respond more to past common information ~lagged market returns!. The
important thing to note here is that the speed of adjustment ~relative to
the market portfolio! is a function of both the contemporaneous beta and
the lagged betas.
B.3. Hypothesis Testing
Note that all the hypothesis tests discussed above are onesided tests involving onesided alternative hypotheses. In tests involving a single restriction, this can be easily handled using a traditional onesided Ztest. However,
in tests involving more than one restriction ~as in the case of joint tests
involving a system of equations!, the regressions have to be estimated under 922 The Journal of Finance the constrained alternative hypothesis.12 This is what we do in this paper.
The resulting Wald test statistic, however, is not distributed as the traditional x 2 with the appropriate number of degrees of freedom but as a mixture of chisquare distributions ~see Gourieroux, Holly, and Monfort ~1982!!.
Specifically, a onesided test with m restrictions has the following distribution:
m Wm ; ( wj xj2 , ~4! j0 1. The complication is that wj is a complex, nonlinear funcwhere 0
wj
tion of the data and depends on the particular alternative hypothesis. Therefore, there are no general closedform solutions for the weight function.
However, as pointed out by Gourieroux et al., a onesided test that takes into
account the constrained alternative hypothesis ought to have better power
characteristics than a twosided test. This suggests that hypothesis tests
that use the distribution in equation ~4! should be able to reject the null
hypothesis more often than those that use the traditional chisquare distribution. This in turn suggests that if we are able to reject the null hypothesis
against the onesided alternative hypothesis using the traditional chisquare
distribution, then we should most likely be able to reject the null hypothesis
using the mixture of chisquare distributions.13 This is the approach we adopt
for the purpose of hypothesis testing.
In the next section, we discuss three pieces of evidence: ~a! own autocorrelations and crossautocorrelations, ~b! results from VAR regressions and
granger causality tests, and ~c! results from Dimson beta regressions.
II. Empirical Results
A. CrossAutocorrelations and Own Autocorrelations
Table II presents crossautocorrelations for sizevolume portfolio returns.
Panel A presents crossautocorrelations for daily portfolio returns and Panel
B presents crossautocorrelations for weekly portfolio returns with weeks
ending on a Wednesday. The correlations are computed using only the extreme trading volume portfolios within each size quartile. The results show
that, in every size quartile, the correlation between lagged high volume portfolio returns, ri 4, t 1 , and current low volume portfolio returns, ri1, t , is always larger than the correlation between lagged low volume portfolio returns,
ri1, t 1 , and current high volume portfolio returns, ri 4, t . For instance, in the
largest size quartile, using daily returns ~see Panel A!, the correlation be12 We thank the referee for pointing this out.
Gourieroux et al. ~1982! provide results on the power characteristics of the constrained
test only for the case of the single constraint. They also provide critical statistics only for the
twoconstraint case and that too for limited parameter values. Computing the critical statistics
or examining the power characteristics for tests involving more than two constraints is beyond
the scope of this paper.
13 Trading Volume and CrossAutocorrelations 923 Table II SizeVolume Portfolio CrossAutocorrelations
rij, t refers to the time t return of a portfolio corresponding to the ith size quartile and the jth
volume quartile within the ith size quartile. The number of daily observations between 1963
and 1996 is 8,560. The number of nonoverlapping weekly observations between 1963 and 1996
is 1,774. Each week ends on a Wednesday. Panels A and B report crossautocorrelations at the
first lag.
r11, t r14, t r21, t 0.22
0.35
0.31
0.34
0.30
0.33
0.27
0.31 0.24
0.30
0.27
0.36
0.28
0.36
0.27
0.35 0.29
0.39
0.36
0.44
0.39
0.45
0.39
0.45 0.39
0.43
0.40
0.40
0.37
0.38
0.30
0.34 0.25
0.29
0.28
0.32
0.26
0.32
0.22
0.29 r24, t r31, t r34, t r41, t r44, t 0.10
0.14
0.11
0.19
0.13
0.22
0.17
0.25 0.12
0.17
0.16
0.23
0.19
0.26
0.25
0.30 0.06
0.09
0.06
0.13
0.08
0.16
0.12
0.19 0.11
0.12
0.13
0.15
0.13
0.17
0.12
0.16 0.05
0.08
0.10
0.12
0.12
0.14
0.13
0.15 0.04
0.05
0.06
0.08
0.07
0.10
0.06
0.10 Panel A: Daily Returns
r11, t
r14, t
r21, t
r24, t
r31, t
r34, t
r41, t
r44, t 1
1
1
1
1
1
1
1 0.14
0.21
0.17
0.26
0.19
0.29
0.22
0.30 0.25
0.35
0.34
0.41
0.37
0.44
0.42
0.46 Panel B: Weekly Returns
r11, t
r14, t
r21, t
r24, t
r31, t
r34, t
r41, t
r44, t 1
1
1
1
1
1
1
1 0.28
0.32
0.33
0.35
0.33
0.36
0.30
0.34 0.15
0.19
0.19
0.22
0.19
0.24
0.17
0.23 0.20
0.24
0.27
0.28
0.27
0.31
0.27
0.30 tween lagged high volume portfolio returns, r44, t 1 , and the contemporaneous low volume portfolio returns, r41, t , is 0.30 while the correlation between
lagged low volume portfolio returns, r41, t 1 , and the contemporaneous high
volume portfolio returns, r44, t , is only 0.12. Similarly, using weekly returns
~see Panel B!, the correlation between r44, t 1 and r41, t is 0.15 and the correlation between r41, t 1 , and r44, t is only 0.06. The fact that we observe
these leadlag patterns in the largest size quartile using both daily and weekly
returns suggests that nonsynchronous trading cannot be the only source of
these lead–lag patterns.
Based on a simple AR ~1! model of portfolio returns suggested by Boudoukh et al. ~1994!, we examine whether crossautocorrelations are simply
an inefficient way of describing the high autocorrelations of low volume
portfolios.14 In the context of the sizevolume portfolios, the AR ~1! model
14
Boudoukh et al. ~1994! specify an AR~1! model for the returngenerating process for each
size portfolio where the AR~1! parameter is positive and declines monotonically with size. The
shocks to the AR~1! process are assumed to be white noise but are contemporaneously correlated across size portfolios. It is important to point out that the AR~1! model, by assumption,
rules out independent crossautocorrelations between portfolio returns. 924 The Journal of Finance would predict that the correlation between the lagged returns of the high
volume portfolio, ri 4, t 1 , and the current returns of the low volume portfolio,
ri1, t , should be less than or equal to the autocorrelation in the returns of the
low volume portfolio, ri1, t ; that is, corr ~ ri1, t , ri 4, t 1 !
corr ~ ri1, t , ri1, t 1 !. In
other words, the model predicts that the low volume portfolio returns’ autocorrelations should be larger than their crossautocorrelations with lagged
high volume returns.
The results in Table II show that in every size quartile, for low volume portfolios Pi1, crossautocorrelations with lagged high volume portfolio returns
corr ~ ri1, t , ri1, t 1 !.
exceed own autocorrelations; that is, corr ~ ri1, t , ri 4, t 1 !
For instance, in Panel B, in size quartile 1, corr ~ r11, t , r14, t 1 ! is 0.43 and
corr ~ r11, t , r11, t 1 ! is 0.39. The same pattern is seen in every size quartile
regardless of whether we use daily or weekly returns. These results clearly
indicate that crossautocorrelations contain independent information about
differences in speed of adjustment. We establish this more formally in the
next section using vector autoregression tests.
Contrast the above result with crossautocorrelations related only to size
differences as seen in Panel B of Table II. Consider portfolios P11 and P41,
which are extreme size quartile portfolios. In examining the leadlag patterns between the returns of these two portfolios, we find that the autocorrelation in the returns of P11, corr ~ r11, t , r11, t 1 ! 0.39, exceeds the correlation
between lagged returns of P41 and current returns of P11, corr ~ r41, t 1, r11, t !
0.30. This is what Boudoukh et al. ~1994! report in their paper and why they
conclude that crossautocorrelations are not as important as own autocorrelations in sizesorted portfolios.
B. Vector Autoregressions
We estimate the VAR using daily or weekly returns of the two extreme
volume portfolios in each size quartile: ~P11, P14!, ~P21, P24!, ~P31, P34!,
and ~P41, P44!. With daily returns, the VAR is estimated using five lags,
1!
K
5.15 With weekly returns, the VAR is estimated with one lag ~K
because additional lags only add noise. All regressions are estimated with
the White heteroskedasticity correction for standard errors. The White correction and the use of lagged dependent variables as regressors result in the
use of asymptotic statistics for making statistical inferences. Table III summarizes the results from the four VAR regressions. Low and High represent
the sum of the slope coefficients of the lagged returns on the low volume
portfolio and the lagged returns on the high volume portfolio, respectively.
L1 and H1 represent the slope coefficients of the onelag returns of the low
volume portfolio and the high volume portfolio ~ a 1 and b1 or c1 and d1 !,
respectively. Panel A presents VAR results using daily returns and Panel B
presents VAR results using weekly returns. 15 The results for 10 lags are similar. Trading Volume and CrossAutocorrelations 925 Table III Vector Autoregressions for the SizeVolume Portfolios
The following VAR is estimated using daily or weekly data from 1963 to 1996:
K r A, t a0 ( a k r A, t
k1 rB, t c0 ( c k r A, t
k1 K ( b k r B, t
k1 k K ut , k K ( d k r B, t
k1 k vt . k The LHS variable is the return on the lowest ~ rA, t ! or the highest ~ rB, t ! volume portfolio within
K
K
each size quartile. The portfolios Pij are defined in Table 1. Low refers to ( k 1 a k or ( k 1 ck
K
K
and High refers ( k 1 bk or ( k 1 dk as per the dependent variable. Similarly, L1 denotes a 1 or c1
O
and H1 denotes b1 or d1 . R 2 is the adjusted coefficient of determination. NOBS refers to the
number of daily or weekly returns used in the regressions. Z ~ A! is the Zstatistic corresponding
K
K
to the crossequation null hypothesis ( k 1 bk ( k 1 ck in each bivariate VAR. The alternative
K
K
hypothesis is ( k 1 bk . ( k 1 ck . K 5 ~ K 1! for regressions involving daily ~weekly! returns.
U
C
The significance levels for Z ~ A! are based on uppertail tests. WA, m ~ WA, m ! is the Wald test
statistic corresponding to the jointtest of null hypothesis across all equations against an unconstrained ~inequality constrained) alternative hypothesis. m is the number of constraints
~degrees of freedom! of the test. All statistics are computed based on White heteroskedasticity
corrected standard errors.
Panel A: Daily Returns ~NOBS
LHS 8,555! Low L1 H1 High O
R2 Z ~ A! P11
P14 0.0308***
0.0767* 0.1524*
0.1565* 0.3053*
0.2466* 0.4511*
0.3289* 0.16
0.11 5.05* P21
P24 0.0343
0.1798** 0.1942*
0.0912 0.2429*
0.3310* 0.2507*
0.4067* 0.22
0.08 3.05* P31
P34 0.0240
0.2645* 0.1633***
0.3154** 0.1943*
0.3157* 0.2129*
0.4541* 0.21
0.06 3.18* P41
P44 0.0161
0.2160* 0.0111
0.3758* 0.1706*
0.3032* 0.1993*
0.4371* 0.09
0.05 3.97* U
Joint Test: WA,4 C
WA,4 75.15* Panel B: Weekly Returns ~NOBS
LHS 1,773! H1 L1 O
R2 Z ~ A! P11
P14 0.1195**
0.0512 0.2423*
0.2610* 0.19
0.08 1.92** P21
P24 0.0867
0.0563 0.1506*
0.2477* 0.12
0.05 1.18 P31
P34 0.0477
0.0889 0.1374*
0.2045* 0.09
0.03 1.37*** P41
P44 0.0088
0.1413 0.0836*
0.1704* 0.02
0.01 1.66** U
Joint Test: WA,4 C
WA,4 24.21* *, **, and *** denote significance at the 1, 5, and 10 percent levels, respectively. 926 The Journal of Finance
B.1. Daily Returns We first focus on the daily results in Panel A of Table III. The evidence
indicates that lagged returns on the high volume portfolio strongly predict
current returns on both the low volume and the high volume portfolios in
each size quartile. The sum of the slope coefficients corresponding to lagged
returns of the high volume portfolio is positive and significant at the 1 percent level in every regression. Though the individual coefficients show
that most of the impact occurs at lag one, there is also significant predictability beyond lag one. Furthermore, the results in Panel A indicate
that the ability of ri 4, t 1 to predict ri1, t is better than the ability of ri1, t 1
to predict ri 1, t . These results suggest that portfolio crossautocorrelations are more important than own autocorrelations in determining differences in the speed of adjustment of security prices to economywide
information.
An examination of adjusted R 2 s in Panel A reveals that, in each size quartile, low volume portfolio returns are more predictable than high volume
portfolio returns. The adjusted R 2 in regressions involving low volume portfolio returns as the dependent variable ~returns of portfolios P11, P21, P31,
and P41! is in the range of 0.09 to 0.22. Each adjusted R 2 is higher than the
square of the firstorder autocorrelation of the corresponding low volume
portfolio return, which provides further evidence that crossautocorrelation
patterns are not driven ~solely! by own autocorrelations.
The results in Panel A indicate that lagged returns on the low volume
portfolio can also predict future returns on the high volume portfolio ~see
the L1 or Low columns for P14, P24, P34, and P44!. Therefore, as discussed
earlier, we test formally whether the ability of lagged high volume portfolio
returns, ri 4, t 1 , to predict current low volume portfolio returns, ri1, t , is better than the ability of lagged low volume portfolio returns, ri1, t 1 , to predict
5
current high volume portfolio returns, ri 4, t . In other words, is ( k 1 bk .
5
( k 1 ck ? In each size quartile, the asymptotic Zstatistic, Z ~ A!, tests the
null hypothesis that the sums of the slope coefficients across equations are
5
5
equal; that is, ( k 1 bk ( k 1 ck . The null is rejected in each size quartile at
the one percent level, indicating that returns on the high volume portfolio
lead returns on the low volume portfolio.16 In a joint test of the crossequation null hypothesis, since the inequality constraints under the alter5
5
native hypothesis, ( k 1 bk . ( k 1 ck , are satisfied in all four pairs of 16
Notice that in size quartiles 2, 3 and 4, low volume portfolio returns predict high volume
portfolio returns with a negative sign. This is simply a result of the fact that we are measuring
relative speed of adjustment between two portfolios. Brennan et al. ~1993! show that if returns
on the low volume portfolio adjust more slowly to common information than returns on the high
volume portfolio then in regressions involving the high volume portfolio return as the dependent variable, the slope coefficient corresponding to the lagged return on the low volume portfolio could be negative. Trading Volume and CrossAutocorrelations 927 regressions, the unconstrained Wald test statistic and the constrained Wald
test statistic are the same; that is, WA, U
WA, C
75.15. The Wald test
statistics reject the joint null hypothesis at the one percent level. Overall,
the results provide strong evidence that returns on high volume portfolios
lead returns on low volume portfolios.
A brief discussion of the economic significance of the results in Panel A is
in order here. Focusing on the P41 regression in the largest size quartile
~because these are the most liquid stocks!, on average, a one percent increase in today’s return of high volume stocks, P44, all else equal, leads to a
0.1706 percent increase in tomorrow’s return of low volume stocks, P41. The
daily standard deviation of the high volume portfolio return is 1.10 percent.
Therefore, a one percent increase is within one standard deviation. The
0.1706 percent increase in the returns of the low volume portfolio is approximately three times above its daily mean of 0.05 percent. This suggests that
these leadlag crossautocorrelations effects could be economically significant. Similarly a one percent increase in the low volume portfolio return,
P41, leads to a 0.2160 percent decrease ~conditionally! in the high volume
portfolio return, P44, which is again economically significant given its daily
mean of 0.05 percent.
B.2. Weekly Returns
Foerster and Keim ~1998! report that since 1963 less than one percent of
the stocks in the three largest size deciles in the NYSE and AMEX did not
trade on a given day. The results in Panel A show that the leadlag crossautocorrelations between high volume and low volume portfolio returns are
as strong in the largest size quartile as they are in the smallest size quartile. This makes it unlikely that these results could be due to nonsynchronous trading.
In order to allay any remaining concerns about nonsynchronous trading,
however, we repeat the VAR tests using weekly portfolio returns. The results involving weekly portfolio returns are presented in Panel B of Table III.
The VAR is estimated with one lag because additional lags only add noise.
The results in Panel B show that high volume portfolio returns lead low
volume portfolio returns even at the weekly frequency. In every size quartile, lagged returns on the high volume portfolio exhibit statistically and
economically significant predictive power for future returns on the low volume portfolio. In contrast, lagged returns on the low volume portfolio exhibit little or no ability to predict future returns on the high volume portfolio
and only weak ability to predict returns on the low volume portfolio. Once
again the joint test statistic for the crossequation null hypothesis A is significant at the 1 percent level. Overall, the weekly results closely parallel
the daily results and make it unlikely that nonsynchronous trading could be
the primary explanation for the leadlag crossautocorrelations reported in
this paper. 928 The Journal of Finance
B.3. Additional Robustness Checks As a final check to see if nontrading inf luences our results, we estimate
the VAR at both the daily and the weekly frequencies using only post1980
data. The results ~not reported in the paper! are similar to those in Table III
and strongly support the hypothesis that returns on the high volume portfolio lead returns on the low volume portfolio.
One potential criticism of these results, given the positive correlation between firm size and volume ~a correlation of 0.15 in our sample!, is that
trading volume simply proxies for firm size. We address this issue in two
ways. First, recall that volume and size are negatively correlated in size
quartile 4 ~see Table I!. Therefore, if the crossautocorrelation results with
respect to volume are being driven by firm size, we should see returns on
portfolio P41 lead returns on portfolio P44. Yet the crossautocorrelations in
Table II indicate that the correlation is higher between lagged returns of
P44 and current returns of P41 than between lagged returns of P41 and
current returns of P44. Moreover, the VAR results in Table III confirm that
returns on P44 lead returns on P41.
Next, we choose high and low volume portfolios from adjacent size quartiles to ensure that portfolio size and volume are negatively correlated. Consider the following three pairs of portfolios: ~P21, P14!, ~P31, P24!, and ~P41,
P34!. In each of these pairs, firm size and volume are negatively correlated.
For instance, the average size of P21 is about four times that of P14 ~see
Table I! but the average volume of P21 is only about onefifth that of P14.
The negative correlation between size and volume allows us to see whether
the volume effect is independent of the size effect in determining leadlag
crossautocorrelations. Now let us return to the crossautocorrelation evidence in Table II. In both Panel A and Panel B, the correlation between
lagged returns of the high volume portfolio ~P14, P24, or P34! and current
returns of the low volume portfolio ~P21, P31, or P41! is higher than the
correlation between lagged returns of the low volume portfolio ~P21, P31, or
P41! and current returns of the high volume portfolio ~P14, P24, or P34!.
This suggests that the volume effect is independent of the size effect. We
also perform VAR tests involving the three pairs of low and high volume
portfolios from adjacent size quartiles. The regression results ~not reported!
are similar to those in Table III.
C. Dimson Beta Regressions
As discussed in Section B.2, we use zero investment portfolios in the Dimson beta regressions. The zero investment portfolios are constructed by subtracting low volume portfolio returns from high volume portfolio returns.
Since we expect high volume portfolio returns to adjust faster to common
factor information than do low volume portfolio returns, the contemporaneous betas from these regressions, bO, 0 , should be positive and the sum of
K
lagged betas, ( k 1 bO, k should be negative. The intuition behind these restrictions is as follows. If the return on the high volume portfolio responds Trading Volume and CrossAutocorrelations 929 more rapidly to common information than the return on the low volume
portfolio then its sensitivity to today’s common information ~market return!
should be greater than that of the low volume portfolio. Therefore, the
contemporaneous beta of the zero investment portfolio should be positive.
Additionally, since the low volume portfolio responds sluggishly to contemporaneous factor information ~current market returns!, it should respond
more to past common factor information ~lagged market returns!. Therefore,
the lagged betas of the zero investment portfolio should be negative.
We estimate the Dimson beta regressions in equation ~3! using the NYSE0
AMEX equalweighted portfolio return as a proxy for the common factor.17
All standard errors are corrected for generalized heteroskedasticity using
the White correction. Table IV presents results from Dimson beta regressions. Panel A reports results using daily returns and Panel B reports results using weekly returns. We use five leads and lags of market returns in
daily Dimson beta regressions and two leads and lags of market returns in
weekly Dimson beta regressions.18
First, we focus on the daily results in Panel A. The contemporaneous betas
of the zero investment portfolio, bO, 0 , are positive and significant at the one
percent level in each size quartile. Also, the sum of the lagged betas is significantly negative in each size quartile. These results indicate that, in each
size quartile, the returns on the low volume portfolio adjust more slowly to
marketwide information than the returns on the high volume portfolio. Not
surprisingly, both the constrained and the unconstrained Wald test statistics
strongly reject the joint null hypothesis that the sum of the lagged betas is
zero in each size quartile, at the one percent level. The sum of leading betas
indicates that current returns on the zero investment portfolios in size quartiles 2, 3, and 4 are able to predict future returns of the equalweighted
market index. This suggests that returns on high volume portfolios in the
larger size quartiles lead returns on the equalweighted market index. The
weekly results in Panel B are similar to the daily results and reveal significant differences in speed of adjustment related to trading volume. Overall,
the results indicate that the leadlag crossautocorrelations observed between high volume and low volume stocks are driven by differences in the
speed of adjustment to common factor information. III. Speed of Adjustment of Individual Stocks
Up to this point our empirical tests use portfolio returns to examine the
relationship between crosssectional differences in trading volume and speed
of adjustment to common information. We find that returns of high volume
portfolios adjust faster to marketwide information than do those of low vol17 We also perform all regressions reported in Table IV using the CRSP valueweighted
market index and the results are similar.
18
For daily returns, the results with 10 leads and lags are similar. For weekly returns, the
use of additional lags only adds more noise to statistical inference. 930 The Journal of Finance
Table IV Dimson Beta Regressions for SizeVolume Portfolio Returns
The following regression is estimated using daily or weekly data from 1963 to 1996:
K r O, t aO ( K b O, k r m, t
k k u O, t , where rO, t is the difference between returns on the highest volume and the lowest volume
portfolios within each size quartile and rm, t k refers to CRSP ~NYSE0AMEX! equalweighted
K
market returns. ( k 1 bO, k refers to the sum of lagged betas, ( k K 1 bO, k refers to the sum of
O
leading betas, and bO, 0 refers to the contemporaneous beta. R 2 is the adjusted coefficient of
determination. NOBS refers to the number of daily or weekly returns used in the regressions.
K
The individual equation statistical tests corresponding to ( k 1 bO, k are lower tail ~onesided!
U
tests. Wm is the Wald test statistic corresponding to the joint null hypothesis ~across all equaK
C
tions! that ( k 1 bO, k 0 against an unconstrained ~twosided! alternative hypothesis. Wm is the
K
Wald test statistic corresponding to the joint null hypothesis ~across all equations! ( k 1 bO, k 0
K
against an inequality constrained ~onesided! alternative hypothesis that ( k 1 bO, k 0. m is the
number of constraints ~degrees of freedom!. All statistics are computed based on White heteroskedasticity corrected standard errors. The significance levels for both the constrained and the
unconstrained Wald test statistics are based on standard x 2 distribution ~see the text for
details!.
Panel A: Daily Returns ~NOBS
Size
1
2
3
4 LHS
P14
P24
P34
P44 P11
P21
P31
P41 (k 5 1 b O, k 0.4832*
0.7941*
0.8584*
0.5600*
W4C Size
1
2
3
4 LHS
P14
P24
P34
P44 P11
P21
P31
P41 (k 1 b O, k Joint Test: W4U 0.2688*
0.3652*
0.4239*
0.2752* 0.5114*
0.6848*
0.6871*
0.4872*
W4C O
R2
0.13
0.60
0.65
0.50 1,769! b O, 0 0.0067
0.0301
0.0391***
0.0372*** 1 b O, k 493.31* Panel B: Weekly Returns ~NOBS
2 5
(k b O, 0 0.0252
0.0489*
0.1122*
0.1286*
Joint Test: W4U 8,549! 2
(k 1 b O, k
0.1929*
0.1858*
0.1971*
0.1311* O
R2
0.35
0.63
0.60
0.46 101.94* *, **, and *** indicate significance at the 1, 5, and 10 percent levels, respectively. ume portfolios. In this section, we use data on individual stocks to examine
the relationship between trading volume and the speed of adjustment. Specifically, we identify stocks that contribute the most or the least to portfolio
autocorrelations and crossautocorrelations and examine their ex ante firm
characteristics. We want to determine if trading volume emerges as an important characteristic in explaining the differences in the speed of adjustment across the two groups of stocks. Trading Volume and CrossAutocorrelations 931 The sample used in this section contains all stocks available at the intersection of CRSP NYSE0AMEX files and annual IBES files from 1976 to
1996. We use the IBES files in order to obtain the number of analysts making annual earnings forecasts. The sample contains a total of 24,704 firm
years, or an average of approximately 1,200 firms per year.
To identify stocks that contribute the most ~or least! to portfolio autocorrelations and crossautocorrelations we use a measure of speed of adjustment based on contemporaneous and lagged betas from Dimson beta
regressions. Each year, from 1977 to 1996, the following Dimson beta regression is estimated for each stock in the sample:
5 ri, t ai
k ( 5 b i, k r m, t u i, t , k ~5! where ri, t is the daily return on the stock, rm, t is the daily return on the
market index, and bi, k is the beta with respect to the market return at lag
k. We use the NYSE0AMEX equalweighted market index as a proxy of the
market portfolio. Tests involving NYSE, AMEX, and Nasdaq valueweighted
market indexes provide similar results.
Recall our discussion in Section B.2 that the speed of adjustment ~relative
to the market portfolio! is a function of both contemporaneous and lagged
betas. For simplicity consider a Dimson beta regression with just one lag
and one lead. In comparing the speed of adjustment of two stocks A and B,
returns of stock B are said to adjust more rapidly to common information
than do returns of stock A if and only if stock B’s contemporaneous beta,
bB, 0 , is greater than stock A’s contemporaneous beta, bA, 0 , and stock B’s
lagged beta, bB,1 , is less than stock A’s lagged bA,1 . We can state this result
in a more parsimonious way as follows. Returns of stock B adjust more
rapidly to common information than do returns on stock A if and only if
bB,1 0bB, 0 , is less than bA,1 0bA, 0 .
For a Dimson beta regression with five leads and five lags, the speed of
5
adjustment ratio is defined to be ( k 1 bj, i, k 0bj, i,0 . We use a logit transformation of this ratio as our measure of speed of adjustment:
1 DELAYi 1 x e , ~6! where
5 x ( b i, k
k1
bi,0 . Our measure is a modification of a measure proposed by McQueen et al.
~1996!. If x is the ratio of lagged beta to contemporaneous beta then the
measure proposed by McQueen et al. is equal to the logit transformation of
x 0~1 x !. Though this measure is monotonic in x for x
1, it is nonmono 932 The Journal of Finance tonic in x for x
1. x is often less than one when measuring the speed of
adjustment of large stocks relative to the equalweighted market index. This
is because large stocks adjust faster to common information than the equalweighted market index. As a result, for a large stock the contemporaneous
beta tends to be greater than one and the lagged beta tends to be negative
and less than one. This creates a problem in comparing a positive value of x
to a negative value of x or in comparing two negative values of x. For x
0
our DELAY measure provides values greater than 0.5, and for x
0 our
measure provides values less than 0.5.
The logit transformation has several appealing properties. First, it is monotonic in x. Secondly, the transformation moderates the inf luence of outliers
and yields values between zero and one. Values closer to zero imply a faster
speed of adjustment and values closer to one imply a slower speed of adjustment. Therefore, stocks with high ~low! DELAY are likely to contribute most
~least! to portfolio autocorrelations and crossautocorrelations. We use this
measure to examine the crosssectional relation between trading volume and
the speed of adjustment of individual stocks.
Next, for each firm in the sample, we match the DELAY measure computed in year t with firm characteristics as of year t 1. The firm characteristics are Volume, defined as the average number of shares traded per day
during year t
1; Turnover, defined as the average daily turnover in percentage during year t
1; Size, which is the market capitalization in millions of dollars as of the December of year t
1; Price, which is the stock
price as of December of year t 1; Stdret, defined as the standard deviation
of daily returns in percentage during year t 1; Nana, which is the number
of security analysts making annual forecasts as of the September of year
t 1; and Spread, defined as the average of the beginning and endofyear
relative spread in percent.19 The data on relative spread are the same as
those used in Eleswarapu and Reinganum ~1993!; they are available only for
the 1980 to 1989 time period and cover only NYSE stocks.
Finally, each year, we form four size quartiles and then divide each size
quartile into four quartiles based on DELAY. We focus our attention on the
extreme DELAY quartiles, High and Low, within each size quartile. High
represents 25 percent of stocks within each size quartile that are likely to
contribute the most to delayed reaction to common factor information, Low
represents 25 percent of stocks that are likely to contribute the least to
delayed reaction to common factor information. For each portfolio, each year,
we compute the median ex ante firm characteristic and then average the
annual medians over time.
The results are reported in Table V. In general, in each size quartile,
both raw trading volume ~Volume! and relative trading volume ~Turnover!
differ significantly across the two DELAY portfolios, High and Low. On
average, the raw trading volume for the high DELAY portfolio, High, is 25 per19 Relative spread is defined as the ratio of the dollar bidask spread to the average of the
bid and ask prices. Trading Volume and CrossAutocorrelations 933 Table V Speed of Adjustment and Ex Ante Firm Characteristics
This table provides timeseries averages of the annual portfolio medians of the speed of adjustment measure DELAY and other ex ante firm characteristics. The sample period is 1976–1996
and the sample size is 24,704 firmyears. The speed of adjustment measure, DELAY, defined in
equation ~6!, is computed by running the Dimson beta regression in equation ~5! for each stock
each year. DELAY is constructed to be between zero and one where higher values represent
those stocks contributing the most to portfolio crossautocorrelations ~slower speed of adjustment! and lower values represent those stocks contributing the least to portfolio crossautocorrelations ~faster speed of adjustment!. The NYSE0AMEX equalweighted market index
is used as the proxy of the market index. At the beginning of each year all stocks available at
the intersection of NYSE0AMEX and annual IBES files are divided first into four quartile
portfolios based on firm size as of the December of the previous year. Size 1 represents the
smallest size quartile and size 4 represents the largest size quartile. Each size quartile is
further divided into four quartile portfolios based on DELAY computed from daily returns for
that year. In each size quartile we focus our attention on the extreme DELAY quartiles. High
represents 25 percent of stocks with the highest DELAY measure and Low represents 25 percent of stocks with the smallest DELAY measure within each size quartile. Each DELAY portfolio contains, on average, 77 stocks. The ex ante portfolio characteristics for these portfolios
are reported below. Size is the market capitalization as of the December of the previous year in
millions of dollars, Volume is the average number of shares traded per day over the previous
year, Turnover is the average daily turnover in percentage over the previous year, Nana is the
number of security analysts making annual earnings forecasts as of the September of the previous year, Price is the stock price as of the December of the previous year, Stdret is the standard deviation of daily returns over the previous year in percentage, and Spread is the average
relative spread for the stock in the previous year also in percentage.
Delayed
Reaction DELAY 1 ~Small! Low
High 0.35
0.70 11777
6422 2 Low
High 0.34
0.65 3 Low
High 4 ~Large! Low
High Size Volume Turnover Size Price Stdret Nana Spread 0.193
0.132 64.22
54.11 9.64
12.02 2.83
2.39 1.93
1.85 2.36
2.08 27342
15663 0.217
0.146 243.43
223.76 18.59
22.16 2.30
1.83 5.18
4.38 1.43
1.34 0.33
0.58 61394
46238 0.206
0.169 742.95
664.29 25.58
29.31 1.85
1.71 11.15
8.93 1.04
0.98 0.30
0.50 209481
128966 0.187
0.194 3662.73
2214.57 39.11
39.30 1.56
1.62 21.48
16.95 0.63
0.71 cent to 45 percent lower than the raw trading volume for the low DELAY
portfolio, Low. Similarly, the turnover for the high DELAY portfolio is,
on average, 20 percent to 35 percent lower than the turnover for the low
DELAY portfolio. An exception is size quartile 4, in which there is not much
difference in turnover across the two DELAY portfolios. This probably results
from the fact that in size quartile 4, turnover and size tend to be negatively
correlated ~see Table 1!. Additionally, Dimson beta estimators are likely to
be very noisy for individual stocks. This can be seen from the results in
Table IV where, using portfolio returns, we find significant differences in
the speed of adjustment between high turnover and low turnover portfolios. 934 The Journal of Finance To allay any remaining concerns that our results are driven by the small
illiquid stocks, we focus our attention on the results for the smallest size
quartile–highest DELAY portfolio. The timeseries average of the median
daily trading volume for the smallest size quartile–highest DELAY portfolio
is 6,422 shares. The timeseries average of the 25th percentile ~on average
there are fewer than 20 stocks below this cutoff! daily trading volume of the
above portfolio is 3,244 shares. The timeseries average of the fifth percentile ~fewer than four of the 77 stocks are below this cutoff! daily trading
volume is 1,103 shares. For comparison, the fifth percentile daily trading
volume for size quartiles 2, 3, and 4 ~the larger size portfolios! are 3,764
shares, 8,705 shares, and 30,593 shares respectively. All these show that our
results are not driven by extremely illiquid stocks.
Stocks with high DELAY also tend to be smaller, have fewer analysts, are
higher priced, and have lower volatility. Differences in relative spread across
high and low DELAY stocks do not seem economically significant. In sum,
the univariate statistics based on the speed of adjustment of individual stocks
confirm our earlier findings and strongly support the hypothesis that trading volume is a significant determinant of how slowly or rapidly stock prices
adjust to new information. IV. Conclusion
In this paper, we find that trading volume is a significant determinant
of leadlag crossautocorrelations in stock returns. Specifically, returns of
portfolios containing high trading volume lead returns of portfolios comprised of low trading volume stocks. Additional tests establish that the
source of these leadlag crossautocorrelations is the tendency of low volume stock prices to react sluggishly to new information. While nontrading
may be a part of the story, the magnitude of the autocorrelations and
crossautocorrelations indicate that nontrading cannot be the sole explanation of our results.
At first glance these results may suggest some market inefficiency; however, it is not clear that investors could profitably trade on these patterns
because transaction costs are likely to overwhelm any potential profits. This
might explain why these patterns do not get arbitraged away. Nevertheless,
the results are interesting since they indicate a market in which trading
volume plays a major role in the speed with which prices adjust to information, yielding insights into how stock prices become more informationally
efficient.
REFERENCES
Badrinath, Swaminathan G., Jayant R. Kale, and Thomas H. Noe, 1995, Of shepherds, sheep
and the crossautocorrelations in equity returns, Review of Financial Studies 8, 401–430.
Bessembinder, Hendrik, and Michael G. Hertzel, 1993, Return autocorrelations around nontrading days, Review of Financial Studies 6, 155–190. Trading Volume and CrossAutocorrelations 935 Boudoukh, Jacob, Matthew P. Richardson, and Robert F. Whitelaw, 1994, A tale of three schools:
Insights on autocorrelations of shorthorizon stock returns, Review of Financial Studies 7,
539–573.
Brennan, Michael J., Tarun Chordia, and Avanidhar Subrahmanyam, 1998, Alternative factor
specifications, security characteristics, and the crosssection of expected stock returns, Journal of Financial Economics 49, 345–374.
Brennan, Michael J., Narasimhan Jegadeesh, and Bhaskaran Swaminathan, 1993, Investment
analysis and the adjustment of stock prices to common information, Review of Financial
Studies 6, 799–824.
Campbell, John Y., Sanford J. Grossman, and Jiang Wang, 1993, Trading volume and serial
correlation in stock returns, Quarterly Journal of Economics 107, 907–939.
Connolly, Robert A., and Chris T. Stivers, 1997, Timevarying leadlag of equity returns in a
world of incomplete information, Working paper, University of North Carolina at Chapel
Hill.
Conrad, Jennifer, and Gautam Kaul, 1988, Time varying expected returns, Journal of Business
61, 409–425.
Datar, Vinay, Narayan Naik, and Robert Radcliffe, 1998, Liquidity and asset returns: An alternative test, Journal of Financial Markets 1, 203–220.
Dimson, Elroy, 1979, Risk measurement when shares are subject to infrequent trading, Journal
of Financial Economics 7, 197–226.
Eleswarapu, Venkat R., and Marc R. Reinganum, 1993, The seasonal behaviour of the liquidity
premium in asset pricing, Journal of Financial Economics 34, 373–386.
Foerster, Stephen R., and Donald B. Keim, forthcoming, Direct evidence of nontrading of NYSE
and AMEX stocks; in W. Ziemba and D. B. Keim, eds: Security Market Imperfections in
Worldwide Equity Markets ~Cambridge University Press, New York!.
Gallant, Ronald A., Peter E. Rossi, and George Tauchen, 1992, Stock prices and volume, Review
of Financial Studies 5, 199–242.
Gourieroux, Christian B., Alberto Holly, and Alain Monfort, 1982, Likelihood ratio test, Wald
test and KuhnTucker test in linear models with inequality constraints on the regression
parameters, Econometrica 50, 63–80.
Hameed, Allaudeen, 1997, Time varying factors and crossautocorrelations in shorthorizon stock
returns, Journal of Financial Research 20, 435–458.
Jain, Prem C., and GunHo Joh, 1988, The dependence between hourly prices and trading
volume, Journal of Financial and Quantitative Analysis 23, 269–283.
Karpoff, Jonathan, 1987, The relation between price changes and trading volume: A survey,
Journal of Financial and Quantitative Analysis 22, 109–126.
Keim, Donald B., and Robert F. Stambaugh, 1984, A further investigation of the weekend effect
in stock returns, Journal of Finance 39, 819–834.
Lo, Andrew, and Craig MacKinlay, 1990, When are contrarian profits due to stock market
overreaction?, Review of Financial Studies 3, 175–205.
McQueen, Grant, Michael Pinegar, and Steven Thorley, 1996, Delayed reaction to good news
and crossautocorrelation of portfolio returns, Journal of Finance 51, 889–920.
Mech, Timothy S., 1993, Portfolio return autocorrelation, Journal of Financial Economics 34,
307–344. ...
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This note was uploaded on 12/08/2011 for the course CIS 625 taught by Professor Michaelkearns during the Spring '12 term at Pennsylvania State University, University Park.
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