The behavior of daily stock market trading volume

Journal of Accounting and Economics 11 (1989) 331-359. North-Holland

THE BEHAVIOR OF DALLY STOCK MARKET TRADING VOLUME* Bipin B. AJINKYA University" of Florida, Gainesville, FL 32611, USA P r e m C. J A I N

University of Pennsylvania, Philadelphia, PA 19104, USA Received May 1988, final version received April 1989 This paper documents the empirical distributions of daily trading volume prediction errors for several commonly used volume measures and expectation models for individual firms and for portfolios. The prediction errors for raw volume measures are significantly positively skewed, with thin left tails and fat right tails. However, natural log transformations of the volume measures are approximately normally distributed. For longer than one-day prediction intervals, recognition of autocorrelation in daily trading volume is advantageous for detecting abnormal trading. Results of analysis for clustering of events and for different size firms are also presented.

1. Introduction

This paper examines properties of daily trading volume for common stocks on the New York Stock Exchange. Although economic information can lead to both trading and price changes, researchers are discovering that trading volume has potential for testing hypotheses not addressable by the typical returns methodology alone. Beaver (1968), Foster (1973), Morse (1981), Bamber (1986), and others have examined trading volume around earnings announcements. James and Edmister (1983), Lakonishok and Smidt (1984), and Lakonishok and Vermaelen (1986) have also studied trading volume to examine other informational characteristics of the equity markets. A major objective in these studies is to assess the extent to which trading volume around the event is abnormal. While empirical distributions of common stock *This work has benefited from suggestions of Ray Ball (the editor), Mike Barclay (the referee), Linda Bamber, Bryan Church, Greg Clinch, Dan Elnathan, Gun-Ho Joh, Andy Lo, Craig Mackinlay, Jens Stephan, Rex Thompson, Senyo Tse, Jerry Warner, and workshop participants at the Georgia Institute of Technology and Texas A & M University. This work was completed while the first author had a summer grant from the Fisher School of Accounting, University of Florida, and the second author was a Peat Marwick Research Fellow. 0165-4101/89/$3.50©1989, Elsevier Science Publishers B.V. (North-Holland)

332

B.B. Ajinl~va and P.C. Jain, Daily trading ~,olume

returns and abnormal returns have been studied extensively [e.g., Fama (1965), Brown and Warner (1980, 1985), Dyckman, Philbrick, and Stephan (1984), and Jain (1986)], analogous information on trading volume is indeed sparse. We study the statistical properties of both actual daily common stock trading volume and of abnormal trading volume using a number of trading volume measures and different expectation models for determining abnormal trading volume. The resulting specifications of the empirical distributions should enable researchers to: (i)

assess the nature and extent of misstatements in achieved significance levels when conducting typical statistical tests based on the t-distribution in trading volume event studies, (ii) evaluate how the statistical tests are affected by the length of the estimation period, the length of the prediction interval, clustering of events, and the size of the firm, and (iii) identify measures for which the distributions of abnormal trading volume approximate normality. The three volume measures chosen for our analysis are: (i) daily number of common shares traded, (ii) daily dollar value of shares traded, and (iii) fraction of shares outstanding traded. The alternative volume expectation models used to determine the prediction errors are based on the simple average of the volume measure over an estimation period, and the linear regression of firm trading volume on market trading volume. The analysis is extended to examine the effects of three conditioning variables on the properties of prediction error distributions: the number of days in the estimation period, the number of days in the prediction interval, and the size of the firm. a The highlights of the results are as follows. For the New York Stock Exchange sample of randomly selected firms and event dates, the distributions of the prediction errors for the untransformed volume measures indicate that means are significantly larger than medians, left tails are very thin, and right tails are fat. Although departure from normality reduces as the number of firms per portfolio is increased, even fifty-firm portfolios exhibit substantial departure from normality. Since a normally distributed variable lends itself to the use of standard parametric statistical tests, we employ square root and t For this exploration, we have restricted the analysis to the various volume measures and models chosen. Other variables that may affect volume prediction errors (and could be focus of future research) include: type of stock exchange, time of trading (e.g., tax/December, January,, day of the week) effects, degree of institutional ownership/trading and block trading, etc. Also the statistical relationship between volume and price variability - the so-called 'mixture of distributions' hypothesis - is not addressed here [see, e.g., Epps and Epps (1976), Tauchen and Pitts (1983), Harris (1986), and Jain and Joh (1988)].

B.B. Ajinkya and P.C. Jain, Daily trading volume

333

natural logarithm transformations of the three basic variables. The natural log transformed volume measures exhibit more symmetry in prediction error distributions. The means nearly equal the medians, and both the tails are reasonably well specified relative to the normal distribution. In detecting the presence of abnormal trading for one-day prediction intervals, the trading volume market model exhibits slightly higher power than the mean adjusted trading model. Furthermore, to account for the autocorrelations in residuals, we use an EGLS (Estimated Generalized Least Squares) procedure. For the one-day prediction interval, there is only a small increase in the power of the tests when autocorrelations in residuals are taken into account. For longer prediction intervals, both the use of the market trading volume and the autocorrelation adjustment are advantageous. As expected, the power of the tests decreases when the interval over which the abnormal trading occurs is increased. When the events are not clustered, the tests are more powerful than when they are. However, under clustering, the mean adjusted model is less powerful than other models. The rest of this paper is organized as follows. The next section specifies the variables, the expectation models, and the mode of analysis used in the study. Section 3 describes data sources, sample selection, and empirical methods. Results of the various analyses are discussed in section 4 and concluding comments appear in section 5.

2. Variables, expectation models and mode of analysis The three basic trading volume measures studied here include: V1, the number of common stock shares traded [see James and Edmister (1983), Grundy (1985), and Harris (1986)], V2, the dollar value of common shares traded [see James and Edmister (1983) and Lakonishok and Vermaelen (1986)], and 1/3, the dollar value of shares traded as a fraction of the total dollar value of the company's shares outstanding [see Beaver (1968) and Morse (1981)]. Since some researchers [e.g., Morse (1980), Pincus (1983), Ajinkya, Atiase, and Gift (1988)] used log transformations of these variables, such transformations are also included as alternative measures. Information event studies focus on the unique firm-specific reaction to the event where the variable of interest is usually the unexpected or abnormal component, i.e., after adjustment for the effects of phenomena not associated with the particular event being examined. Such adjustment can be accomplished by subtracting from observed volume during the event period an expected volume based on either (i) the average trading volume for the same firm during an arbitrary estimation period or (ii) the market model for trading volume analogous to the market model for returns. Motivations for using a trading volume market model are discussed in the next subsection. This study employs both types of expectations as detailed below.

B.B. Ajinl~va and P.C. Jain, Daily trading t,olume

334

2.1. Prediction errors based on expectation models

The first expectation model is based on the simple average volume over a specified estimation period. For n u m b e r of shares traded, the model is m

nit = nit + eiz,

(1)

for dollar value of shares traded, the model is nitpi t = nitpi t + eit,

(2)

for fraction of outstanding value traded, the model is n , t p J S i , p, t = ( n i , p i t / S i t p i , ) + eit,

(3)

where nit = number of firm i shares traded on day t, Pit = price per share for firm i on day t, S,t = n u m b e r of shares outstanding on day t, e , = error terms or residuals from the respective models, i = 1 . . . . . I (number of firms), and t = 1 . . . . . N (number of days in the estimation period). The bars above the variables represent averages over N days. The second expectation model is based on a volume market model regression line using ordinary least squares estimation. The major purpose in specifying an appropriate model is to reduce the variance of the abnormal volume. Although a well-developed economic theory such as the CAPM for returns does not seem to exist for trading volume, a number of theoretical papers linking trading volume to information releases can be used to motivate a trading volume market model. G r u n d y and McNichols (1988) model the response of trading volume to a public information release in the noisy rational expectations framework extending the earlier work of Diamond and Verrecchia (1981) and others. If the information announcement affects the entire market (e.g., interest rate changes, m o n e y supply, etc.), then trading volume of the entire market is affected. Since individual firms are components of the market, trading volume of individual firms is expected to be correlated with the market trading volume. Admati and Pfleiderer (1988) develop a model consisting of a variety of traders including informed and discretionary liquidity traders. The informed traders trade on their private information whereas the discretionary liquidity traders trade for liquidity reasons. They show that both types of traders prefer to trade at the same time so that the effect of an individual's trading on prices is minimized.

B.B. Ajinkya and P.C Jain, Daily trading volume

335

Thus, the individual firm trading volume in their model also is expected to be correlated with market trading volume. 2 Finally, to the extent transfer of information across firms as depicted in Foster (1981) and Penman (1988) affects trading volume across firms, correlations among trading across firms and hence between individual firm trading volume and market trading volume should exist. The market model for trading volume can also be motivated by assuming a multivariate normal distribution for the cross-section of securities in a manner similar to the development of the market model for returns as a statistical model. 3 In general, the extant theoretical and empirical research suggests that by bringing to bear additional information about the market, the market model could increase the power of the tests over the mean-adjusted trading method. The respective equations for the market model for the three volume measures are n i t = ai + biY~nit + e . ,

(4)

nitPi t = ai + biY'. ( nitPi,) + ei,,

(5)

n itPit/Si, Pi t = a i + b i ( Y ' . ( n i t P , t ) /

Y'. (S.p,,)) +

e,,,

(6)

where a i and bi are the intercept and the slope coefficient from the respective regressions. The alternative market indices for each volume measure are formed by summing the variables over I firms as shown. The summation across i is designated by E. Note that for the fraction measure V3, the index represents the fraction of the outstanding value of the market traded during the day. 2.2. Autocorrelation in trading volume We document that unlike returns, both raw trading volume and residuals are significantly autocorrelated. Autocorrelations in trading volume could arise when all the traders do not trade within one day on information they use to rebalance their portfolios. Some investors could adjust their holdings later than others either because they come to know of the information later or they choose to trade only periodically to minimize transaction costs. Karpoff (1985) develops a model in which an increase in trading volume lasts beyond one 2Barclay et al. (1988) suggest that the private information revealed through trading could have a systematic component as well as a firm-specific component. If informed trading reflects private information about an industry or the market, trading will increase for a number of firms at the same time. Thus, a relation between firm-specific trading and market trading will be observed. 3See Fama (1976, ch. 3) for the developmentof the market model for security returns assuming the multivariate normal distribution for a cross-section of securities. We show later that the log transformed volume measures are approximately normal.

336

B.B. Ajinkva and P.C. Jain, Daily trading volume

period when investors react heterogeneously to information. Huffman (1987) develops a dynamic equilibrium model in which exogeneous shocks give rise to serially correlated transaction volume. This occurs as the portfolio decisions made in one period influence transactions in successive periods. In general, even without a well-developed theory of multiperiod trading volume, reasonable scenarios are consistent with autocorrelations in trading volume. Our analysis in this paper examines the presence of autocorrelations and incorporates them in the statistical analysis.

2.3. Calculation of test statistics After estimating the various models over an estimation period, the prediction error (representing excess or abnormal trading volume) for security i on a hypothetical event day t is computed as Pi,t = (actual trading),,, - (expected trading),,.

(7)

For a portfolio of securities, the test statistic is the ratio of the day-zero mean excess trading volume to its estimated standard deviation. First, we describe the computation of the test statistic when the autocorrelations are assumed to be zero. The test statistic for any event day t is given by

(8) where

1 P, = ( 1 / 1 ) Y'~ Pi, t,

(9)

i=1

S(fft)

= ~ft=l ~ (Pt- if)2/( N -

1),

(10)

with N = number of days in the estimation period and I = number of securities in the portfolio whose trading volumes are available on day t. For prediction intervals longer than one day, the test statistic for average cumulative prediction error from day K to day L is given by

PKL/S(Pm.),

(12)

B.B. Ajinkya and P.C. Jain, Daily tradingvolume

337

where L

PrL= ~_, P t / ( L - K +

1),

(13)

t=K

S( PKL) = S( Pt)(( L - K + 1).

(14)

This method is similar to the approach developed by Brown and Warner (1985). 4 In the presence of autocorrelations, the least squares estimates of the intercept and the slope coefficient [in eqs. (4) to (6)] are still unbiased but not efficient. In the present study, we are also concerned with the variance estimator since it is used to compute significance levels for abnormal trading volume. In the presence of positive residual autocorrelations, the least squares estimator given by eqs. (10) or (14) for standard deviation will be biased downwards. Of course, the extent of bias is an empirical question. If the true autocorrelations in residuals are known, the regression coefficients and residual variances can be estimated using a generalized least squares estimation technique. In practice, however, we can only estimate the autocorrelations and use them in place of true values. Since the true autocorrelations are not known, an asymptotic procedure is used to incorporate autocorrelations in estimating the regression parameters. Following Judge et al. (1982, 1985), this procedure is named Estimated Generalized Least Squares (EGLS) model. In the first step, ordinary least squares method is used to estimate the intercept and the slope coefficient. Since these estimates are unbiased, they are used to estimate the required autocorrelations. The next step uses the estimated autocorrelations to obtain efficient estimates of the regression parameters to compute prediction errors. Additional explanations are provided later.

2.4. Logarithm and square root transformations Since the normal distribution has dominated statistical practice as well as theory, researchers have suggested various transformations when the variable is not distributed normally. Snedecor and Cochran (1979) report that the logarithm and the square root transformations are often used. Maddala (1976, ch. 13) recommends that if the distribution is skewed and has thick tails, it may be possible to make the distribution symmetric by taking logs. The following natural logarithm transformations of the three basic volume mea4Note that by using a time-seriesof average excesstrading volume, the test statistic takes into account the cross-sectionaldependencein the security-specificexcess trading volume.

B.B. Ajinl~va and P.C. Jain, Daily trading t,olume

338

sures are also examined: 5

LV1 : ln(1.0 + n,) = ln(a .0 + Vl ),

(15)

LV2 = ln(1.0 + n,p,) = ln(1.0 + W2),

(16)

LV3 = ln(1.0 + nipi)/ln(1.O + S, pi)

(17)

= In(1.0 + V2)/ln(1.0 + value of outstanding shares). We also used a square root transformation in which transformed variable is the square root of the raw variable. 3. Data and procedure Daily trading volume data are retrieved from the Media General tape. The daily CRSP (Center for Research in Security Prices) master tape provides information on the number of outstanding shares each day. All the volume and price series are adjusted for the effects of stock splits and stock dividends. The Media General tape available to us contains trading volume data on 1,237 N Y S E firms for a maximum of 1,536 trading days from December 26, 1974 to January 23, 1981. Of these firms, 59 firms were not used because their Cusip numbers on the Media General could not be found on the CRSP tape, or because they had more than twenty consecutive days of missing observations or nontrading. Thus the final sample consists of 1,178 firms traded on the NYSE. The sampling procedure consists of selecting securities and event dates randomly. Define day zero as the event day for a given security. One half of the estimation period is from a period prior to the event period, and the other half subsequent to the event period. For example, for an eleven-day event period, an estimation period consisting of 100 days would span a period from day - 55 to - 6 and from day + 6 to + 55. Most of our analysis is conducted separately for 100-, 170-, and 238-day estimation periods and three-, seven-, and eleven-day event periods. A new random sample is selected each time a new analysis is performed. 4. Results

4.1. Selected summary statistics of daily trading volume Table 1 shows the mean, median, standard deviation, and certain other summary statistics for the three volume measures without and with log 5Richardson, Sefcik, and Thompson (1986) analyzed Box-Cox (1962) type transformations and found that log transformations of trading volume after adding a small positive constant term yielded approximate normality. In order to handle the problem of log transformation in the event of zero volume, we added 1.0 to all volume measures before applying the log transformation. Given the reported trading volume averages in ensuing tables, 1.0 is a small number and is not likely to affect the inherent distributional properties.

19 [19] (57)

36136] (3.5)

58159] (7.7) 86186] (10.2)

134 [134] (15.6)

199 [194] (25,5)

312 [299] (48.5)

53515411 (78.9)

872 [845] (133.5)

3625 [2015] (5336.0)

Smallest

2

3

5

6

7

8

9

Largest

25.3 (21.1) [19.4] 2.9 16.2

86.3 [73.91 (52.1)

47.6 [36.8] (38.5)

32.4 [24.0] (29.3)

22.0 [15.6] (22.0)

20.9 [14.61 (21.1)

143 [9.7] (15.5)

8.6 [5.91 (9.5) 9.4 [6.41 (10.5)

6.6 [4.41 (7,5)

4.4 [2.81 (53)

3

Mean [median] IStd dev.)

675 (519) [537] 3.0 16 4

3368 [2913] (1974)

1172 [907] (958)

785 [581] (714)

454 [318] (459)

361 1247] (384)

227 [151] (253)

116 [78] (131) 157 [103] (182)

70146] (83)

331201 (43)

4

Mean [medianl (Std. de',)

Value of shares traded in $ thousands

0.145 (0.15) [0.012] 29 163

0.116 [0097] (0.08)

0.126 [0.0971 (0.11)

0.137 [0 100l (0.13)

0133 [0 0941 (0.14)

0.157 [0.1081 (0.16)

0.150 [0.1011 (0.17)

0.169 [0.l 14] (0.19) .151 [101] (0.17)

0.161 [0.107] (0.18)

0.148 [0.0951 (018)

5

Mean [median I (Std. dev.)

Percent of outstanding shares traded

Daily trading volume without any transformation

No. of shares traded in thousands

~Summary statistics for all firms are averages of 17,670 values (fifteen 100-day subperiods for 1,178 firms).

A I1firnts Mean Standard deviation Median Skewness Kurtosis

588.8 (1981.0) [164.0] 12.9 212.8

2

1

4

Mean [median I (Std. dev.)

Deciles based on market value of outstanding common shares

(market valuc of common shares outstanding in $ millions)

Firm size

2.34 (0.67) [2.30~ 0.46 3.58

4.1 ]4.01 (0.57)

3.4 [3.4] (0.67)

3.0 [3.0] (0.72)

25 [2 5] (0.73)

2.3 [2.31 (0.72)

2.0 [2.0] (0.70)

17 [1.61 (0.65) 1.7 [1 7] (0.67)

1.5 [1.4] (0.63)

1.2 [1.21 (0.62)

6

Mean [median] (Std. dev)

No, of shares traded

4.87 (0.86) [4.88] 0.15 3.63

76 [76] (0.60)

6.5 [6.5] (0.73)

6.0 [6.0] (0.80)

53 [531 (0.87)

5.0 [5.01 (0.89)

4.5 [4.5] (0.92)

3.8 [3.81 (0.91) 4.1 [4.1] (0.95)

3.3 [3.4] (0.94)

2.6 [2.61 (0.95)

7

Mean [median] (Std dev.)

Value of shares traded

0.395 (0.07) [0.395] 0.004 3.64

0.51 [0.51] (0.04)

0.48 [0.48] (0.05)

0.46 ]0.46] (0.06)

0.42 [0.42] (0.07)

0.41 10.411 (0.02)

0.38 [0.381 (0.08)

0.35 [0.351 (0.08) .36 [.361 (0.08)

0.32 I0.321 (0.09)

0.27 [0.271 (0.10)

8

Mean ]median] (Std. dev.)

Percent of outstanding shares traded

Daily trading volume v, ith natural log transformation

Summary statistics of daily trading volume without and with natural log transformation for 1,178 NYSE firms from December 26, 1974 to January 23, 1981. a

Table 1

,,D

~'~ ~

I~

~.~

"~

~'~ ~-

.~ "~

340

B.B. Ajinkya and P.C. Jain, Da#v trading t,olume

t r a n s f o r m a t i o n . 6 The statistics are presented for deciles according to firm size m e a s u r e d in terms of the value of c o m m o n stock outstanding. As expected, the n u m b e r of shares a n d the dollar value t r a d e d increase as the firm size increases. T h e firms in the smallest decile, on average, t r a d e d a b o u t $33~000 ( m e d i a n = $20,000) per d a y (column 4), whereas the firms in the largest decile t r a d e d a b o u t $3,368,000 ( m e d i a n = $1,974,000) per day. T h e m e a n t r a d i n g across deciles in terms of percent of shares o u t s t a n d i n g ( c o m m n 5) varies between 0.169% and 0.116% per day. T h e m e a n s of p e r c e n t of o u t s t a n d i n g shares t r a d e d across deciles are significantly inversely related to firm size. T h e S p e a r m a n R a n k correlation is significant at 1% level of significance. 7 A l t h o u g h the m e a n percent t r a d e d variable across deciles is related to firm size, the deviations from the overall mean are not very large. Overall, a t y p i c a l N Y S E firm t r a d e d 0.145% of its o u t s t a n d i n g shares every d a y d u r i n g this p e r i o d . It is r e a s o n a b l e to c o n c l u d e that in a p o r t f o l i o of m o r e than a few r a n d o m l y selected securities, the p o r t f o l i o average is not likely to be influenced b y a small n u m b e r of firms. F o r the overall sample, the r e p o r t e d m e d i a n , skewness, a n d kurtosis (last three rows) indicate that daily t r a d i n g volume w i t h o u t a n y t r a n s f o r m a t i o n is not d i s t r i b u t e d normally. F o r brevity, skewness a n d k u r t o s i s statistics for each decile are not p r e s e n t e d separately which also i n d i c a t e s similar d e p a r t u r e s from normality. T h e f r e q u e n c y d i s t r i b u t i o n of value t r a d e d (over 1.7 million o b s e r v a t i o n s a c r o s s firms a n d time) is p l o t t e d in fig. 1 along with the n o r m a l density function. Fig. 2 depicts the cumulative d i s t r i b u t i o n function of the empirical t r a d i n g v o l u m e a n d the n o r m a l density. Since these graphs indicate that the t r a d i n g v o l u m e d i s t r i b u t i o n is highly skewed to the right, it would seem i n a p p r o p r i a t e to a p p l y statistical tests b a s e d on n o r m a l theory. We, therefore, e m p l o y e d two c o m m o n t r a n s f o r m a t i o n s to find out whether the d i s t r i b u t i o n s of the t r a n s f o r m e d variables are closer to normality. ~ A l t h o u g h the square root t r a n s f o r m e d variables were closer to n o r m a l i t y t h a n the u n t r a n s f o r m e d variables, they were still quite different from the n o r m a l d i s t r i b u t i o n a n d thus will not be described. The second t r a n s f o r m a t i o n was the n a t u r a l log transformation. T a b l e 1 (columns 6 to 8) also presents selected s u m m a r y statistics for this t r a n s f o r m e d variable, a n d figs. 3 and 4 *The various statistics reported in table 1 are computed as follows. For each firm, first 1,500 trading days were divided into fifteen subperiods of 100 days each. These subperiods were used to compute the average trading per day, standard deviation, median, skewness, and kurtosis statistics. Thus a total of 17,670 values (15 × 1,178 firms) for each of these statistics are computed. The average statistics for all firms reported in the table are averages of these 17,670 values. 7The Spearman-rank correlation is described in Lehmann (1975, ch. 7). Bamber (1986) also reports a similar association between firm size and trading volume around earnings announcements. ~Plots for the other two variables (number of shares traded and fraction of outstanding shares traded) were similar and are not shown. Since a large number of observations are available, we are able to use small intervals of 0.1 on the x-axis. The plots therefore appear to be continuous.

B.B. Ajinkya and P.C. Jain, Daily trading volume

341

VALUE TRADED Row Volume

10

Empirical o

©

Normal

--4

--2

0

2

4

Standard Deviation from the Mean Fig. 1. Density of trading volume.

depict the density and cumulative distribution functions, respectively. The figures indicate that the empirical distribution of log transformed trading volume is quite close to the normal distribution. 9 From the summary statistics presented in the last five rows of columns 6 to 8, we note that the medians are almost the same as the means and the skewness and kurtosis values are quite close to those for the normal distribution, x° Similar to these results, Ball and 9The percentage of observations below the mean for the log transformed variable (fraction of outstanding shares traded) is 50.5% (expected 50.0% for normal distributions). The empirical distribution is quite close to the normal distribution in the sense that at such specific points as one, two, and three standard deviations above the mean, the cumulative empirical distribution is within 1% of the theoretically expected percentages under normality. The selected empirical (theoretical) percentages are 84.85 (84.13), 97.56 (97.72), and 99.75 (99.87). l(~lthough the plot of the log transformed variables shows remarkable closeness to normality, the various chi-square goodness-of-fit tests rejected the null hypothesis of normality. This occurs because for a very large number of observations such as in the present case, even small deviations from the expected percentages in various cells result in large chi-square statistics. For example, for a sample of 1.7 million observations, if the actual fraction around the mean deviates merely by 0.0005 from the expected, the contribution of one cell in the calculation of the chi-square would be approximately 10.0. If the entire distribution is divided into twenty cells, the computed chi-square statistic could easily be over 200, which would be highly significant. The tests of normality used here are described in Snedecor and Cochran (1979, p. 84). The normality results are similar for the other two log transformed measures of trading volume.

342

B.B. Afinl, ya and P.C. Jain. Daily trading t,olume

VALUE TRADED R a w Volu rn ~ 1 O0

9C'

-

Empirical /

/

80

70

/ / Normal

!

60

50

,

/

//

4-0

.30

20

10

C,

~

--4

I --2

0

,i,, ,,,,i, 2

JllJ J J ' l l l l i

Illrllll 4

Standard "Deviation from the Mean Fig. 2. Cumulative distribution of trading v o l u m e .

Finn (1989) have reported that the empirical distribution of trading volume on the Sydney Stock Exchange is approximately lognormal. Most of the ensuing analysis was performed for both the untransformed and the log transformed measures, but the results are presented for certain selected cases only. 4.2. Results for excess trading volume The previous section presented an analysis of the actual trading volume variable. We now focus on daily prediction errors or excess trading. It is possible that the distributional characteristics of the actual trading volume and the prediction errors could differ substantially. Tables 2 and 3 show the properties of prediction errors for individual securities and for portfolios (when no abnormal trading is introduced artificially) for the raw and the log transformed variable, respectively. Results are reported only for the fraction of outstanding shares traded when the estimation period is 100 days. The qualitative results do not change when the other two measures are used or when the length of the estimation period is changed.

B.B. Ajinkya and P.C. Jain, Daily trading volume

343

VALUE TRADED Log Transformed Volume 4.5

-

Empirical

4

3.5-

0

25

.D

o 2

1.5

1

(3.,5

0

~ --4

--2

0

2

Standard Deviation from the Mean Fig. 3. Density of trading volume.

W e first discuss the results based on the mean-adjusted trading model. The c o m p a r i s o n of these results to those based on the trading volume market model (OLS) and the E G L S model is presented subsequently. F r o m table 2 it is evident that the distribution of prediction errors is highly nonnormal. For individual firms, the average skewness measure is 6.3, the kurtosis measure is 87.2, and the studentized range is 28.0. These values are significant at any reasonable levels of significance. As the n u m b e r of securities per portfolio increases, the empirical distribution is closer to normal; but even for fifty security portfolios, the departure from normality is substantial. Even larger portfolios are unlikely to exhibit normality since we find that the distribution of daily m a r k e t volume itself departs significantly from normality. 11 liThe average skewness and kurtosis statistics for the market index defined by the percent of outstanding shares traded are 0.73 and 3.78, respectively. Although somewhat smaller than those for fifty security portfolios, they are statistically significant at 1% level of significance. These averages are from fifteen values which are computed from fifteen 100-day periods. The departure of the other two market indices from normality is even larger.

344

B.B. Ajinkva and P.C. Jain Daily trading uolume

VALUE TRADED Log T r a n s f o r m e d

Volume

1 OO

90

80

Empirical and Normal

70 0 60

50

,.c ©

40

E ,o t~

50

2_0

10

0 --4-

--2

C'

2.

4

Standard Deviation from the Mean

Fig. 4. Cumulative distribution of trading volume.

Table 3 indicates that the distribution of prediction errors for the log transformed measures is much closer to normality. The skewness measure is quite close to zero for individual securities and for portfolios of different sizes. The kurtosis measure of over 5.0 for individual securities is significant but still rather small, and decreases to 3.0 (exactly the expectation under normality) for portfolios of fifty securities. Besides examining the skewness and kurtosis, an examination of the entire distribution is informative since our analysis in presence of abnormal trading can be visualized as a lateral shift of the cumulative distribution function. Similar to the previous figures, we plotted graphs of various portfolios of the untransformed and the log transformed variables, respectively (not shown). The untransformed variable exhibits substantial departures from normality. On the other hand, for the log transformed variables, the empirical distribution of prediction errors is remarkably close to normality (similar to figs. 3 and 4). This suggests that the inferences based on normal theory for the log transformed variables would be appropriate for prediction error analysis.

2

20,000

20,000

10,000

5,000

2,000

1

1

5

10

20

50

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model d

3

Model

- 0.00002 - 0.00002 - 0.000(~

- 0.00001 - 0.00001 - 0.00000

- 0.00001 - 0.00001 - 0.00000

- 0.00001 - 0.00000 0.00000

- 0.00004 -0.00004 - 0.00002

4

Mean excess trading a

0.00028 0.00028 0.00027

0.00046 0.00046 0.00043

0.00064 0.00063 0.00060

0.00096 0.00095 0.00088

0.00187 0.00186 0.00177

5

Standard deviation~

1.4 1.4 1.2

3.0 3.0 2.4

2.8 2.8 2.6

5.1 5.2 5.2

6.3 b 5.8 6.0

6

Skewness a

9.0 8.9 10.7

35.1 36.3 40.1

22.8 23.1 25.8

75.0 79.2 89.2

85.2 98.6

87.2 b

7

Kurtosis a

11.7 11.5 16.4

21.0 21.2 31.1

19.7 20.0 24.9

30.4 32.6 39.5

28.0 ~ 33.6 36.2

8

Studentized range a

0.127 0.081 0.057 0.023

2,000 5,000 10,000 20,000

3.255 3.161 3.114 3.081

Kurtosis

~Upper 1% and 0.5% points for studentized range (N = 1000) are 7.80 and 7.99, respectively. dEstimated generalized least squares model uses autocorrelation coefficients estimated from the residuals, as described in Judge et al. (1982, ch. 15).

Skewness

Sample size

"Standard deviation, skewness, kurtosis, and studentized range are averages from the number of portfolios depicted using the estimation period of 100 days. bUpper 1% points for samples drawn from a normal population:

No. of portfolios (replications)

No. of securities in each portfolio

P r o p e r t i e s of daily excess t r a d i n g v o l u m e m e a s u r e s w h e n no a b n o r m a l t r a d i n g v o l u m e is i n t r o d u c e d . T h e e v e n t d a t e s and securities are selccted r a n d o m l y and s e p a r a t e l y for each portfolio grouping. T h e t r a d i n g v o l u m e m e a s u r e is fraction of o u t s t a n d i n g c o m m o n shares traded, without any transformation.

Table 2

e~

4.

e,

20,000

10,000

5,000

2,000

1

5

10

20

50

Mean-adj usted trading OLS (market) model EGLS model

Mean-adiusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model d

3

Model

0.00180 0.00063 0.00055

- 0.00029 0.00016 0.00028

0.00026 0.00084 0.00069

0.00074 0.00006 0.00008

0.00039 0.00020 0.00045

4

Mean excess trading ~

0.0111 0.0107 0.0104

0.0174 0.0168 0.0162

0.0246 0.0237 0.0230

0.0344 0.0331 0.0322

0.0769 0.0740 0.0722

5

Standard deviation"

0.02 0.07 0.09

0.06 0.03 0.03

0.03 0.00 0.02

0.09 0.06 0.07

0.15 h 0.12 0.13

6

Skewness ~

3.0 3.0 3.1

3.1 3.2 3.2

3.2 3.2 3.3

3.3 3.4 3.5

5.2 h 5.6 6.2

7

Kurto~is"

7.3 6.8 7.3

7.4 7.6 7.8

9.0 9.3 9.4

87 8.9 9.9

14.1' 14.7 16.1

8

Studentized range a

0.127 0.081 0.057 0.023

Skewness

3.255 3.161 3.114 3.081

Kurtosis

CUpper 1% and 0.5% points for studentized range (N = 1000) are 7.80 and 7.99, respectively. dEstimated generalized least squares model uses autocorrelation coefficients estimated from the residuals, as described in Judge et al. (1982, ch. 15).

2,000 5,000 10,000 20,000

Sample s i z e

~Standard deviation, skewness, kurtosis, and studentized range are averages from the number of portfolios depicted using the estimation period of 100h days. Upper 1% points for samples drawn from a normal population:

2

20,000

1

No. of portfolios (replications)

No. of securities in each portfolio

of outstanding common shares traded.

P r o p e r t i e s o f d a i l y excess t r a d i n g v o l u m e m e a s u r e s w h e n n o a b n o r m a l t r a d i n g v o l u m e is i n t r o d u c e d . T h e event d a t e s a n d securitics are selected r a n d o m l y a n d s e p a r a t e l y for e a c h p o r t f o l i o g r o u p i n g . T h e t r a d i n g v o l u m e m e a s u r e is n a t u r a l logarithm transformation of the fraction

Table 3

,q%

,~

.~ ~' ~

~, .~

~-~ =,

B.B. Ajinkya and P.C Jain, Daily trading volume

347

Tables 2 and 3 also show results when the trading volume market model (OLS) and the Estimated Generalized Least Squares (EGLS) model is used. The average market model R-square is about 11% for the log transformed measures (and about 8% for the raw measures), which is similar to the returns case. The estimated first-order autocorrelation in the OLS residuals is, on average, 0.3. A first-order autocorrelation of 0.3 is significantly different from zero since the two standard deviation value is about 0.2 when 100 observations are used in the estimation period. 12 The EGLS model takes the autocorrelation structure in the residuals into account. The use of first-order autocorrelation substantially eliminates all of the autocorrelation present in the residuals. Thus, for the EGLS model, AR(1) structure is imposed on the residuals. The EGLS model when residuals are AR(1) is described in Maddala (1976, p. 278) or Judge et al. (1982, p. 445) and is similar to the Cochrane-Orcutt (1949) procedure. As described therein, the first step in the EGLS model is the same as the OLS model. Subsequently, autocorrelations are estimated from the OLS residuals and are incorporated to obtain more efficient estimates of the requisite parameters and prediction errors. The results indicate that for one-day prediction intervals, there is only a small improvement in explaining the variation in daily trading volume with the use of the OLS and the EGLS models. For example, for individual firms, the average standard deviation of prediction errors in table 3 decreases from 0.0769 for the mean adjusted model to 0.0740 for the OLS model, and finally to 0.0722 when the EGLS model is used. 13 In the next subsection, we present an analysis of rejection percentages (power) when abnormal trading is artificially introduced.

4.3. The power of the tests for one-day event period Given that the empirical distribution of the prediction errors for the log transformed variable does not deviate substantially from the normal distribution, we extend our analysis by introducing abnormal trading on day zero and examine the rejection percentages of one-tailed tests of size 0.05. In our initial simulation analysis, we define the event period to be only one day. Depending upon the nature of the event, abnormal trading could occur in anticipation of an event a n d / o r after the event. Since we do not have a theory (like the efficient markets hypothesis) to guide us how trading volume should behave around an event, the effective event period may be longer than one day when an event occurs on a single day. In an actual study of trading volume, the 12The expected value of a first-order autocorrelation is approximately - 1 / ( N - 1) and the variance is 1/( N - 1), where N is the number of observations in the estimation period [see Fama (1976, p. 118)]. ~3Instead of AR(1), we used two other residual autocorrelation structures specified by AR(2) and ARMA (1,1) for a sample of 2,000 replications. Since the additional reduction in the standard deviations was less than 1%, more sophisticated ARIMA models of residuals do not seem to be useful. For the rest of the paper, we use only AR(1) residual autocorrelation structure for the EGLS model.

348

B.B. Ajinkya and P.C. Jain, Daily trading ~,olunle

researcher would have to subjectively decide the appropriate length of the event period. Longer event periods are examined in the next subsection. The approach for introduction of abnormal trading is as follows. Levels of abnormal trading from 0.1 to 0.5 of the average trading in the estimation period are added to the trading on day zero. For example, if a security, on average, trades 5,000 shares per day (for the number of shares traded measure) during the estimation period, an introduction of 0.1 level of abnormal trading means that we artificially add 500 shares of trading on day zero. Table 4 reports rejection percentages for individual securities and for portfolios of up to fifty securities. These percentages are based on a large number of replications in that even half as many replications resulted in rejection percentages to be almost similar to those reported. Thus, the reported percentages are reliable for generalizing the results for most circumstances. As expected, fewer replications are required to obtain generalizable results as portfolio size is increased because the rejection percentages converge faster as the number of securities per portfolio is increased, t4 While many information content studies use portfolios of more than fifty securities, the portfolio size in this study does not exceed fifty firms. Although additional simulation analysis with larger portfolios would provide us with a more detailed rejection frequency tables, the nature of the patterns in the various ensuing tables is not likely to be altered in a substantially useful manner. For brevity and for keeping the computing expenses under control we, therefore, did not study larger portfolios) 5 When no abnormal trading is present (column 4), the rejection percentages are close to the theoretically expected 5%, ranging between 4.8% and 6.8%. For individual securities, the rejection percentage increases to a maximum of 12.2% when the level of abnormal trading is 0.5. As expected, the rejection percentage increases as the number of securities per portfolio increases. With fifty securities in the portfolio, 0.1 level of abnormal trading (column 5) is detected between 41% and 52% depending on the model, and it requires the level of abnormal trading to be only 0.3 (column 7) for the rejection percentages to be close to 100%. 16 ~4For individual securities, we found thal a sample sizc of 100,000 randomly selccted securities v, as more than adequate to obtain reliable rejection pcrcentages. For deciding the number of randomly selected portfolios, we started with the number 100,000 divided by the portfolio size. In all cases, the rejection percentages were reliable as described in the text. Thus the portfolio size times the number of replications from table 4 onwards is constant at 100,000. ~SThe computation of each of many of the rejection percentages reported in tables 4 to 8 takes more than one hour of CPU (implying approximately three to tire times connect time) on the VAX 8600 computer at the Wharton School computing facility. ~6Although wc do not simulate rejection percentages for larger portfolios, it is possible to conjecture about the increase in rejection percentages by extrapolating the numbers in various columns. For example, if we extrapolate the numbers in column 5 of table 4, it would take a portfolio of about 100 securities to detect the abnormal trading of 0.1 level for almost 100% of the time.

Table 4

100,000

20,000

10,000

5,000

2,000

1

5

10

20

50

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model

Mean-adjusted trading OLS (market) model EGLS model b

3

Model

5.4 6.8 6.2

5.6 5.7 6.0

5.3 6.3 6.3

4.8 5.4 5.6

5.4 5.8 5.8

4

0.0 6

0.2 7

0.3 8

0.4

41.2 48.2 52.0

19.4 22.0 24.0

12.9 14.5 15.4

8.0 9.3 10.0

5.9 6.5 6.6

87.5 91.5 93.5

46.8 53.4 57.1

25.2 29.3 31.7

12.9 15.4 16.9

6.5 7.3 7.6

99.3 99.9 100.0

77.1 82.7 86.5

43.1 49.4 53.5

20.3 24.2 27.2

7.2 8.3 8.7

100.0 100.0 100.0

94.1 96.9 97.9

64.0 70.8 75.1

42.2 36.2 40.7

8.0 9.5 10.2

Rejection percentages of the null hypothesis a

5

0.1

100.0 100.0 100.0

99.4 99.8 99.9

82.3 87.5 90.4

50.3 55.7

8.9 10.9 12.2

9

0.5

Actual levels of abnormal trading added on day zero, represented as a fraction of mean raw trading in the estimation period

a One-tailed test, a = 0.05. bEstimated generalized least squares model uses autocorrelation coefficients estimated from the residuals, as described in Judge et al. (1982, ch. 15).

2

No. of portfolios (replications)

1

No. of securities in each portfolio

Comparison of rejection percentages of the null hypothesis that mean abnormal trading is zero. The event period is one day. The trading volume measure is natural log of the fraction of outstanding c o m m o n shares traded.

c~

g~

350

B.B. Ajinkva and P.C. Jain, Daily trading ~2olume

A comparison across models indicates some advantage in going from the mean-adjusted model to the OLS (market) model and then to the EGLS model. The advantage is not substantial. Most of the power gain for one-day prediction errors is from increasing the number of securities per portfolio. Thus, fine tuning the measurement of excess trading may not bring appreciable benefits but an increase in sample size could. Given a choice, an increased sample could substantially improve the power of the test.

4.4. Longer event periods: Distribution of excess trading One important aspect by which daily trading volume is different from daily stock returns is the presence of first-order autocorrelation of about 0.3 in residuals, as noted earlier. We noted from table 4 that for single-day excess trading, ignoring the presence of autocorrelation does not appreciably affect the power of the tests. However, longer periods could exhibit more striking differences since the number of covariances for computing multi-day variance increases with increasing prediction interval. A positive autocorrelation in the residuals implies that if the variance of multi-day residuals (or prediction errors) is computed assuming independence across days, the computed variance will be biased downwards. We now address the issue of computing the variance of multi-day prediction errors. We first examine whether ignoring autocorrelations in residuals biases the variance [i.e., standard deviation given by eq. (14)] downwards in a way that can be detected empirically. If the standard deviation is biased downwards, the test statistic given by eq. (12) will be overstated and the null hypothesis will be rejected more often than it should be under a properly specified test statistic. Thus, we examine whether the rejection percentages are higher than expected under the normality assumption when autocorrelations are not taken into account. For brevity, in this subsection we discuss only the analysis when market trading volume is used. Row 1 in table 5 presents rejection percentages for multi-day (three, seven, and eleven days) prediction periods, and for three estimation periods when no abnormal trading is introduced. The residuals are calculated using the OES model (without autocorrelation adjustment) and the rejection percentages are for the test statistic given by eq. (12). If the autocorrelations in residuals were inconsequential, the rejection percentages should be 5% across the entire first row. Instead, the rejection percentages are quite large. The null hypothesis is rejected between 9.4% and 16.4%, depending on the number of days in the prediction interval and the estimation period. The rejection percentages increase as the prediction interval increases from three to seven to eleven days which is consistent with positive autocorrelation in residuals. In general, we conclude that for multi-day prediction intervals, ignoring autocorrelations in residuals biases the test statistic significantly.

B.B. Ajinkya and P.C. Jain, Daily trading volume

351

Table 5 Rejection percentages when the event periods are three, seven, and eleven days, the estimation periods are 100, 170, and 238 days, and two different methods for estimating the variance of cumulative prediction errors are used. The null hypothesis is that the cumulative mean abnormal trading in the event period is zero. The results are for individual securities when no abnormal trading is introduced. The trading volume variable is the natural log transformation of the fraction of outstanding common shares traded, a Estimation period (days): Prediction interval (days):

238

170

100

3

7

11

3

7

11

3

7

11

1

OLS (market) model

9.4

13.2

16.1

9.7

13.3

16.4

9.5

13.9

15.4

2

EGLS model 2 b

5.6

5.8

6.1

6.0

6.2

6.5

6.1

6.9

7.7

~One-tailed test, a = 0.05. bThe OLS model assumes residuals to be independent across days. The EGLS model uses the AR(1) residual autocorrelation structure to estimate the regression parameters and computes the variance of the multi-day prediction interval by using the variance-covariance matrix of single-day residuals.

The computation of variance for multi-day prediction errors accounting for autocorrelations in residuals is outlined below. In this procedure, we first use the EGLS method [assuming AR(1) residuals] to obtain efficient estimates of the intercept and the slope coefficient. For computing prediction errors for any day (say, day t + 1) while using the EGLS procedure, we need actual trading volume on the previous day t [see Judge et al. (1982, sect. 15.4)]. For the first day in the prediction interval, this does not cause any problem since the previous day t is assumed to be a nonevent day. However, for the subsequent days (i.e., day t + 2 and beyond) in the prediction interval, we do not use the previous days' (day t + 1 and so on) actual firm-specific trading volume. Instead we use expected trading volume on day t + 1 and so on computed from using all the data in the estimation period. This is desirable because the firm-specific trading volume for all days during the prediction interval is assumed to be affected not only by randomness but also by the specific event under examination. 17 The cumulative prediction errors are then given by the sum of daily prediction errors. The conditioning independent variable, characterized by the market trading volume, is assumed to be exogenous and known for computing prediction errors. We assume that the effect of the firm-specific event on the market trading volume can be ignored for our analysis because of a large number of firms comprising the market. The variance of multi-day prediction errors is computed from the sum of single-day prediction error variances and the necessary covariances, and is used to compute the test statistics. 17Essentially, the computation procedure is similar to that used in the time-series analysis when estimation period data are used for multi-period ahead forecasts.

352

B.B. A/inkya and P.C. Jain, Daily trading t,olume

Whether the above EGLS approach is satisfactory or not can be evaluated by examining the empirical distribution of the test statistic when the null hypothesis is true. If the distribution is approximately the same as the normal distribution, we can consider the method to be satisfactory. 18 Row 2 of table 5 presents rejection percentages for this method. For the estimation period of 238 days, the rejection percentages are between 5.6% and 6.1%, which are quite close to the expected 5%. Only for the extreme case of the 100-day estimation period and l l - d a y prediction interval does the rejection percentage exceed 7.0%. This suggests that for cumulation over large number of days (approximately 11), an estimation period of 100 days is too small but a period of 238 days seems adequate. Although table 5 results are based on the 5% significance level, plots similar to the earlier ones were drawn for the entire distribution of the test statistics. The plots indicated that the test statistics are close to normal when autocorrelations are taken into account, but the test statistics are misspecified when autocorrelations are ignored. Conclusions are similar for portfolios of five to fifty securities. For the remainder of the analysis, our results use the above procedure to estimate the variance of multi-day prediction errors. 19

4. 5. Longer event periods with abnormal trading Table 6 presents results for the mean-adjusted trading model and the EGLS model when the estimation period is 238 days and the prediction interval is seven days. The main difference between the mean-adjusted model and the EGLS model is that the mean-adjusted model does not take the market trading volume into account in estimating the parameters and the prediction errors, whereas the EGLS model does. Thus, we are able to compare the effect of using the market trading volume by examining the two consecutive rows in table 6. For both these models, autocorrelations in residuals have specifically been taken into account in computing variance for multi-day prediction intervals as described in subsection 4.4. Rejection percentages using the OLS (market) model are not computed since our earlier results (row 1 in table 5) indicated that test statistics for multi-day prediction errors without autocorrelation adjustment are biased. ~ O u r main concern in this analysis has been to calculate the variance of the multi-day prediction errors by taking the residual covariances into account. Effects of other possible adjustments [e.g., Theil (1971, p. 122)] due to estimation errors in various parameters are assumed to be negligible. Since we find that the empirical distribution of the test statistic after adjusting for the autocorrelations is close to the theoretical one, additional adjustments are not likely to be very useful for this study. 19We employed two other methods to account for autocorrelations in residuals for computing multi-day prediction error variances. These approaches formed multi-day residuals from single-day residuals and computed the variances using multi-day residuals. The results were similar to those obtained using method 2 described in the text and the details are available from the authors.

Table 6

100,000

20,000

10,000

5,000

2,000

1

5

10

20

50

aOne-tailed test, a = 0.05. bEstimated generalized least squares model.

No. of portfolios (replications) 2

No. of securities in each portfolio 1

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model b

Model 3

4.7 5.0

4.7 5.1

4.7 4.7

4.9 5.6

5.5 5.8

0.0 4

0.2 6

0.3 7

0.4 8

7.0 10.3

6.7 8.5

6.5 6.3

6.0 6.8

5.8 6.1

10.7 17.7

9.4 10.8

8.7 8.6

6.6 7.8

6.0 6.4

15.0 23.7

10.3 13.3

9.7 10.7

7.6 8.9

6.2 6.7

2L7 30.7

13.0 15.7

11.3 12.9

8.3 9.9

6.4 7.1

Rejection percentages of the null hypothesis ~

0.1 5

26.0 38.0

16.0 19.6

13.2 14.9

8.9 11.1

6.6 7.4

0.5 9

Actual levels of abnormal trading, represented as a fraction of m e a n raw trading in the estimation period

Rejection percentages of the null hypothesis at the 5% significance level when the event period is seven days, the estimation period is 238 days, and different levels of abnormal trading are introduced. The null hypothesis is that the cumulative m e a n abnormal trading in the event period is zero. The variable used for trading volume is natural log transformation of fraction of outstanding c o m m o n shares traded.

o~

354

B.B. Ajinkya and P.C. Jain, Dai(v trading [,olume

As expected, the power of the tests decreases when abnormal trading occurs over an interval (seven days) rather than a single day (day zero). For example, when level of abnormal trading is 0.4 (column 8), the EGLS model rejection percentage of the null hypothesis for the portfolio of fifty securities is 30.7%, compared to the earlier 100.0% rejection percentage for a single-day prediction interval reported in table 4. Another important observation from this table is the usefulness of the market volume for multi-day prediction intervals. The EGLS model usually rejects the null more often than the mean-adjusted model. Thus, the usage of market volume increases the power of the tests. This tendency existed for single-day analysis in table 4 but it is more apparent as the prediction interval increases from one day to three, seven, and eleven days. 2° For brevity, only seven-day prediction interval results are reported.

4.6. Event date clustering

An examination of power in the presence of clustering (identical event date) is interesting for at least two reasons. First, differences between the EGLS model (which uses market volume) and the mean-adjusted trading model should be more evident in the clustering case. This follows from the fact that market trading volume would explain increasingly larger variation in portfolio trading volume as the number of securities per portfolio increases when clustering is present. This is because, as the number of securities increases in a portfolio, the portfolio is more like the market. Second, since the presence of clustering implies larger positive cross-sectional dependence, the diversification effect (i,e., the decrease in portfolio variance) from adding more securities to portfolios is less for the clustering case than for the nonclustering case. 2~ Thus, for a given level of abnormal trading and a given portfolio size, it should be more difficult (i.e., low power) to identify the presence of abnormal trading in the clustering case. 2°We also employed a nonparametric sign test in which the null hypothesis is that the proportion of sampled securities having positive abnormal trading is 0.5. The test statistic is well-behaved in the absence of abnormal trading but, as expected, the rejection frequencies in the presence of abnormal trading for the nonparametric test are usually smaller than those reported in table 6 (and table 4). For example, for fifty securities in a portfolio and abnormal trading of 0.5, the sign test rejects the null 25.8% of the time compared to 38.0% reported in table 6. Also, the average t-statistics are larger than the corresponding z-statistics (computed from sign tests) for different levels of abnormal trading. However, if the abnormal trading is very large, even the less powerful nonparametric test will reject the null hypothesis nearly 100% of the time, and thus the parametric test will not necessarily dominate the nonparametric test in all circumstances. For additional discussion and the details of the tests employed, see Brown and Warner (1980, p. 217/. 21The average correlation coefficient between trading of two randomly selected firms with the same event date is approximately 0.1, whereas it is close to zero (0.003) when the event dates are different for the pair. These results were obtained by randomly selecting 5,000 pairs of firms with 100 trading days in the estimation period.

Table 7

2

1

10,000

2,000

10

50

I0,000

2,000

10

50

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model

Mean-adjusted trading EGLS model b

3

Model

4.8 4.5

5.5 5.6

4.8 6.0

4.8 6.6

4.9 5.8

5.0 5.9

4

0.0 6

0.2 7

0.3

8

0.4

24.4 75.7

14.6 28.4

10.6 17.6

45.0 94.7

23.6 48.1

15.3 26.9

63.0 99.0

35.2 67.6

21.4 38.7

5.8 7.3

6.4 7.2

5.4 6.6

6.5 10.8

7.4 9.1

6.0 7.1

7.0 13.3

8.4 10.5

6.5 8.0

8.3 16.0

9.0 11.6

7.1 9.0

Rejection percentages of the null hypothesis ~

11.3 33.0

8.8 13.9

7.4 10.1

Rejection percentages of the null hypothesis ~

5

0.1

9.0 18.5

9.7 13.6

7.5 9.9

75.7 99.9

49.1 82.3

29.4 52.3

9

0.5

Actual levels of abnormal trading, represented as a fraction of mean raw trading in the estimation period

aPanels A and B are based on calculations similar to those in tables 4 and 6, respectively. The tests are one-tailed, a = 0.05. bEstimated generalized least squares model.

20,000

5

Panel B: Prediction period is seven days

20,000

5

Panel A : Prediction period is one day

No. of portfolios (replications)

No. of securities in each portfolio

Clustering: Rejection percentages of the null hypothesis for detecting abnormal trading using different estimation models. The null hypothesis is that the m e a n abnormal trading in the event period is zero. The trading volume measure is natural log of the fraction of outstanding common shares traded. The event dates for all firms in each portfolio are identical.

.=

e,

5"

356

B.B. Ajinkya and P.C. Jain, Daily trading volume

Table 7 presents rejection percentages when all the securities in the portfolio have an identical event date. For brevity, results are presented for selected cases only. A comparison of the mean adjusted trading model to the E G L S model can be made directly. For example, given a one-day prediction period and portfolios of fifty securities with 0.3 level of abnormal trading (column 7, panel A), the rejection percentages are 45.0% for the mean-adjusted trading but 94.7% for the E G L S model. Similar results are found for longer prediction periods. Thus, in presence of clustering, we r e c o m m e n d the use of the E G L S model instead of the mean-adjusted model. T o address the second point about the comparison of power between the clustering case and the nonclustering case, contrast table 7 to tables 4 and 6. In particular, the last two rows of tables 6 and 7 are directly comparable. The rejection percentages for the clustering case (table 7) are smaller than that for the nonclustering case (table 6) at each level of abnormal trading.

4. 7. Sample formation by firm size T h e average R-square for regression of individual firm trading volume on m a r k e t trading volume is about 11%, as reported earlier. It is reasonable to expect this association to vary somewhat across different firm sizes. For r a n d o m l y selected firms from the deciles with the largest and the smallest firms, the average R-squares are 18.5% and 7.7%, respectively. 22 Given this empirical observation, it is expected that the use of the market through the E G L S model would be more beneficial for the larger firms than for the smaller firms. Results (not reported for brevity) similar to those reported in tables 4 and 6 were obtained separately for the firms in the largest and the smallest deciles. Various comparisons indicated that similar to all firms, the E G L S model detects the abnormal trading more often than the mean-adjusted trading across the two extreme deciles examined, but the gains are usually m o r e p r o n o u n c e d for the larger firms. 23

5. Summary and conclusions This paper examines characteristics of daily trading volume of c o m m o n stocks traded on the New York Stock Exchange. Using simulation procedures with actual data, the paper investigates three trading volume definitions and

22The autocorrelation structure is similar across different size firms. The average first-order autocorrelation across firm-size decries is close to the earlier reported overall average of 0.3. The percentages of nontrading days for the firms in the sample are somewhat different across decries. For the entire sample, an average firm does not trade for 1.12% of the days. This increases to 3.81% for the smallest decile and declines to 0.2% for the largest decile. 23As expected, for very large abnormal trading levels (i.e., when the rejection percentages are close to 100%), even the simpler mean-adjusted model rejects the null hypothesis most of the time, and the improvement from using the EGLS model is not substantial.

B.B. Ajinkya and P.C. Jain, Daily trading volume

357

two expectation models to measure excess (abnormal) trading, and evaluates properties of various statistical tests that might be used in event studies of trading volume. The issues addressed include (i) nonnormality of trading and excess trading, (ii) advantage of using the market over mean-adjusted trading, (iii) effects of autocorrelation on multi-day excess trading (cumulative average prediction errors), (iv) effects of event clustering, and (v) effects of firm size. The paper's findings are discussed below. The distributions of untransformed trading volume measures for individual securities and for portfolios exhibit significant departures from normality. Natural log transformations of the volume measures lend greater symmetry to trading volume and to prediction errors such that the entire empirical cumulative distribution function is remarkably close to the normal cumulative distribution function. This is true regardless of the number of securities in each portfolio. Thus, the empirical power of the event study procedures closely approximates the theoretical power implied under normality. The raw trading volume can be considered to be approximately lognormally distributed. The average R-square for the trading volume market model is 11%, which is similar to the returns case. Unlike returns, the first-order autocorrelation in trading volume residuals is about 0.3, which should be taken into consideration to detect excess trading volume. We compare the rejection percentages of the null hypotheses when the mean-adjusted trading model is used to those when the market model (i.e., OLS without autocorrelation adjustment) or the EGLS (estimated generalized least squares) model is used. The EGLS model takes the autocorrelations into account by incorporating an AR(1) residual autocorrelation structure. For a one-day prediction interval, there is only a small advantage in using the OLS model or the EGLS model over the meanadjusted trading model. For multi-day prediction intervals, it is inappropriate to ignore autocorrelations to detect abnormal trading volume. If autocorrelations are ignored, the null hypothesis is rejected too often when no abnormal trading is introduced in the data. For this reason, we do not use the models without autocorrelation adjustment (i.e., OLS) for the analysis involving multi-day prediction errors. The simple procedure to incorporate autocorrelations is satisfactory in that the test statistics under the null hypothesis are properly specified. The advantage of using the EGLS model over the mean-adjusted trading model increases as the length of the prediction interval increases, and also as the firm size increases. In the presence of event date clustering, it is more difficult (i.e., the power is lower) to identify the existence of abnormal trading due to positive cross-sectional correlation across securities. However, the use of market trading volume is more advantageous in the presence of clustering than in its absence. Using the natural log of trading volume, and given one- to three-day prediction intervals, estimation periods of 100 days seem adequate for estimat-

358

B.B. Afinkva and P.C. Jain Daily trading volume

ing the expectation models. For longer prediction intervals, longer estimation periods lead to improved specification of the test statistics. In general, the use of daily trading volume data in event studies is straightforward. The methodologies based on the EGLS model and the standard parametric tests are well specified in a variety of conditions. Similar to the returns case, there are substantial gains to more precise pinpointing of an event.

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