X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 377
theory. A main objective of our study is to see if JKL’s results hold at the intraday interval. Also, it will be interesting to see whether the volatility–transaction relation
varies over intraday periods. Second, we use the generalized method of moments GMM in empirical estimation.
The GMM model invokes much weaker distributional assumptions on stock returns. Within the GMM framework, the disturbance terms of the volatility regressions can
be both serially correlated and conditionally heteroskedastic. The GMM thus provides more robust tests than least squares regressions that rely heavily on the assumption
of serial independence and homoskedasticity. Because returns and volatility are poten- tially correlated across firms, we jointly estimate the parameters of the volatility model
for the system of equations involving all firms in the sample. By taking into account cross-sectional correlation between individual firms, we obtain an efficient estimate
of the regression parameter.
Third, in empirical estimation we introduce an ad hoc adjustment for camouflaged trades, where one transaction may be strategically split into several trades to minimize
their impact on market price. Order splitting tends to attenuate the relation between return volatility and average trade size while boosting the effect of trade frequency.
To mitigate this effect, we adjust the frequency and size of trades reported in the raw data to account for the effect of camouflaged trades.
Our result shows that the frequency of transaction significantly affects intraday return volatility. This result is consistent with JKL’s finding based on the interday
data. In addition, our study documents several interesting findings. First, we found that the size of trade is significantly related to return volatility, particularly for more
frequently traded stocks. Because trade frequency is positively correlated with firm size, the size of trade is also positively related with return volatility of large firms.
This result contrasts JKL’s interday finding that there is virtually no positive relation between average trade size and return volatility for large firms. Second, we found
significant variations in the transaction–volatility relation over the intraday periods. Consistent with Wood et al. 1985, Jain and Joh 1988, and Stoll and Whaley 1990,
we found that stock return volatility varies over intraday intervals, with the highest volatility occurring at the opening of the market. Moreover, we found that the effect
of the frequency of trades on return volatility is much stronger in the market opening period. Finally, we provide some evidence that order splitting tends to attenuate the
empirical relation between average trade size and return volatility, while boosting the effect of trading frequency on return volatility.
The remainder of this article is organized as follows. In section 2, we describe the model and estimation problems. We discuss the time pattern of return volatility and
possible variations in the volatility–volume relation over the intraday periods. In section 3, we describe the empirical methodology and estimation procedures. We then
discuss our data and present our results in section 4. Finally, we summarize our findings in section 5.
2. The model
In testing the relation between volatility and volume, we measure the volatility of intraday returns using a method suggested by JKL 1994.
3
This method defines return
378 X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397
volatility as the absolute percentage price change conditional on past returns and day- of-the-week effects. Specifically, we first estimate the residual of the following return
model:
R
it
5
o
5 k
5
1
a
ik
W
kt
1
o
14 n
5
1
b
in
R
it 2 n
1 e
it
1 where R
it
is the return of security i over a half-hour interval t; W
kt
’s are weekday indicator variables; R
it2n
is the return of security i at time t 2 n; and e
it
is the residual of security i.
4
Similar to JKL 1994, lagged 14 half-hourly returns are used to capture the short-term movements in conditional expected returns. We then multiply the
absolute residual from the above model, |e
it
|, by 1,000 and use it as a measure of return volatility VOL
it
of security i in period t. Return volatility for each stock is then regressed against its frequency of transactions
NT, average trade size ATS, the dummy variables for the first half hour OPEN and the closing half hour CLOSE, Monday dummy MONDAY, and 14 lagged
volatility terms. Specifically, we estimate the following volatility regression:
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
2 where m
it
is the disturbance term. Wood et al. 1985 and Jain and Joh 1988 documented that NYSE trading volume
follows a U-shaped intraday pattern. In addition, Stoll and Whaley 1990 and Foster and Viswanathan 1993 show that the volatility at market opening is much higher
because of asymmetry information effects arising from non-trading overnight. The half-hourly dummy variables OPEN and CLOSE in the above regression model are
used to detect any differences in intraday return volatility in the opening and closing intervals after accounting for the information content in volume. The Monday dummy
variable is introduced to reflect the trading-gap effect at the beginning of the week. This dummy variable equals 1 for Mondays and 0 otherwise. The l coefficients reflect
the persistence in the volatility of security i. The lagged VOL terms for each security are included to correct for heteroskedasticity in return volatility.
As the pattern of intraday return volatility varies over time, the volatility–volume relation may differ across intraday trading periods. To examine whether the relation
of return volatility with the frequency of transactions and average trade size changes over intraday periods, we also estimate the following regression:
VOL
it
5 a
i
1 g
i
ATS
it
1 b
i
NT
it
1 h
i
OPEN
t
3 ATS
it
1 v
i
OPEN
t
3 NT
it
1 p
i
MONDAY
t
1
o
14 n
5
1
l
in
VOL
it 2 n
1 m
it
2a where the third and the fourth regressors are slope dummy interaction variables
added to capture the sensitivity of return volatility to the size and frequency of trades in the opening period.
5
X.E. Xu, C. Wu International Review of Economics and Finance 8 1999 375–397 379
Another issue is related to the specification in the volatility regression. In JKL’s study, they separated volume into the components of average trade size and the
number of transactions. The average trade size is defined as the total number of shares traded in a period day divided by the number of transactions. By definition, total
trading volume is simply the product of average trade size and the number of transac- tions. Thus, one cannot linearly separate total trading volume into these two compo-
nents. One possible consequence is that the total effect of trading volume may not be completely captured by the sum of the effects of the size and frequency of trades.
An alternative specification is to take a log transformation of each transaction variable so that the log total volume is exactly equal to the sum of the log values of average
trade size and the number of transactions. In principle, this log transformation may provide a better decomposition of the total volume effect. In our empirical investiga-
tion, we use the variables of size and frequency of trades with and without the log transformation and examine the sensitivity of estimation results to these variable
specifications.
3. Empirical methodology
