Data and empirical results

380 X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 where Z t is a vector of variables included in the regression model, u is a vector of parameters to be estimated, N is the number of observations, and the disturbance m t is stationary. For each regression, there are 20 orthogonal conditions and 20 parameters to be estimated. 8 For example, for 141 cross-sectional units, there are a total of 141 3 20 5 2,820 parameters to be estimated. Since the number of parameters to be estimated equals the number of orthogonal conditions, the model is just identified. We estimate the true parameter vector u by the value of uˆ that minimizes the following quadratic function: S u,V 5 [Nm N u]9V 2 1 [Nm N u]9 5 where V 5 Cov[Nm N u ],[Nm N u ]9. We impose the moment condition [Eq. 3] in estimating the volatility regressions of individual securities. The procedure of the GMM estimation involves selecting an estimator to set the linear combination of the moment conditions to 0 while minimizing [Eq. 5]. We apply the GMM to the system of equations involving all securities in the sample. More specifically, we estimate Eqs. 2 or 2a simultaneously for all securities to provide efficient estimates of parameters by accounting for cross-correla- tion in error terms.

4. Data and empirical results

Data for price, size, time, and date for each stock transaction were obtained from the Trades and Quotes TAQ database of the New York Stock Exchange NYSE over the period from April to June 1995. Over this period, the NYSE was open from 9:30 a.m. to 4:00 p.m. Eastern Time. Transaction data in each day are divided evenly into 13 half-hour intervals for 63 trading days. This results in 819 time series units observations for each stock. We use the midquotes at the beginning and the end of each 30-minute interval to compute the returns for each stock for that interval. Using midquotes avoids measurement errors caused by bid-ask bounce, which induce spuri- ous volatility in stock returns. For each half-hour interval, we count the number of transactions and calculate the average trade size for each stock. Average trade size and the number of trades are all assumed to be 0 for no-trade intervals. The sample includes 141 randomly selected stocks from the TAQ database. 9 We then evenly divide these 141 stocks, based on the mean daily trading frequency during the sample period, into three groups, each having 47 stocks: the most, average, and least frequently traded stocks. Since the frequency of trades is often correlated with return volatility, it would be interesting to see whether the volatility–volume relation varies with stocks of different trading frequencies. Grouping the sample by trading frequency thus allows us to detect any major differences in the volume–volatility relation for firms with different trading activities. X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 381 4.1. Empirical evidence Table 1 lists the firms ticks included in the sample by the order of average daily trading frequency, and summarizes the characteristics of stocks included in the sample. As shown in Table 1, there is a sharp difference in the average frequency of trades between the three groups of stocks. The average daily number of transactions is 229 for the most frequently traded stocks, whereas it averages only 16 transactions per day for the least traded stocks. The average market values of the firms at the end of the second quarter of 1995 are reported in the last column. As expected, the trade frequency is positively correlated with firm size. The cross-sectional average and standard error of firm-specific intraday trade statis- tics at 30-minute intervals i.e., volatility [VOL], the number of trades [NT], average trade size [ATS], correlation between ATS and NT, and the first-order autocorrelation of NT and ATS are provided in Table 2. The volatility, average trade size, and the number of trades are all higher for more frequently traded stocks. Not surprisingly, the number of trades is lower, while the average trade size is higher after the adjustment for camouflaged trades. The low correlation between NT and ATS ensures that there is no serious multicollinearity between NT and ATS in the regression. The first-order autocorrelation of ATS is about 0.06, while that of NT is 0.29. For brevity, we only report the results of estimation for the volatility–volume regressions. 10 Table 3 summarizes the results of volatility regressions for the entire sample without an adjustment for camouflaged trades. Columns 2–4 include the cross- sectional median, mean of the t statistics of parameter estimates, and their Z statistics. 11 Columns 5 and 6 show the average coefficient estimates and their test statistics T, which measure the significance of the respective aggregate estimates. 12 The pattern for the coefficient estimates of the number frequency of transactions is quite similar to those obtained by JKL. The average t value for the number of transactions is relatively large 15.210 and significant at the 1 level. However, contrary to JKL’s finding, the average t value for average trade size is 2.018, which is significant at the 5 level. The Z statistic for the average trade size is 23.962, which shows an even stronger result for the importance of average trade size in explaining the return volatility at intraday hour–hour intervals. The aggregate T statistics exhibit a similar pattern with the coefficients of both the size and frequency of trades being significant at the 1 level. Empirical results from the full sample indicate that the number of trades and average trade size both have significant relationships with the return volatility. However, the impact of changes in the number of trades on return volatility is still stronger than that on average trade size. The average coefficient for NT is 0.761, which indicates that one standard deviation increase in NT i.e., 3.407 trades would result in a 2.593 unit increase in return volatility. On the other hand, the average coefficient for ATS is 0.0076, indicating a 0.240 unit increase in return volatility for each standard deviation increase in ATS i.e., 31.544 hundred shares. The results show that the volatility of returns is significantly higher in the opening half hour. On the other hand, the coefficient of the dummy variable for the closing → X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 Table 1 The TAQ random NYSE sample: 141 stocks Most frequently traded stocks Average frequently traded stocks Least frequently traded stocks Average Market value Average Market value Average Market value trades millions of trades millions of trades millions of Ticker per day dollars Ticker per day dollars Ticker per day dollars DIS 829 44861 FNQ 85 424 POP 31 210 NSM 771 2666 GR 80 2273 RGR 30 505 PFE 635 53159 TRN 77 1444 CPT 28 403 ATI 510 13043 IFN 70 238 HGI 28 444 TX 453 26857 RHS 69 182 GMI 26 298 UIS 435 1093 KMT 68 908 WIC 23 656 CMB 419 37477 OS 68 408 CMT 23 772 MRO 359 6289 WLM 65 594 MTX 21 888 FON 343 16662 IGL 65 3462 KLU 21 797 AIT 323 30299 EGG 62 834 COA 20 423 SEG 313 7123 TDW 62 2712 EN 20 1716 NUE 260 4154 WEG 61 466 BLU 20 104 BK 240 12593 FAY 61 268 SWP 19 307 GIS 216 8969 RHI 59 2357 FLY 18 69 WAG 214 9292 HDL 54 218 CMC 18 467 UCL 211 9065 AWK 53 1584 JPR 18 358 TEN 209 8458 HTT 51 402 YES 17 238 NAV 200 691 REL 48 941 CLB 16 246 WLA 198 17255 OSU 48 196 CES 16 517 CSX 192 9355 MRY 48 768 APR 16 102 SRM 191 1182 LE 46 709 IF 15 40 ROK 190 12007 PAR 43 160 WTX 15 114 NKE 175 17029 ECL 43 2348 PIA 15 236 PX 175 6889 SDW 43 474 CCG 14 355 ANN 170 416 SBO 42 307 JHS 14 154 IR 170 4548 BZL 41 56 VIT 14 93 PKN 141 2300 LDF 39 148 NPY 14 195 HMY 136 632 EF 38 157 CKC 13 397 continued → X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 383 Table 1 Continued Most frequently traded stocks Average frequently traded stocks Least frequently traded stocks Average Market value Average Market value Average Market value trades millions of trades millions of trades millions of Ticker per day dollars Ticker per day dollars Ticker per day dollars NWL 129 4506 DUC 38 332 WDV 13 88 CPL 126 5489 TTC 37 376 CLC 12 323 HRP 126 1200 FTC 37 13 SOR 12 299 OLS 123 1578 TSA 37 763 TH 12 135 JR 121 2678 DDR 37 726 DVI 11 146 SME 121 586 WPH 37 866 TNO 11 566 UEP 121 3945 MUR 36 2215 III 11 63 TRW 114 5830 DF 36 1165 OMM 11 218 WB 113 8965 KCP 35 216 NCP 10 197 FLE 111 1284 LEO 35 555 CSI 10 190 ASA 110 366 CCC 35 404 MKS 10 242 NCC 104 9609 FOE 34 716 AP 10 119 ASO 104 2624 WAC 34 1295 LNV 9 116 MMC 103 7580 EOG 34 4118 SWH 9 30 BTT 94 835 SHR 34 1088 NMP 8 103 SPC 92 4537 JFC 33 90 GNL 8 161 STT 91 5103 BUR 32 712 ACO 8 400 NWK 88 278 JEC 32 570 LBI 8 55 PIR 86 634 LGE 32 1533 ALM 7 87 Average daily Average market value Sample No. of stocks number of trades millions of dollars Most frequently traded stocks 47 229 9191.28 Average frequently traded stocks 47 48 910.42 Least frequently traded stocks 47 16 311.55 The third column indicates the average daily trading frequency including trades in NYSE and other markets for each sample group. The forth column summarizes the mean market value of the companies during the second quarter of 1995. 384 X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 Table 2 Summary mean statistics of intraday half-hour transactions for the random NYSE sample Most Average Least Sample statistics Full random frequently frequently frequently April-June, 1995 sample traded traded traded Volatility VOL 1.770 2.552 1.993 0.825 0.0052 0.0191 0.0179 0.0072 Number of Trades NT—wo adjustment 4.399 9.701 2.585 0.910 for camouflaged trades 0.0244 0.1260 0.0202 0.0078 Number of Trades NT—adjusted for 4.339 9.543 2.566 0.908 camouflaged trades 0.0237 0.1224 0.0199 0.0078 Average Trade Size ATS—wo 15.934 25.359 15.425 7.017 adjustment for camouflaged trades 0.0553 0.2113 0.1705 0.1009 Average Trade Size ATS—adjusted for 16.129 25.783 15.557 7.047 camouflaged trades 0.0530 0.2172 0.1724 0.1016 Correlation of ATS and NT—wo 0.1849 0.0799 0.1556 0.3154 adjustment for camouflaged trades 0.00097 0.00156 0.00167 0.00272 Correlation of ATS and NT—adjusted for 0.1833 0.0786 0.1553 0.3159 camouflaged trades 0.00098 0.00157 0.00166 0.00271 First Order Autocorrelation of NT—wo 0.29099 0.41727 0.30581 0.15537 adjustment for camouflaged trades 0.00126 0.00304 0.00356 0.00249 First Order Autocorrelation of NT— 0.29177 0.41640 0.30453 0.15438 adjusted for camouflaged trades 0.00127 0.00304 0.00355 0.00248 First Order Autocorrelation of ATS— 0.05703 0.07338 0.05049 0.04629 wo adjustment for camouflaged trades 0.00041 0.00142 0.00121 0.00101 First Order Autocorrelation of ATS— 0.05626 0.07070 0.05162 0.04645 adjusted for camouflaged trades 0.00040 0.00133 0.00120 0.04717 This table reports the cross-sectional average of firm-specific intraday 30-minute interval trade statistics with the standard error of the cross-sectional average in parentheses. VOL stock return volatility measure is the absolute value of the return conditional on its 14 lags and weekday dummies; NT Number of Trades is the number of NYSE transactions for each 30-minute trading period; and ATS Average Trade Size is the mean number of shares in hundreds traded per transaction for each 30-minute interval. half hour is not significant. We also checked the volatility pattern for other intraday trading periods. None of the remaining trading intervals has a significant dummy coefficient. Thus, after considering the volume effects, it appears that only the opening half hour exhibits a unique pattern of volatility. Table 4 reports the results for the three trade frequency groups. The results in panel A for the most frequently traded stocks show a stronger pattern than the results in Table 3 for the full sample. The average t statistics for average trade size, the number of trades, and the first half-hour dummy are all significant at the 1 level. The mean t value of average trade size for the most frequently traded stocks is 2.845 which is higher than the mean t value 2.018 for the entire sample reported in Table 3. In addition, the mean t value of the lagged-one volatility term is significant at the 10 level or at the 5 level for the one-sided test, suggesting that the heteroskedasticity of residual variance is more pronounced for frequently traded stocks. The Z statistics in the forth column and aggregate T statistics in the last column show that coefficients X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 385 Table 3 GMM estimates of volatility regressions without an adjustment for camouflaged trades: The full sample Median of Mean of the cross- the cross- Cross- Cross-sectional T values of Explanatory sectional sectional sectional average the average variables t values t values Z values coefficients coefficients Intercept 2 1.337 2 1.281 2 15.211 2 0.1432 2 3.3072 Average Trade Size 1.942 2.018 23.962 0.0076 6.2975 Number of Trades 14.900 15.210 180.609 0.761 14.4677 First Half Hour Dummy 4.144 4.306 51.131 1.786 10.5059 Last Half Hour Dummy 2 0.653 2 0.664 2 7.885 2 0.2579 2 7.2648 Monday Dummy 2 0.074 0.02 0.237 0.0839 0.9051 Lag 1 Volatility 0.891 1.009 11.981 0.0324 6.3720 Lag 2 Volatility 2 0.075 0.056 0.665 0.0016 0.5842 Lag 3 Volatility 2 0.002 0.108 1.282 0.0029 0.9832 Lag 4 Volatility 2 0.064 0.089 1.057 0.0032 1.0557 Lag 5 Volatility 0.11 0.301 3.574 0.0091 2.4543 Lag 6 Volatility 0.087 0.102 1.211 0.0032 1.2299 Lag 7 Volatility 0.251 0.246 2.921 0.008 2.2914 Lag 8 Volatility 2 0.054 0.073 0.867 0.0024 0.6847 Lag 9 Volatility 2 0.327 2 0.087 2 1.033 2 0.0027 2 0.9331 Lag 10 Volatility 2 0.097 2 0.032 2 0.380 2 0.0006 2 0.1778 Lag 11 Volatility 0.222 0.26 3.087 0.0074 2.5188 Lag 12 Volatility 0.276 0.294 3.491 0.0088 3.5749 Lag 13 Volatility 2 0.171 0.042 0.499 0.0013 0.4693 Lag 14 Volatility 2 0.205 2 0.055 2 0.653 2 0.0016 2 0.5688 Estimates of the following regression for the full random sample 141 stocks: VOL it 5 a i 1 g i AV it 1 b i NT it 1 h i OPEN t 1 v i CLOSE t 1 p i MONDAY t 1 o 14 n 5 1 l in VOL it 2 n 1 m it where VOL it stock return volatility measure is the absolute value of the return on security i at time t conditional on its 14 lags and weekday dummies, NT it is the number of transactions for security i at time t , ATS it is the average trade size total share volume divided by the number of transactions for security i at time t, OPEN t is a dummy variable that is equal to 1 for the open half hour and 0 otherwise, CLOSE t is a dummy variable that is equal to 1 for the close half hour and 0 otherwise, MONDAY t is a dummy variable that is equal to 1 for Mondays and 0 otherwise, and the coefficients l in measure the persistence in volatility of security i. The sample period is from April 1995 to June 1995. Each regression has 819 time series observations. indicates a significance level of 5 or better for two-tailed test whereas indicates a significance level of 5 for one-tailed test. The cross-sectional Z value in the column 4 is calculated as the mean t value multiplied by √ 141. The T value in the last column is calculated as the cross-sectional average coefficient in column 5 divided by the standard error of the cross-sectional coefficient. Mean Median R 2 5 32.77 32.61. of the size and frequency of trades, the dummy variables of opening and closing intervals, and the first lag of the volatility are on average significant at the 1 level. Table 4 panel B reports the results for the stocks with an average trading frequency. The results for this group are very close to those for the entire sample in Table 3. The mean median t value of average trade size is 2.292 2.417, which is significant 386 X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 at the 5 level. Again, contrary to JKL’s finding, average trade size appears to contain a significant amount of information. The Z and T values further support this hypothesis. Table 4 panel C reports the results for the stocks with the lowest frequency of trades. Interestingly, the results for this group are much closer to the finding of JKL’s study. The coefficient of the number frequency of transactions is highly significant, whereas the coefficient of average trade size is not significant. However, the mean coefficient of average trade size is still positive, and the t statistics are significant for a little more than 25 of the stocks in this group. The Z statistics and aggregate T statistics show a similar pattern of statistical significance for both average trade size and frequency. It is worth noting that the average daily number of transactions is quite small for the least traded group, as shown in Table 1. Although the asymmetric information effect at transaction level might be stronger for infrequently traded stocks see Easley et al., 1996, the contemporaneous half-hour return volatility–size effect is shown to be weaker than that of more frequently traded stocks. One possible reason behind this perplexing result is that infrequently traded stocks typically have a slower price discovery process. Also, from the empirical standpoint, infrequent trading produces a data discreteness problem in the estimation. This problem could cause the estimates of the average trade size coefficient to be biased toward 0 for the infrequently traded stocks. This statistical problem associated with infrequent trading is well documented in the finance literature e.g., Campbell et al., 1997. The problem of infrequent trading or non-trading tends to obscure the empirical relation between volatility and average trade size. The average coefficients differ significantly across the three sub-samples. For in- stance, one standard deviation change in average trade size would result in a 0.330 unit change in volatility for the most frequently traded stocks, 0.417 for the average frequently traded stocks, and 0.052 for the least frequently traded stocks. On the other hand, one standard deviation change in the number of trades would result in a 1.599 unit change in volatility for the most frequently traded stocks, 1.992 for the average frequently traded stocks, and 1.537 for the least frequently traded stocks. The impact of changes in the number of trades on volatility is generally stronger than the impact of changes in average trade size. Also, the impact of average trade size on volatility appears to be much weaker for least frequently traded stocks. Both the ATS and NT coefficients are significantly different among the three sub- samples since the p value for the ANOVA of ATS and NT coefficients are all less than 1. Overall, the less frequently traded stocks have lower ATS coefficients and higher NT coefficients. Table 5 reports the ANOVA of coefficients and t values among three sub-samples for two variables ATS and NT. Results from the ANOVA also indicate that less frequently traded stocks have lower t values for ATS coefficients and higher t values for NT coefficients. In summary, the results above are generally in favor of the argument that average trade size contains important information for return volatility. This finding is much stronger for stocks with a higher or average frequency of trades. The result supports the contention that average trade size has information content beyond that contained X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 387 Table 4 GMM estimates of volatility regressions without an adjustment for camouflaged trades: A summary of three random samples Median of Mean of the cross- the cross- Cross- Cross-sectional T values of Explanatory sectional sectional sectional average the average variables t values t values Z values coefficients coefficients Panel A The Most Frequently Traded Stocks Mean Median R 2 5 28.21 28.93 Intercept 0.057 2 0.025 2 0.171 0.095 0.95 Average Trade Size 2.530 2.845 19.504 0.009 6.3688 Number of Trades 12.441 12.232 83.858 0.2502 10.6017 First Half Hour Dummy 5.814 5.599 38.385 2.649 6.2922 Last Half Hour Dummy 2 0.799 2 0.761 2 5.217 2 0.3144 2 5.0628 Monday Dummy 0.175 0.149 1.021 0.258 0.9416 Lag 1 Volatility 1.682 1.662 11.394 0.0537 5.9513 Panel B The Average Frequently Traded Stocks Mean Median R 2 5 35.01 34.76 Intercept 2 1.319 2 1.255 2 8.604 2 0.2201 2 3.2949 Average Trade Size 2.417 2.292 15.713 0.0115 5.3832 Number of Trades 16.401 16.167 110.835 0.7592 13.5089 First Half Hour Dummy 4.877 5.132 33.435 2.095 11.7697 Last Half Hour Dummy 2 0.571 2 0.65 2 4.456 2 0.2736 2 3.9538 Monday Dummy 2 0.096 0.08 0.548 0.0172 0.4468 Lag 1 Volatility 1.077 0.852 5.841 0.0273 3.1645 Panel C The Least Frequently Traded Stocks Mean Median R 2 5 35.09 36.61 Intercept 2 2.710 2 2.564 2 17.578 2 0.3046 2 11.9451 Average Trade Size 0.765 0.917 6.287 0.0024 0.9711 Number of Trades 17.795 17.232 118.137 1.274 12.6139 First Half Hour Dummy 1.921 2.187 14.993 0.6134 7.1409 Last Half Hour Dummy 2 0.648 2 0.58 2 3.976 2 0.1856 2 3.5959 Monday Dummy 2 0.272 2 0.17 2 1.165 2 0.0233 2 0.759 Lag 1 Volatility 0.451 0.514 3.524 0.0161 2.0201 Estimates of the following regression for the three random sample 47 stocks each: VOL it 5 a i 1 g i ATS it 1 b i NT it 1 h i OPEN t 1 v i CLOSE t 1 p i MONDAY t 1 o 14 n 5 1 l in VOL it 2 n 1 m it where VOL it stock return volatility measure is the absolute value of the return on security i at time t conditional on its 14 lags and weekday dummies, NT it is the number of transactions for security i at time t , ATS it is the average trade size total share volume divided by the number of transactions for security i at time t, OPEN t is a dummy variable that is equal to 1 for the open half hour and 0 otherwise, CLOSE t is a dummy variable that is equal to 1 for the close half hour and 0 otherwise, MONDAY t is a dummy variable that is equal to 1 for Mondays and 0 otherwise, and the coefficients l in measure the persistence in volatility of security i. The sample period is from April 1995 to June 1995. Each regression has 819 time series observations. indicates a significance level of 5 or better for two-tailed test whereas indicates a significance level of 5 for one-tailed test. The cross-sectional Z value in the column 4 is calculated as the mean t value multiplied by √ 47. The T value in the last column is calculated as the cross-sectional average coefficient in column 5 divided by the standard error of the cross-sectional coefficient. 388 X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 in the number of transactions. Furthermore, return volatility exhibits significant varia- tions over the intraday periods. The volatility is significantly higher in the opening period. The finding that average trade size has a significant effect on return volatility at the intraday level may not be surprising. Since volume is basically the aggregate of trades with different size, as the time interval gets shorter, the average trade size should contain more important information impounded in volume. In the limit, when the width of intervals approaches 0 continuity, volume and average trade sizes are essentially equivalent. Hence, average trade size contains most of the information in volume at very short intervals. On the other hand, as the time interval gets wider, the average trade size effect could be washed out because of the aggregation, rendering more information to the frequency of trades. Thus, the empirical relation between return volatility and average trade size may depend on the time interval of data measurement. 4.2. Tests of the sensitivity to camouflaged trades A familiar phenomenon in stock transactions, which may cause a problem for our empirical estimation, is that a large block trade may be divided into several smaller trades in the final transaction report. A rationale for this rearrangement of the trade report is to reduce the impact of large trades on market price. Alternatively, informed traders may camouflage their trading activities by strategically making several small- sized trades rather than one large trade. Such trading or reporting pattern may attenu- ate the relation between the size of trades and return volatility. For example, if a large order is divided into five smaller orders, the frequency of trades increases from one to five while the average trade size reduces to one fifth of the original level. Any price changes resulting from this trading would then appear to be associated more with trade frequency. As an order is divided into a greater number of smaller orders, the frequency of trades will become larger and average trade size smaller. In the extreme case where trade frequency is very large and average trade size is very small, price changes would be attributed to trade frequency alone. To assess the potential effect of order splitting, we combined trades that exhibit this type of pattern using the following procedure. We first identified any trades that had a size larger than 1,000 shares and were reported less than 5 seconds apart. We then checked whether these trades had the same transaction price traded at the same side and were traded at the same quotes. We combined these trades into one trade when these conditions were met. This ad hoc data adjustment allows us to assess the potential effect of camouflaged trades on empirical estimation. Tables 6 and 7 report the results of GMM estimation for the data adjusted for camouflaged trades. The results for the entire sample and sub-samples all indicate that the coefficients of the average trade size become slightly more significant, whereas the coefficients of trade frequency become less significant. For example, for the most frequently traded stocks, the mean t value of average trade size increases from 2.845 to 2.947, while the mean t value of trade frequency decreases from 12.232 to 12.099. Similarly, for the stocks with average frequency of trades, the mean t value of average X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 389 Table 5 ANOVA for average coefficients and t values among three sub-samples Samples Avg. ATS coefficient Avg. ATS t value Avg. NT coefficient Avg. NT t value A. Without Adjustment for Camouflaged Trades MFT 0.0090 2.8455 0.2502 12.2320 AFT 0.0115 2.2923 0.7592 16.1665 LFT 0.0024 0.9173 1.2737 17.2315 F 5.3892 9.2728 56.6125 17.3480 p value 0.5577 0.0167 0.0000 0.0000 B. With Adjustment for Camouflaged Trades MFT 0.0092 2.9474 0.2533 12.0990 AFT 0.0110 2.3124 0.7451 16.0056 LFT 0.0025 0.9292 1.2804 17.2336 F 4.7809 9.8450 56.9892 18.0655 p value 0.9828 0.0101 0.0000 0.0000 MFT is the most frequently traded stock sample, AFT is the average frequently traded stock sample, and LFT is the least frequently traded stock sample. A is based on the estimated results from Table 4 under the case without an adjustment for camouflaged trades, while B is based on the estimated results from Table 7 under the case with an adjustment for camouflaged trades. ATS stands for the average trade size, while NT is the number of trades at the 30-minute interval. The F statistics test the hypothesis that the coefficients or t values among three sub-samples are equal. indicates a significance level of 5 or better for two-tailed test. trade size rises to 2.312, while the mean t value of trade frequency drops to 16.006. The results hence provide some evidence that trade camouflage may attenuate the volatility–volume relation. We hesitate to add that our data adjustment method is ad hoc and to some extent may underestimate the number of camouflaged trades. This identification problem may not be completely resolved until a better-coded data set is available. Nevertheless, our results reveal that order splitting can weaken the empirical relation between average trade size and return volatility. 4.3. Tests using log transaction variables In principle, a better way to decompose the total volume effect would be to take a log transformation of the trade-related variables. Algebraically, total volume V is simply the product of average trade size ATS and the number of transaction NT. With a log transformation, log volume log V is equal to the sum of log average trade size log ATS and log trade frequency log NT. The log transformation thus yields a linearly separable decomposition for the total volume effect. We re-estimate all regressions using the log transformed variables. In general, we find that log transformation modestly increases the explanatory power of average trade size. However, we found that log transformation is problematic for less frequently traded stocks at the intraday level. At intraday 30-minute intervals, many infrequently traded stocks have no trade hence ATS 5 0, NT 5 0 for certain intervals. The log ATS and log NT are undefined under these cases, resulting in a significant number 390 X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 Table 6 GMM estimates of volatility regressions adjusting for camouflaged trades: The full sample Median of Mean of the cross- the cross- Cross- Cross-sectional T values of Explanatory sectional sectional sectional Average the average variables t values t values Z values coefficients coefficients Intercept 2 1.493 2 1.329 2 15.781 2 0.1548 2 3.5916 Average Trade Size 1.944 2.063 24.497 0.0076 6.2397 Number of Trades 14.885 15.113 179.457 0.7596 14.4137 First Half Hour Dummy 4.185 4.327 51.380 1.788 10.4561 Last Half Hour Dummy 2 0.647 2 0.665 2 7.896 2 0.258 2 7.2472 Monday Dummy 2 0.033 0.033 0.392 0.0871 0.9396 Lag 1 Volatility 0.851 1.038 12.326 0.0333 6.5881 Lag 2 Volatility 2 0.06 0.072 0.855 0.0021 0.7794 Lag 3 Volatility 2 0.002 0.105 1.247 0.0028 0.9495 Lag 4 Volatility 2 0.055 0.094 1.116 0.0034 1.0948 Lag 5 Volatility 0.119 0.324 3.847 0.0098 2.6361 Lag 6 Volatility 0.12 0.115 1.366 0.0035 1.3511 Lag 7 Volatility 0.276 0.263 3.123 0.0085 2.4245 Lag 8 Volatility 2 0.015 0.081 0.962 0.0027 0.7664 Lag 9 Volatility 2 0.319 2 0.072 2 0.855 2 0.0022 2 0.788 Lag 10 Volatility 2 0.083 2 0.028 2 0.332 2 0.0005 2 0.1429 Lag 11 Volatility 0.234 0.257 3.052 0.0074 2.5292 Lag 12 Volatility 0.199 0.28 3.325 0.0085 3.4431 Lag 13 Volatility 2 0.204 0.033 0.392 0.001 0.3732 Lag 14 Volatility 2 0.188 2 0.041 2 0.487 2 0.0012 2 0.4312 Estimates of the following regression for the full random sample 141 stocks: VOL it 5 a i 1 g i ATS it 1 b i NT it 1 h i OPEN t 1 v i CLOSE t 1 p i MONDAY t 1 o 14 n 5 1 l in VOL it 2 n 1 m it where VOL it stock return volatility measure is the absolute value of the return on security i at time t conditional on its 14 lags and weekday dummies, NT it is the number of transactions for security i at time t , ATS it is the average trade size total share volume divided by the number of transactions for security i at time t, OPEN t is a dummy variable that is equal to 1 for the open half hour and 0 otherwise, CLOSE t is a dummy variable that is equal to 1 for the close half hour and 0 otherwise, MONDAY t is a dummy variable that is equal to 1 for Mondays and 0 otherwise, and the coefficients l in measure the persistence in volatility of security i. The sample period is from April 1995 to June 1995. Each regression has 819 time series observations. indicates a significance level of 5 or better for two-tailed test whereas indicates a significance level of 5 for one-tailed test. The cross-sectional Z value in the column 4 is calculated as the mean t value multiplied by √ 141. The T value in the last column is calculated as the cross-sectional average coefficient in column 5 divided by the standard error of the cross-sectional coefficient. Mean Median R 2 5 32.70 32.17. of missing values for the average frequently traded and least frequently traded stock samples. This may cause a problem of inefficiency in our empirical estimation. There- fore, we decide to report only the log results for most frequently traded stocks. Table 8 A reports the results of GMM estimation for the data set without an adjustment for camouflaged trades. The results show an increase in the explanatory X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 391 power of the log ATS. The average t statistic of average trade size increases from 2.845 to 2.938, while that of trade frequency decreases from 12.232 to 9.358. Table 8 B shows the results of GMM estimation for the data adjusted for camouflaged trades. The results indicate a further increase in the average t value of average trade size to 3.084 and a decrease in the average t value of frequency of trades to 9.262. The aggregate T statistics for log ATS also increase slightly for both data sets with and without an order adjustment. The results indicate that log transformation seems to reduce the effect of trade frequency more than to bring up the explanatory power of average trade size. Overall, the results reinforce our preceding conclusion that average trade size contains nontrivial trade information. 4.4. The volatility–volume relation at the market opening To examine whether the relation between return volatility and the size and frequency of trades varies over intraday periods, we estimate an equation [Eq. 2a] that incorpo- rates slope dummies to allow the coefficients of trade-related variables in the opening period to be different from those of the remaining trading periods. Although estimates were obtained for all data samples, in the interest of brevity we only report the results for the data adjusted for camouflaged trades in Table 9. The results for the data without an adjustment for camouflage trades are very similar to those with an adjustment. Also, to provide a more concise summary, we highlight only the estimation results for the trade-related variables. The overall results show that the effect of the frequency of trades NT on return volatility is much higher in the opening period. The coefficients of the slope dummy variables OPEN 3 NT are all significant at the 1 level for the full sample as well as for the three frequency groups. On the other hand, the effect of average trade size does not appear to be significantly different in the opening period. Table 9 A summarizes the estimates for the full sample. Column 5 shows that the average coefficient of the frequency of trades is 0.697. But the average value of this coefficient increases by 0.527 in the market opening half-hour period, giving a total value of 1.223 for the coefficient of trade frequency at the market opening. Table 9, B–D, reports the results for the three trading frequency groups. The coefficient of trade frequency is smaller for the most frequently traded stocks. The average value of the coefficient is 0.230, and it increases by 0.245 to reach a value of 0.475 in the opening period. For the group with an average frequency of trades, the coefficient rises from 0.682 to 1.193 in the opening period, with an increment of 0.511. For the group with the lowest frequency of trades, the coefficient rises from 1.180 to 2.004, with an increment of 0.824. Thus, the effect of trade frequency on return volatility ranges from 70 to 107 higher in the opening period than that of the remaining periods. These results are consistent with the previous finding that information flow is much heavier at the market opening. The number of transactions is usually much larger in the opening period. Our result supports the contention that a relatively large number of transactions is accompanied by a higher level of information content at the market opening. We also introduced the slope dummy variables for other intraday trading periods. However, none of these dummy variables were significant. Thus, it 392 X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 Table 7 GMM estimates of volatility regressions adjusting for camouflaged trades: A summary of three random samples Median of Mean of the cross- the cross- Cross- Cross-sectional T values of Explanatory sectional sectional sectional average the average variables t values t values Z values coefficients coefficients Panel A The Most Frequently Traded Stocks Mean Median R 2 5 28.02 28.99 Intercept 0.076 2 0.084 2 0.576 0.077 0.77 Average Trade Size 2.937 2.947 20.204 0.0092 6.4577 Number of Trades 12.517 12.099 82.947 0.2534 10.5583 First Half Hour Dummy 5.751 5.596 38.364 2.651 6.2820 Last Half Hour Dummy 2 0.809 2 0.766 2 5.251 2 0.3181 2 5.0978 Monday Dummy 0.18 0.148 1.015 0.258 0.9416 Lag 1 Volatility 1.794 1.706 11.696 0.0552 6.1368 Panel B The Average Frequently Traded Stocks Mean Median R 2 5 34.96 34.29 Intercept 2 1.311 2 1.321 2 9.056 2 0.2343 2 3.5286 Average Trade Size 2.396 2.312 15.850 0.011 5.3610 Number of Trades 15.670 16.006 109.732 0.7451 13.6967 First Half Hour Dummy 5.114 5.207 35.697 2.104 11.8202 Last Half Hour Dummy 2 0.43 2 0.642 2 4.401 2 0.2673 2 3.8460 Monday Dummy 2 0.085 0.119 0.816 0.0268 0.6961 Lag 1 Volatility 1.014 0.895 6.136 0.0286 3.3863 Panel C The Least Frequently Traded Stocks Mean Median R 2 5 35.12 35.83 Intercept 2 2.711 2 2.582 17.701 2 0.3073 2 11.9109 Average Trade Size 0.734 0.929 6.369 0.0025 0.992 Number of Trades 17.809 17.234 118.150 1.28 12.5490 First Half Hour Dummy 1.904 2.177 14.925 0.6091 7.0908 Last Half Hour Dummy 2 0.666 2 0.587 2 4.024 2 0.1885 2 3.6460 Monday Dummy 2 0.272 2 0.169 2 1.159 2 0.0235 2 0.768 Lag 1 Volatility 0.451 0.514 3.524 0.0161 2.0201 Estimates of the following regression for the three random samples 47 stocks each: VOL it 5 a i 1 g i ATS it 1 b i NT it 1 h i OPEN t 1 v i CLOSE t 1 p i MONDAY t 1 o 14 n 5 1 l in VOL it 2 n 1 m it where VOL it stock return volatility measure is the absolute value of the return on security i at time t conditional on its 14 lags and weekday dummies, NT it is the number of transactions for security i at time t , ATS it is the average trade size total share volume divided by the number of transactions for security i at time t, OPEN t is a dummy variable that is equal to 1 for the open half hour and 0 otherwise, CLOSE t is a dummy variable that is equal to 1 for the close half hour and 0 otherwise, MONDAY t is a dummy variable that is equal to 1 for Mondays and 0 otherwise, and the coefficients l in measure the persistence in volatility of security i. The sample period is from April 1995 to June 1995. Each regression has 819 time series observations. indicates a significance level of 5 or better for two-tailed test whereas indicates a significance level of 5 for one-tailed test. The cross-sectional Z value in the column 4 is calculated as the mean t value multiplied by √ 47. The T value in the last column is calculated as the cross-sectional average coefficient in column 5 divided by the standard error of the cross-sectional coefficient. X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 393 Table 8 GMM estimates of volatility regressions with log average trade size and trade frequency: Most frequently traded stocks Median of Mean of the cross- the cross- Cross- Cross-sectional T values of Explanatory sectional sectional sectional average the average variables t values t values Z values coefficients coefficients Panel A Without adjustment for camouflaged trades, Mean Median R 2 5 25.44 25.89 Intercept 2 5.791 2 5.708 2 39.132 2 2.546 2 8.8711 Log Average Trade Size 2.513 2.938 20.142 0.3229 7.5798 Log Number of Trades 10.020 9.358 64.155 1.782 15.6316 First Half Hour Dummy 5.547 5.482 37.583 2.653 6.3469 Last Half Hour Dummy 2 0.667 2 0.664 2 4.552 2 0.2956 2 4.1810 Monday Dummy 0.178 0.163 1.117 0.251 0.9161 Lag 1 Volatility 2.320 2.374 16.275 0.0768 8.5183 Panel B With adjustment for camouflaged trades, Mean Median R 2 5 25.36 25.73 Intercept 2 5.824 2 5.725 2 39.249 2 2.561 2 8.8924 Log Average Trade Size 9.956 9.262 63.497 1.772 15.5439 Log Number of Trades 5.487 5.479 37.562 2.653 6.3469 First Half Hour Dummy 2 0.687 2 0.662 2 4.538 2 0.2954 2 4.2020 Last Half Hour Dummy 0.201 0.162 1.111 0.251 0.9161 Monday Dummy 2.304 2.397 16.433 0.0775 8.6410 Lag 1 Volatility 2 0.127 0.043 0.295 0.0013 0.2746 Estimates of the following regression for the most frequently traded stock sample 47 stocks: VOL it 5 a i 1 g i LATS it 1 b i LNT it 1 h i OPEN t 1 v i CLOSE t 1 p i MONDAY t 1 o 14 n 5 1 l in VOL it 2 n 1 m it where VOL it stock return volatility measure is the absolute value of the return on security i at time t conditional on its 14 lags and weekday dummies, NT it is the number of transactions for security i at time t , LATS it is the average trade size total share volume divided by the number of transactions for security i at time t, OPEN t is a dummy variable that is equal to 1 for the open half hour and 0 otherwise, CLOSE t is a dummy variable that is equal to 1 for the close half hour and 0 otherwise, MONDAY t is a dummy variable that is equal to 1 for Mondays and 0 otherwise, and the coefficients l in measure the persistence in volatility of security i. The sample period is from April 1995 to June 1995. Each regression has 819 time series observations. indicates a significance level of 5 or better for two-tailed test whereas indicates a significance level of 5 for one-tailed test. The cross-sectional Z value in the column 4 is calculated as the mean t value multiplied by √ 47. The T value in the last column is calculated as the cross-sectional average coefficient in column 5 divided by the standard error of the cross-sectional coefficient. appears that only the opening period exhibits a distinct relation between the return volatility and trade frequency.

5. Summary