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Unformatted text preview: THE JOURNAL OF FINANCE • VOL. LV, NO. 2 • APRIL 2000 Trading Volume and Cross-Autocorrelations in Stock Returns TARUN CHORDIA and BHASKARAN SWAMINATHAN* ABSTRACT This paper finds that trading volume is a significant determinant of the lead-lag 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 cross-autocorrelation patterns in short-horizon 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 lead-lag cross-autocorrelations in stock returns. Our investigation indicates that trading volume is a significant determinant of the cross-autocorrelation 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 cross-autocorrelations initiated by Lo and MacKinlay ~1990!. Lo and MacKinlay find that positive autocorrelations in portfolio returns are due to positive cross-autocorrelations 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 cross-autocorrelation patterns beyond that contained in firm size. The explanations that have been proposed for these cross-autocorrelation patterns ~see Mech ~1993!! can be classified into three groups. The first group of explanations claims that cross-autocorrelations are the result of time-varying expected returns ~see Conrad and Kaul ~1988!!. A variant of this explanation suggests that cross-autocorrelations 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 cross-autocorrelations should disappear. The second group of explanations ~see Boudoukh et al.! suggests that portfolio autocorrelations and cross-autocorrelations are the result of market microstructure biases such as thin trading. The final explanation for the lead-lag cross-autocorrelations claims that these lead-lag effects are due to the tendency of some stocks to adjust more slowly ~underreact! to economy-wide 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 lead-lag patterns not get arbitraged away? Most likely because of the high transaction costs that any trading strategy designed to exploit these short-horizon 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 post-1980 time period. These results show that own autocorrelations and nonsynchronous trading cannot fully explain the observed lead-lag 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 Cross-Autocorrelations 915 Next, in order to examine the source of these cross-autocorrelations, 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 lead-lag 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 cross-autocorrelations. The evidence indicates that there are striking differences in trading volume across stocks that contribute the most and the least to portfolio autocorrelations and cross-autocorrelations. 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 lead-lag effects, but it cannot explain all of them. The lead-lag 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 closed-end 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 equal-weighted portfolio returns are computed for each portfolio by averaging the non-missing 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 cross-autocorrelations, 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 size-volume portfolios. The mean portfolio returns suggest a negative cross-sectional 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 Size-Volume 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 Cross-Autocorrelations Summary statistics for size-volume 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 equal-weighted 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 cross-sectional statistics ~median and mean! of size and volume are computed for each portfolio for each year. Then the yearly cross-sectional 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 first-order 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 first-order 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 first-order 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 first-order daily autocorrelations are somewhat lower than first-order 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 Cross-Autocorrelations 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 cross-sectional 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 cross-autocorrelations 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 cross-autocorrelations. 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. Cross-autocorrelations, on the other hand, do not suffer from this problem. Therefore, in the rest of the paper, we focus our attention on differences in cross-autocorrelations. 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 cross-autocorrelations 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 lead-lag 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 cross-autocorrelations 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 cross-equation 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 cross-autocorrelations 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 Cross-Autocorrelations 921 B.2. Dimson Beta Regressions In the VAR tests, we control for size-related 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 lead-lag 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 one-sided tests involving one-sided alternative hypotheses. In tests involving a single restriction, this can be easily handled using a traditional one-sided Z-test. 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 chi-square distributions ~see Gourieroux, Holly, and Monfort ~1982!!. Specifically, a one-sided 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 closed-form solutions for the weight function. However, as pointed out by Gourieroux et al., a one-sided test that takes into account the constrained alternative hypothesis ought to have better power characteristics than a two-sided 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 chi-square distribution. This in turn suggests that if we are able to reject the null hypothesis against the one-sided alternative hypothesis using the traditional chi-square distribution, then we should most likely be able to reject the null hypothesis using the mixture of chi-square 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 cross-autocorrelations, ~b! results from VAR regressions and granger causality tests, and ~c! results from Dimson beta regressions. II. Empirical Results A. Cross-Autocorrelations and Own Autocorrelations Table II presents cross-autocorrelations for size-volume portfolio returns. Panel A presents cross-autocorrelations for daily portfolio returns and Panel B presents cross-autocorrelations 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 two-constraint 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 Cross-Autocorrelations 923 Table II Size-Volume Portfolio Cross-Autocorrelations 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 cross-autocorrelations 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 lead-lag 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 cross-autocorrelations are simply an inefficient way of describing the high autocorrelations of low volume portfolios.14 In the context of the size-volume portfolios, the AR ~1! model 14 Boudoukh et al. ~1994! specify an AR~1! model for the return-generating 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 cross-autocorrelations 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 cross-autocorrelations with lagged high volume returns. The results in Table II show that in every size quartile, for low volume portfolios Pi1, cross-autocorrelations 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 cross-autocorrelations 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 cross-autocorrelations 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 lead-lag 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 cross-autocorrelations are not as important as own autocorrelations in size-sorted 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 one-lag 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 Cross-Autocorrelations 925 Table III Vector Autoregressions for the Size-Volume 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 Z-statistic corresponding K K to the cross-equation 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 upper-tail tests. WA, m ~ WA, m ! is the Wald test statistic corresponding to the joint-test 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 cross-autocorrelations are more important than own autocorrelations in determining differences in the speed of adjustment of security prices to economy-wide 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 first-order autocorrelation of the corresponding low volume portfolio return, which provides further evidence that cross-autocorrelation 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 Z-statistic, 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 Cross-Autocorrelations 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 lead-lag cross-autocorrelations 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 lead-lag 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 cross-equation 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 lead-lag cross-autocorrelations 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 post-1980 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 cross-autocorrelation 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 cross-autocorrelations 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 one-fifth 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 lead-lag cross-autocorrelations. Now let us return to the cross-autocorrelation 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 Cross-Autocorrelations 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 equal-weighted 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 equal-weighted market index. This suggests that returns on high volume portfolios in the larger size quartiles lead returns on the equal-weighted 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 lead-lag cross-autocorrelations 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 cross-sectional 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 value-weighted 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 Size-Volume 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! equal-weighted 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 ~one-sided! 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 ~two-sided! 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 ~one-sided! 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 cross-autocorrelations 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 Cross-Autocorrelations 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 cross-autocorrelations 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 equal-weighted market index as a proxy of the market portfolio. Tests involving NYSE, AMEX, and Nasdaq value-weighted 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 equal-weighted 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 cross-autocorrelations. We use this measure to examine the cross-sectional 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 end-of-year 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 bid-ask spread to the average of the bid and ask prices. Trading Volume and Cross-Autocorrelations 933 Table V Speed of Adjustment and Ex Ante Firm Characteristics This table provides time-series 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 firm-years. 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 cross-autocorrelations ~slower speed of adjustment! and lower values represent those stocks contributing the least to portfolio crossautocorrelations ~faster speed of adjustment!. The NYSE0AMEX equal-weighted 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 time-series average of the median daily trading volume for the smallest size quartile–highest DELAY portfolio is 6,422 shares. The time-series 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 time-series 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 lead-lag cross-autocorrelations 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 lead-lag cross-autocorrelations 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 cross-autocorrelations 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. 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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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