An empirical investigation of trading on asymmetric information and heterogeneous prior beliefs

Journal of Empirical Finance 7 Ž2000. 417–454 www.elsevier.comrlocatereconbase

An empirical investigation of trading on asymmetric information and heterogeneous prior beliefs Paul Brockman) , Dennis Y. Chung Department of Accountancy, Faculty of Business and Information Systems, Hong Kong Polytechnic UniÕersity, Hung Hom Kowloon, Hong Kong, People’s Republic of China Accepted 5 October 2000

Abstract The purpose of this study is to identify and analyze inter-temporal trading patterns attributable to informed trading. Recent theoretical models posit that heterogeneous prior beliefs provide a source of trading volume in addition to the commonly accepted trading motives of liquidity and asymmetric information. After separating informed from uninformed trading using the estimation procedure of Easley et al. wJournal of Finance 51 Ž1996. 1405x, we test for the presence of trading on heterogeneous beliefs as opposed to asymmetric information. The empirical findings confirm the existence of trading on heterogeneous prior beliefs and generally support the inter-temporal patterns proposed by Wang wJournal of Financial Markets 1 Ž1998. 321x. q 2000 Elsevier Science B.V. All rights reserved. JEL classification: G10; G14 ŽGeneral Financial Markets, Information and Market Efficiency. Keywords: Informed trading; Asymmetric information; Heterogeneous beliefs; Market microstructure

1. Introduction The purpose of this study is to identify and analyze inter-temporal trading patterns attributable to informed traders. In particular, we distinguish between )

Corresponding author. Tel.: q852-2766-7032; fax: q852-2330-9845. E-mail address: [email protected] ŽP. Brockman..

0927-5398r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 5 3 9 8 Ž 0 0 . 0 0 0 2 0 - 7

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trading due to asymmetric information and trading due to heterogeneous prior beliefs. Although there is general consensus in the literature that investors trade in order to rebalance portfolios Ži.e., liquidity-motivated trading. and to profit from private information Ži.e., asymmetric information-motivated trading., recent theoretical models suggest that investors trade because of heterogeneous beliefs Že.g., Varian, 1985; Harris and Raviv, 1993; Wang, 1998.. Heterogeneous beliefs, or differences of opinion, may be due to disagreements over the processing of public announcements or the interpretation of private signals. If such a trading motive describes actual order flows, then heterogeneous beliefs could be a source of considerable trading volume and may be responsible for previously documented trading patterns. It is important to understand the underlying motives for trading in the financial markets because these motives impact the way in which prices are generated and the means by which liquidity is provided. To date, there is little, if any, empirical evidence of trading volume due to heterogeneous beliefs. This is probably due to the difficulty in distinguishing informed from uninformed order flow, and then isolating the trading based on heterogeneous beliefs from trading based on asymmetric information. In this paper, we overcome these difficulties by applying the econometric procedures of Easley et al. Ž1996. ŽEKOP hereafter. to the testable hypotheses of Wang’s Ž1998. informed trader model. Wang Ž1998. proposes a theoretical model in which equilibrium prices and order flows are affected by asymmetric information and heterogeneous prior beliefs. An informed trader receives a private signal about terminal asset values, but there is general disagreement among market participants regarding the precision of this signal Ži.e., heterogeneous beliefs.. This market structure gives rise to three distinct motives for trading: liquidity, asymmetric information, and heterogeneous beliefs. In addition to identifying these trading motives, the model also specifies how each source of trading volume behaves over time, thus providing a set of testable hypotheses Žsee Section 2.3... The Wang Ž1998. model, as well as other strategic trading models, can be tested empirically only if one can distinguish empirically between informed and uninformed trading. This is sometimes accomplished on an ex ante basis by using trade size or institutional ownership as proxies for information trading. The validity of such proxies, however, is far from certain. More recently, EKOP develop a statistical procedure for identifying Žand estimating. informed and uninformed order submission rates.2 These estimates can then be used to generate posterior probabilities of information and non-information trading days. With a credible method of identifying informed trading periods, Wang’s Ž1998. overall model can be tested empirically. Of particular interest is the claim that trading volume is generated by heterogeneous prior beliefs. A more general implication of

2

Easley et al. Ž1998a. is the latest of a series of papers applying this same basic model, including Easley et al. Ž1996., Easley et al. Ž1997a,b., and Easley et al. Ž1998b..

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our approach is that theoretical trading models can be tested in a meaningful way using the EKOP estimation procedure. In this study, we estimate EKOP parameters and calculate posterior probabilities using a large intra-day data set of over seven million observations from the Stock Exchange of Hong Kong ŽSEHK.. Parameter estimates are similar to those obtained in previous studies. The posterior probability results demonstrate that informed vs. uninformed classifications are quite robust to the 80% cut-off rule suggested in Easley et al. Ž1998a.. After separating informed trading from uninformed trading, we test seven hypotheses related to Wang’s Ž1998. heterogeneous beliefs model. Volume and order imbalance autocorrelations, as well as volume and price variance cross correlations, are consistent with the presence of asymmetric information and heterogeneous prior beliefs. We show that the heterogeneity parameter is nonzero, and estimate the full measures for heterogeneous prior beliefs and asymmetric information. Although the empirical inter-temporal variations in heterogeneous beliefs are consistent with expectations, the inter-temporal variations in asymmetric information are significantly larger than anticipated by Wang’s Ž1998. model. Lastly, firm-by-firm regression analysis provides additional support for the existence of trading on heterogeneous prior beliefs. We investigate the impact of heterogeneous beliefs and asymmetric information on trading volume. The volume impact of trading on asymmetric information is relatively smooth throughout the day, while the volume impact of trading on heterogeneous beliefs increases at the end of the day. To the best of our knowledge, these findings represent the first empirical evidence of trading volume attributable to heterogeneous beliefs. The remainder of the paper is organized as follows. Section 2 provides an overview of Wang’s Ž1998. strategic trading model and EKOP’s econometric techniques. This section also develops the seven hypotheses to be tested herein. Section 3 discusses the data and presents the estimation procedures, while Section 4 presents the empirical findings and an analysis of their implications. Section 5 contains a brief conclusion.

2. Models of informed trading and testable hypotheses 2.1. Asymmetric information and heterogeneous prior beliefs Wang’s Ž1998. multi-period trading model assumes the existence of three types of risk-neutral market participants: Ž1. a monopolistically informed trader who receives a private signal about the risky asset’s terminal value, Ž2. uninformed liquidity traders who do not act strategically Ži.e., noise traders., and Ž3. market makers who set equilibrium market prices after observing the combined order flow of informed and uninformed traders. The single risky asset’s terminal value, Õ, ˜ is assumed to be normally distributed with mean zero and variance S 0 . Asymmetric

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information is incorporated into the model by providing the informed trader with a private signal, s˜) , about Õ˜ that cannot be observed by the other participants. It is common knowledge that the private signal is a scalar multiple of the terminal value Ži.e., s˜) s scalar = Õ˜ ., but the true magnitude of this scalar is unknown to all participants including the informed trader. Although market makers and liquidity traders do not observe the private signal, they are aware of its existence and are free to disagree about its level of precision. Heterogeneous prior beliefs are incorporated into the model by specifying differences of opinion among traders with respect to the signal’s precision Ži.e., traders assign different scalar magnitudes.. The informed trader believes that s˜) s c i Õ, ˜ and so he expects the terminal value to be drawn from the distribution s˜) ; N Ž0, c i2S 0 .. The market maker believes that s˜) s c m Õ, ˜ and so she expects the terminal value to be drawn from the distribution s˜) ; N Ž0, c m2 S 0 .. Because market makers do not have direct access to the signal, they must infer its value from observing order submissions. It is also possible to model liquidity traders’ beliefs in a similar fashion, but this would not be useful since their order submissions follow a random process Ži.e., Brownian motion. and are unaffected by strategic behavior. Wang Ž1998. then shows that the two belief parameters, c i and c m , can be collapsed into a single measure defined by K s cm rci such that c m acts as the numeraire. The informed trader is considered to be overconfident relative to the market maker if K ) 1 and underconfident if K - 1. Simply stated, the informed trader is overconfident when he believes the private signal to be more precise Ži.e., lower variance. than does the market maker, and underconfident when he believes the signal to be less precise. Trading takes place in an N-period sequential auction market in which informed and liquidity traders submit their orders to the market makers. Market makers set equilibrium prices and absorb any excess demand or supply. Their information set consists of all previous order imbalances and prices. The informed trader’s information set contains the private signal and all previous prices, while the liquidity traders’ information set is irrelevant since it is not used to trade strategically. As in previous strategic trading models, the role of the liquidity trader is to provide a sufficiently noisy environment such that informed traders maintain some degree of unanimity. Beginning with this beliefrinformation structure and assuming that the degree of heterogeneity is bounded by 0 - K - 2, Wang Ž1998. solves for equilibrium pricing rules and order submissions. The market makers’ pricing rule is a linear function of lagged price and current Žcombined. order submissions. The informed trader’s optimal order submission is a function of both the level of asymmetric information and the degree of heterogeneous beliefs. This is perhaps the single most important contribution of the model. Total trading volume can now be partitioned into uninformed liquidity trading and two distinct types of informed trading, asymmetric information and heterogeneous beliefs. And unlike previous heterogeneous beliefs models based on public information Že.g., Harris and Raviv, 1993., this model posits that heterogeneous beliefs are due to disagreements over

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the precision of private information. Previous empirical research finds considerable support for the existence of liquidity-based and asymmetric informationmotivated trading. To the best of our knowledge, no previous study has documented the existence of heterogeneous beliefs, much less investigated their inter-temporal properties. 2.2. The probability of informed trading The EKOP model Žand its variants. is designed to capture the salient features of the trading process. For our purposes, the primary feature of this model is that it provides a method of identifying informed and uninformed trading Žas described below.. After separating information from non-information days, we can then directly test informed trading days for the presence of asymmetric information and heterogeneous beliefs. Our intra-day data set also allows for an investigation of inter-temporal patterns for each of these trading motives. Similar to Wang’s Ž1998. model, the EKOP model incorporates asymmetric information by inserting a private information signal into the trading process. Informed traders submit buy and sell orders conditional on the occurrence of a private information event and the observation of its good-vs.-bad news content.3 Uninformed traders are unaware of private information events and, therefore, do not observe the good-vs.-bad news content Ži.e., uninformed order flow is independent of private information.. This asymmetric information model and its variants have been used to determine the probability of informed trading in high vs. low volume stocks ŽEasley et al., 1996., to extract the information content of trade size and test various market microstructure models ŽEasley et al., 1997a, b., to analyze the effect of analysts’ following on the level of informed and uninformed trading ŽEasley et al., 1998a., and to examine whether informed traders prefer to trade in the stock or options market ŽEasley et al., 1998b.. More specifically, the model provides a method for estimating the probability of a private information event Ž a ., the probability of negative news given the occurrence of a private information event Ž d ., the order arrival rate of informed traders Ž m ., and the order arrival rate of uninformed traders Ž ´ .. At the beginning of every trading day, nature selects whether an information event occurs Žwith probability a . or not Žwith probability 1 y a .. On non-information days, only uninformed traders participate in the market, and buy order arrivals Žwith arrival rate ´ . are equivalent to sell order arrivals Žwith arrival rate ´ .. On private-information event days, both informed and uninformed traders enter the market. If the information event represents bad news Žwith probability d ., then both informed

3

Although all versions of the EKOP model are based on the same underlying trade process, this section specifically describes the recent Easley et al. Ž1998a. version.

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and uniformed traders will issue sell orders Žwith arrival rate m q ´ . but only uninformed traders will submit buy orders Žwith arrival rate ´ .. And if the information event represents good news Žwith probability 1 y d ., then both informed and uniformed traders will issue buy orders Žwith arrival rate m q ´ . but only uninformed traders will submit sell orders Žwith arrival rate ´ .. 2.3. Testable hypotheses Based on the estimation procedures of the EKOP model, we empirically test the following theoretical implications of the Wang Ž1998. model. The first four hypotheses follow directly from Wang’s Ž1998. propositions 1 through 4, although we change their order to suit our empirical orientation. H1. The volume of informed trading Ži.e., asymmetric information plus heterogeneous trading. can be larger than that of uninformed trading. H2. Heterogeneous prior beliefs create serial correlation in volume and order imbalance. H3. Volume and contemporaneous price variance Žor absolute price changes. are positively correlated. H4. Heterogeneous prior beliefs lead to higher levels of trading volume. We test the first hypothesis, or H1, by comparing order arrival rates of informed trading Ž m . to those of uninformed trading Ž ´ . across all firms in our sample. In addition, we compare dollar volumes between informed and uninformed trading days. We test H2 and H3 by calculating serial and cross correlations for volume, order imbalance, and price variance at 5-min intervals throughout the trading day.4 Because information days can be distinguished from non-information days using the EKOP approach, we are also able to compare correlation levels between these two categories. According to the Wang Ž1998. model, we expect higher positive correlations on the information days. The fourth hypothesis can be tested indirectly by comparing trading volume on information and non-information days. Establishing that higher volume occurs on information days, however, is merely consistent with the existence of heterogeneous beliefs and does not rule out the possibility that trading on asymmetric information is the sole source of the volume increase. Wang Ž1998. also provides 4 Order imbalance is defined as the difference between the number of shares in buyer-initiated trades and the number of shares in seller-initiated trades divided by the total number of shares traded in a given interval.

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specifications for measuring trading on asymmetric information and heterogeneous prior beliefs. These measures allow us to test for the presence of trading on heterogeneous beliefs on information days in a regression framework Žsee Eq. Ž13. below.. In addition to Wang’s Ž1998. propositions 1 through 4, we conduct a direct test for the significance of heterogeneous prior beliefs trading by estimating Wang’s Eq. Ž8..5 H5. The heterogeneity parameter is significantly different from zero. Once the existence of trading due to heterogeneous beliefs can be established, then it is possible to investigate its inter-temporal properties. The following two hypotheses are based on the informed trader’s optimal trading strategy according to the Wang Ž1998. model. H6. Asymmetric information-based trading will be uniform or smooth throughout the trading period as the informed trader attempts to hide the private signal content. H7. Heterogeneous prior beliefs-based trading remains low throughout the trading day and then peaks at the close of trading. We test H6 and H7 by placing time-of-day dummy variables into the regression framework used to test H4. Overall, the empirical results will provide direct evidence on the existence and relative importance of liquidity, asymmetric information, and heterogeneous beliefs trading.

3. Data and model estimation 3.1. Data The data set is obtained from the SEHK’s Research and Planning Division and includes intra-day data for 645 companies covering the period from May 1, 1996 to August 29, 1997.6 The full sample represents all firms that traded over the period under investigation. Bid prices and depths, ask prices and depths, and transaction prices and volumes are compiled at 30-s intervals throughout the 5

We thank an anonymous referee for suggesting this as a direct test for the presence of heterogeneous prior beliefs. 6 See Brockman and Chung Ž1998. for a discussion of equity trading on the Stock Exchange of Hong Kong.

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trading day, including a morning session from 10:00 to 12:30 and an afternoon session from 14:30 to 15:55. Companies with incomplete trading information over the 16-month period Ži.e., initial public offerings and delistings. are eliminated from the full sample. This reduces the sample to 532 firms and a total of 7,420,801 observations. 3.2. Model parameters According to the EKOP model, the arrival of buys Ž B . and sells Ž S . within the trading day follows a combined Poisson process. The probabilities of observing B buys and S sells on a non-event day, a bad news day, and a good news day, respectively, are given by: y´

e

ey´

´B B!

´B B!

y´

e

´S S!

,

eyŽ ´q m .

eyŽ ´q m .

Ž 1. Ž ´qm.

Ž ´qm. B!

S

S! B

ey ´

,

Ž 2.

.

Ž 3.

´S S!

As mentioned above, the probability of a non-event day is 1 y a , the probability of a bad news day is ad , and the probability of good news day is a Ž1 y d .. Combining these probabilities with the Poisson processes in Eqs. Ž1., Ž2., and Ž3. yields the following likelihood function: L Ž Ž B,S . < a , d , ´ , m . s Ž 1 y a . ey ´

y´

q Ž ad . e

´B B!

´B B!

y´

e

´S S!

yŽ m q ´ .

e

q a Ž 1 y d . eyŽ m q ´ .

Ž mq´ .

S

S!

Ž mq´ . B!

B

ey ´

´S S!

.

Ž 4.

Parameter estimates for the probability of informed trading Ž a ., the probability of bad news Ž d ., the order arrival of informed traders Ž m ., and the order arrival of uninformed traders Ž ´ . are obtained by maximizing the likelihood function: N

L Ž D < a , d , ´ , m . s Ł L Ž a , d , ´ , m < Bi ,Si . . is1

Ž 5.

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The data set, D, requires only the number of buys Ž B . and sells Ž S . compiled on a daily basis over a total of N days. The overall derivation assumes that B and S are independent across trading days. This assumption is consistent with an informationally efficient market that fully impounds news events by the close of the trading day Žrecall that information events occur at the beginning of the trading day.. In addition, Easley et al. Ž1997a. provide empirical evidence in support of the independence assumption. 3.3. Posterior probabilities Good news, bad news and no news days can be identified using the parameter estimates obtained from the maximum likelihood estimation of Eq. Ž5.. Applying Bayes’ Rule, the posterior probability that a given day is associated with good news is the following:

ž ž

Ž 1 y aˆ . e y´ˆ

´ˆ B i Bi !

e y ´ˆ

´ˆ S i Si !

aˆ Ž 1 y dˆ . e y Žmˆ q ´ˆ .

q aˆ dˆ e y ´ˆ

´ˆ B i Bi!

e

Ž mˆ q ´ˆ . Bi !

Ž mˆ q ´ˆ . y Ž mˆ q ´ˆ .

Bi

e y ´ˆ

´ˆ S i Si !

/

Si

q aˆ Ž 1 y dˆ . e y Žmˆ q ´ˆ .

Si !

Ž mˆ q ´ˆ .

Bi

e y ´ˆ

Bi!

´ˆ S i Si !

,

Ž 6.

/

where the subscript i represents the ith trading day and aˆ , dˆ, m ˆ and ´ˆ are the estimated parameters for the Poisson processes described in Eq. Ž4.. In words, the probability of good news conditional on observing Ž Bi , Si . Ži.e., PrŽgood news < Bi , Si .. is equal to the joint distribution of good news and observing Ž Bi , Si . Ži.e., PrŽgood news l Bi , Si . s PrŽ Bi , Si
ž ž

Ž 1 y aˆ . e y´ˆ

´ˆ B i Bi !

e y ´ˆ

´ˆ S i Si !

q aˆ dˆ e y ´ˆ

ad ˆ ˆ e y´ˆ ´ˆ B i Bi!

ž

ž

Ž 1 y aˆ . e y´ˆ

´ˆ B i Bi !

e y ´ˆ

´ˆ S i Si !

q aˆ dˆ e y ´ˆ

´ˆ B i Bi!

e

´ˆ B i Bi !

e y Žmˆ q ´ˆ .

Ž mˆ q ´ˆ . y Ž mˆ q ´ˆ .

Si !

´ˆ B i Bi !

Si !

/

q aˆ Ž 1 y dˆ . e y Žmˆ q ´ˆ .

e y ´ˆ

Ž mˆ q ´ˆ . y Ž mˆ q ´ˆ .

Si

Si

Si !

Ž 1 y aˆ . e y´ˆ

e

Ž mˆ q ´ˆ .

´ˆ S i Si !

Ž mˆ q ´ˆ .

Bi

e y ´ˆ

Bi!

´ˆ S i Si !

q aˆ Ž 1 y dˆ . e y Žmˆ q ´ˆ .

Ž mˆ q ´ˆ . Bi!

Bi

e y ´ˆ

´ˆ S i Si !

Ž 7.

.

Ž 8.

/

/

Si

,

/

Although Eqs. Ž6., Ž7., and Ž8. provide posterior probability estimates, a decision rule must still be applied in order to classify trading days into good news, bad news, and no news. Following Easley et al. Ž1998a., we set a cut-off rate of 80% for a day to be classified into one of the three groups. For example, a company’s trading day will be labeled Ano newsB only if its Eq. Ž8. value is greater than or

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P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Table 1 Selected market statistics on trading activities of sample companies over the period from May, 1 1996 to August 29, 1997 Number of companies in the sample Average market capitalization per company ŽHK$. Average daily trading volume per company in number of shares Average daily trading volume per company in total dollar volume ŽHK$. Average percentage of trading days Žover the sample period. with one or more shares traded Average percentage of 5-min intervals with one or more shares traded Average percentage of 30-s intervals with one or more shares traded Average share price in 5-min intervals ŽHK$. Average absolute bid–ask spread in 5-min intervals ŽHK$. Average relative bid–ask spread in 5-min intervals Average volume depth in 5-min intervals Average dollar depth in 5-min intervals ŽHK$.

532 $6,069,920,000 5,257,287 $16,686,136 84.83% 33.68% 9.03% $6.137 $0.06444 0.02021 693,588 $1,318,737

The sample is made up of the population of all companies with ordinary shares listed on the Stock Exchange of Hong Kong ŽSEHK. throughout the entire sample period. Two trading days ŽOctober 14, 1996 and December 12, 1996. are not included in the sample period because data on the bid and ask quotes on these two particular days were not available from the SEHK.

equal to 80%. We show below that the 80% cut-off rule can be altered substantially without affecting the results since most days fall clearly into one category or the other.

4. Empirical results and analysis 4.1. DescriptiÕe statistics Table 1 presents summary statistics for the combined sample of 532 companies covering the sample period of 330 trading days. Reported values refer only to the ordinary shares Ži.e., no convertibles, preferred stock, or warrants. of the respective firms. The average Hong Kong company’s market capitalization, price, and daily trading volume are $6,069,920,000, $6.137, and 5,257,287 shares, respectively.7 Average daily dollar volume per company is $16,686,136. The typical Hong Kong company is traded during 84.83% of all trading days, 33.68% of all 5-min intervals, and 9.03% of all 30-s intervals. These numbers show that the SEHK is a relatively large and actively traded stock market. 7

All $ values refer to Hong Kong dollars. The Hong Kong dollar is officially pegged at 7.8 per U.S. dollar, and remained very close to this rate during the period under investigation.

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Average absolute bid–ask spreads Ži.e., ask price minus bid price. are $0.06444 and average relative bid–ask spreads Ži.e., absolute bid–ask spread divided by the bid–ask midpoint. are 0.02021, representing approximately 2% of the stock’s price. Depth, a second dimension of liquidity, may be measured simply as the number of shares at the highest bid price plus the number of shares at the lowest ask price, or the number of shares at the highest bid and lowest ask price times their respective prices. The first measure is referred to as volume depth and the second is referred to as dollar depth. Both measures show that the average Hong Kong stock is relatively liquid with a volume depth of 693,588 shares and dollar depth of $1,318,737. 4.2. Posterior probabilities The daily number of buys Ž B . and sells Ž S . are compiled for each company during 30-s intervals. Identification of buys and sells is straightforward on the SEHK because transactions can take place only at posted bid prices Žsells. or posted ask prices Žbuys.. Next, the Newton–Raphson method with a line search algorithm is used to obtain parameter estimates that maximize the natural log of the following likelihood function: 330

Ł

Ž 1 y a . ey ´

is1

´ Bi Bi !

ey ´

yŽ m q ´ .

qa Ž 1 y d . e

´ Si Si !

q ad ey ´

Ž mq´ . Bi !

Bi

ey ´

´ Bi Bi !

ey Ž m q ´ .

Ž mq´ .

Si

Si !

´ Si Si !

,

Ž 9.

where 330 is the number of days in the estimation period, and Bi and Si are the number buys and sells, respectively, during the ith trading day. As in Easley et al. Ž1996., a and d are restricted to Ž0, 1. through a logit transformation, and m and ´ are restricted to Ž0, `. through a logarithmic transformation. Starting values are determined by an extensive grid search over the parameter space and the estimation is carried out on a firm-by-firm basis. From the original sample of 532 firms, 521 companies generate valid maximum likelihood estimates.8 Table 2 summarizes the results from the posterior probability estimates based on Eqs. Ž6., Ž7., and Ž8.. The number of firm-days and percentages of each classification is reported in Panel A, and some additional evidence on the

8

In total, results for 11 companies are considered invalid due to problems with convergence or estimates that involve corner solutions Ži.e., zero or one. to within seven or more decimal places.

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robustness of the 80% cut-off rule is provided in Panel B. Out of a total of 171,930 company-days, 74.9% are classified as no news days, 11.6% are good

Table 2 Estimation of the trade process model and identification of good news, bad news, and no news days Panel A

Number of firm-days

%

Classified as GOOD NEWS day Classified as BAD NEWS day Classified as NO NEWS day Not classified Ži.e., posterior probability - 0.80. Total Ž521 firms=330 days.

19,859 16,917 128,762 6392 171,930

11.6 9.8 74.9 3.7 100.0

Panel B

Classified as GOOD NEWS day

Classified as BAD NEWS day

Classified as NO NEWS day

Posterior probability Ž p . that the day was a day of the classified news type

Number of firm-days

%

Number of firm-days

%

Number of firm-day

%

0.80 F p- 0.85 0.85F p- 0.90 0.90 F p- 0.95 0.95F pF1.00 Total

334 508 736 18,281 19,859

1.7 2.6 3.7 92.1 100.0

401 534 800 15,182 16,917

2.4 3.2 4.7 89.7 100.0

754 1042 1736 125,230 128,762

0.6 0.8 1.3 97.3 100.0

Maximum likelihood estimation of the Easley et al. Ž1996. and Easley et al. Ž1998a. trade process model Žhereafter EKOP. is carried out for each sample firm. The Newton–Raphson method with the line search algorithm is used to obtain the parameter estimates that maximize the natural log of the likelihood function: 330

Ł is 1

Ž 1y a . ey ´

´ Bi Bi !

ey ´

q a Ž 1y d . eyŽ m q ´ .

´ Si Si !

q ad ey ´

Ž mq ´ . Bi !

Bi

ey ´

´ Bi Bi !

ey Ž m q ´ .

Ž m q ´ . Si Si !

´ Si Si !

over the 330-day sample period. a , d , m , and ´ are the Poisson process parameters representing the probability of an information event, probability of the information being bad news, arrival rate of informed trades, and arrival rate of uninformed trades respectively. Bi and Si are the number of buys and the number of sells respectively on the ith trading day. Transaction and bid–ask data on all the sample firms are compiled at 30-s intervals throughout the trading days. For each interval, trades are identified as a buy if the transaction price is at the posted ask and trades are identified as a sell if the transaction price is at the posted bid. The total number of buys and the total number of sells for each of the 330 trading days are then determined for each sample firm. As in EKOP, a and d are restricted to Ž0, 1. through a logit transform and m , and ´ are restricted to Ž0, `. through a logarithmic transform of the unrestricted parameters. Starting values are determined based on an extensive grid search over the parameter space for each firm to ensure that the global maximum is reached. Of all the sample firms, 11 are eliminated due to estimation and convergence problems encountered during the maximum

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429

news days, 9.8% are bad news days, and 3.7% could not be classified. The findings that most days tend to be no news days, and good news days slightly exceed bad news days, conform rather well to a priori expectations. The results in panel B demonstrate that 92.1%, 89.7%, and 97.3% of the no news, bad news, and good news day classifications, respectively, are unaltered even if the 80% cut-off rule is raised to 95%. The posterior probability estimates seem to provide a definitive and robust method of classification. In subsequent sections, we combine good and bad news days into a single category representing information days.

Notes to Table 2: likelihood estimation process and valid parameter estimates are obtained for the remaining 521 firms. The end-of-day posterior probabilities that the day was a good news day, a bad news day and a no news day can be derived for each day and each firm as: Pr  good newsrBi ,Si 4

ž

s aˆ Ž 1y dˆ . ey Žmˆ q ´ˆ .

q ad ˆ ˆe

y´ˆ

´ˆ B i

Ž mˆ q ´ˆ .

yŽ m ˆ q ´ˆ .

e

Bi !

Bi

ey ´ˆ

Bi !

Ž mˆ q ´ˆ .

´ˆ S i Si !

/ž

Ž 1y aˆ . ey ´ˆ

Si

q aˆ Ž 1y dˆ . e

yŽ m ˆ q ´ˆ .

Si !

´ˆ B i Bi !

ey ´ˆ

Ž mˆ q ´ˆ .

´ˆ S i Si ! Bi y ´ˆ

e

Bi !

´ˆ S i Si !

y1

/

,

Pr  bad newsrBi ,Si 4

ž

s ad ˆ ˆ ey´ˆ

q ad ˆ ˆ ey´ˆ

´ˆ B i Bi !

Ž mˆ q ´ˆ .

ey Žmˆ q ´ˆ .

´ˆ B i Bi !

eyŽ mˆ q ´ˆ .

Si

Si !

Ž mˆ q ´ˆ .

/ž

Ž 1y aˆ . ey ´ˆ

´ˆ B i Bi !

ey ´ˆ

Si

q aˆ Ž 1y dˆ . eyŽ mˆ q ´ˆ .

Si !

´ˆ S i Si !

Ž mˆ q ´ˆ .

Bi

ey ´ˆ

Bi !

´ˆ S i Si !

y1

/

,

Pr  no newsrBi ,Si 4

ž

s 1y aˆ ey´ˆ

q ad ˆ ˆ ey´ˆ

´ˆ

´ˆ B i Bi !

ey ´ˆ

´ˆ S i Si !

Bi

Bi !

eyŽ mˆ q ´ˆ .

/ž

Ž 1y aˆ . ey ´ˆ

Ž mˆ q ´ˆ . Si !

´ˆ B i Bi !

ey ´ˆ

´ˆ S i Si !

Si

q aˆ Ž 1y dˆ . eyŽ mˆ q ´ˆ .

Ž mˆ q ´ˆ . Bi !

Bi

ey ´ˆ

´ˆ S i Si !

y1

/

,

where Bi and Si are defined as before and aˆ , dˆ, m ˆ and ´ˆ are the estimated Poisson process parameters of the EKOP model. As in Easley et al. Ž1998a., a day is classified as a good news day if the posterior probability of the day being a good news day is greater than or equal to 0.8. Similarly, a day is classified as a bad news day or a no news day if the posterior probability for the respective news type is greater than or equal to 0.8. For the 521 firms with valid parameter estimates, a total 171,930 firm-days Ž521 firms=330 days. are classified using the above procedures. Panel A reports the breakdown of the 171,930 firm-days across the different news categories. Panel B shows the distributions of posterior probabilities Žas described above. associated with days that are classified as good news days, bad news days and no news days.

430

Panel A: Arrival rates of informed and uninformed traders

Combined sample 10th trading volume decile 9th trading volume decile 8th trading volume decile 7th trading volume decile 6th trading volume decile 5th trading volume decile 4th trading volume decile 3rd trading volume decile 2nd trading volume decile 1st trading volume decile

Number of firms in the orginal sample

Number of firms with valid estimation results

Mean Žmedian. estimate of arrival rate of informed traders, m

Mean Žmedian. estimate of arrival rates of uninformed traders, ´

Mean Žmedian. difference between arrival rates of informed and uninformed traders, m y ´

t-Statistic w p-valuex of paired t-test for testing m)´

Test statisti w p-valuex of sign test for testing m ) ´

532 53 53 53 54 53 53 54 53 53 53

521 53 53 53 54 53 53 51 50 48 53

36.46 Ž36.67. 62.09 Ž65.08. 55.21 Ž57.71. 50.74 Ž51.51. 48.12 Ž49.73. 40.30 Ž42.73. 38.58 Ž38.27. 26.07 Ž27.33. 22.75 Ž25.18. 11.03 Ž9.22. 5.96 Ž4.34.

13.49 Ž9.46. 41.22 Ž42.33. 26.98 Ž26.92. 20.81 Ž21.57. 14.60 Ž13.05. 11.29 Ž11.22. 8.24 Ž8.48. 5.22 Ž4.90. 3.16 Ž3.24. 0.96 Ž0.86. 0.25 Ž0.18.

22.98 Ž21.38. 20.87 Ž18.52. 28.22 Ž24.88. 29.92 Ž27.47. 33.51 Ž33.82. 29.00 Ž28.64. 30.34 Ž27.73. 20.84 Ž20.62. 19.59 Ž21.53. 10.07 Ž8.37. 5.71 Ž3.96.

33.06 w0.0001x 7.77 w0.0001x 13.90 w0.0001x 14.49 w0.0001x 16.87 w0.0001x 14.50 w0.0001x 16.43 w0.0001x 11.92 w0.0001x 14.96 w0.0001x 9.73 w0.0001x 6.56 w0.0001x

252.50 w0.0001x 18.50 w0.0001x 26.50 w0.0001x 26.50 w0.0001x 27.00 w0.0001x 26.50 w0.0001x 26.50 w0.0001x 25.50 w0.0001x 25.00 w0.0001x 24.00 w0.0001x 26.50 w0.0001x

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Table 3 Comparison of informed and uninformed trading activities

Panel B: Trading volume on information and non-information days Mean Žmedian. daily dollar trading volume on INFO days

Mean Žmedian. estimate of INFO-based trading volume on INFO days

Mean Žmedian. daily dollar trading volume on NON-INFO days

Mean Žmedian. difference between estimated INFO-based trading volume and NON-INFO day trading volume

t-statistic w p-valuex of paired t-test for zero mean difference

Test statistic w p-valuex of sign test for zero median difference

516 53 53 53 54 53 53 51 50 48 48

33,928 Ž8575. 172,725 Ž84,053. 54,152 Ž30,860. 35,601 Ž20,455. 24,850 Ž14,442. 16,029 Ž8852. 12,918 Ž6548. 8744 Ž3882. 4408 Ž2450. 1747 Ž771. 740 Ž353.

26,234 Ž7418. 116,326 Ž67,096. 45,690 Ž24,177. 30,892 Ž16,670. 22,285 Ž12,804. 14,390 Ž7780. 12,012 Ž5949. 7983 Ž3328. 3970 Ž2216. 1516 Ž678. 661 Ž277.

7824 Ž642. 56,399 Ž20,294. 8462 Ž4622. 4709 Ž3011. 2565 Ž1594. 1639 Ž1025. 906 Ž504. 762 Ž437. 438 Ž223. 231 Ž90. 95 Ž48.

18,411 Ž5110. 59,927 Ž30,939. 37,228 Ž17,689. 26,183 Ž13,769. 19,720 Ž11,195. 12,751 Ž6569. 11,106 Ž5524. 7221 Ž3011. 3532 Ž1970. 1285 Ž607. 566 Ž232.

12.17 w0.0001x 5.17 w0.0001x 10.28 w0.0001x 12.21 w0.0001x 11.80 w0.0001x 12.17 w0.0001x 11.77 w0.0001x 3.70 w0.0005x 11.85 w0.0001x 7.56 w0.0001x 3.25 w0.0021x

242.00 w0.0001x 16.50 w0.0001x 26.50 w0.0001x 26.50 w0.0001x 27.00 w0.0001x 26.50 w0.0001x 26.50 w0.0001x 24.50 w0.0001x 25.00 w0.0001x 22.00 w0.0001x 21.00 w0.0001x

Panel A shows the means, medians and their respective differences in the arrival rates of informed traders and uninformed traders for the combined sample as well as for the ten trading volume deciles. The estimated parameters of m and ´ of the Easley et al. Ž1996. and Easley et al. Ž1998a. ŽEKOP. trade process model, representing the arrival rates of informed traders and uninformed traders respectively, are compared on a firm-by-firm basis. The trading volume deciles are formed on the basis of the trading volume of the firms and trading volume is measured by the company’s daily total dollar trading volume averaged across all trading days over the sample period. Panel B shows the results of comparing trading volume on information ŽINFO. days with that on non-information ŽNON-INFO. days. A day is an INFO day if it is classified as either a good news or a bad news day and is a NON-INFO day if it is classified as a no news day using the Easley et al. Ž1996. and Easley et al. Ž1998a. ŽEKOP. trade process model. Trading volume is stated in terms of dollar volume Žin thousands.. For each firm, an estimate of information-based trading is made for each INFO day by taking the difference between the actual trading volume of the day and the mean Žmedian. daily trading volume of the firm across all NON-INFO days. Mean Žmedian. measures of the estimated daily INFO-based trading volume across all INFO days are then compared with the daily non-information-based trading across all NON-INFO days for each firm.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Combined sample 10th trading volume decile 9th trading volume decile 8th trading volume decile 7th trading volume decile 6th trading volume decile 5th trading volume decile 4th trading volume decile 3rd trading volume decile 2nd trading volume decile 1st trading volume decile

Number of firms

431

432 Table 4 Serial correlations in trading volume and order imbalance and contemporaneous correlation between trading volume and price variance and absolute price change on information ŽINFO. days Serial correlation in order imbalance

Contemporaneous correlation Contemporaneous correlation between trading volume between trading volume and and price variance absolute price change

NON-INFO days

INFO days

NON-INFO days

INFO days

NON-INFO days

INFO days

NON-INFO days

INFO days

Mean Žmedian. value of correlation t-statistic w p-valuex of paired t-test for difference in the correlation between INFO and NONINFO days

0.1599 Ž0.1175. 15.05 w0.0001x df s 512

0.3220 Ž0.2828.

0.1469 Ž0.1834. 7.45 w0.0001x df s 496

0.2262 Ž0.2078.

0.0904 Ž0.0586. 8.22 w0.0001x df s 498

0.1627 Ž0.1434.

0.1554 Ž0.1451. 13.03 w0.0001x df s 509

0.2486 Ž0.2517.

Test statistic w p-valuex of sign test for difference in the correlation between INFO and NONINFO days

138.50 w0.0001x

Number of firms

516

515

496

515

475

511

510

514

Mean Žmedian. number of 5-min intervals used in correlation calculation

2392 Ž2242.

2407 Ž1914.

2347 Ž2151.

2368 Ž1886.

1126 Ž743.

1951 Ž1340.

1534 Ž1156.

2160 Ž1537.

66.50 w0.0001x

114.50 w0.0001x

136.00 w0.0001x

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Serial correlation in trading volume

1,234,054

1,239,388

1,168,749

1,219,555

554,338

996,900

782,567

1,110,439

Percentage of firms with correlation ) 0 and correlation significant at 0.05 level

70.0%

85.6%

84.7%

96.7%

41.5%

68.1%

70.2%

84.8%

Percentage of firms with correlation ) 0 and correlation not significant at 0.05 level

24.4%

11.7%

7.8%

3.1%

33.7%

23.1%

20.4%

10.3%

Percentage of firms with correlation - 0 and correlation not significant at 0.05 level

5.6%

2.7%

4.8%

0.2%

24.6%

8.0%

9.4%

4.5%

5.6%

2.7%

4.8%

0.2%

24.6%

8.0%

9.4%

4.5%

Percentage of firms with correlation - 0 and correlation significant at 0.05 level

0.0%

0.0%

2.6%

0.0%

0.2%

0.8%

0.0%

0.4%

433

A day is an INFO day if it is classified as either a good news or a bad news day and is a NON-INFO day if it is classified as a no news day using the Easley et al. Ž1996. and Easley et al. Ž1998a. ŽEKOP. trade process model. Measures of trading volume, order imbalance, price variance and absolute price change are compiled at 5-min intervals on all trading days for each sample firm. Trading volume is the number of shares traded in a 5-min interval. Order imbalance is the difference between the number of shares in buyer-initiated trades and the number of shares in seller-initiated trades divided by the total number of shares traded in an interval. In each interval, trades are identified as a buy if the transaction price is at the posted ask and are identified as a sell if the transaction price is at the posted bid. Price variances are computed for each 5-min interval based on transaction prices recorded 30 s apart. Absolute price change measures the magnitude of price change over an interval. The data set covers all 5-min intervals with posted bid and ask prices Ži.e., a valid bid–ask spread. and recorded transactions for the sample firms over the period from May 1, 1996 to August 29, 1997. Intervals from days that are not calssified as either the INFO or NON-INFO types are excluded from the analysis. All correlations are calculated on a firm-by-firm basis and the summary results are reported in this table.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Total number of 5-min intervals used in correlation calculations

434

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

4.3. Order arriÕal rates, Õolumes, and correlations In this subsection, we test hypotheses H1, H2, and H3. Panel A of Table 3 provides the combined and volume-decile order arrival rates for informed Ž m . and uninformed Ž ´ . traders. The parameters are estimated on a firm-by-firm basis and then combined into their respective decile portfolios. The overall sample mean Žmedian. informed and uninformed arrival rates are 36.46 Ž36.67. and 13.49 Ž9.46., respectively. The mean Žmedian. differences are significant at the 0.0001 level using both the paired t-test and sign test. It is clear from the results that the mean and median arrival rates for informed traders are significantly larger than those for the uninformed traders. This finding is robust to the level of trading activity and the use of parametric vs. nonparametric tests. In addition to the order arrival rates presented in Panel A, we also test for volume differences between information and non-information days in Panel B. According to Wang’s Ž1998. model and H1, informed volume trading can exceed uninformed volume trading. Dollar trading volumes are estimated on a firm-by-firm basis and then combined into their respective decile portfolios.9 The overall sample mean Žmedian. informed and uninformed dollar volumes Žin thousands. are 26,234 Ž7418. and 7824 Ž642., respectively. The mean Žmedian. differences are significant at the 0.0001 level using both the paired t-test and sign test. The results in Panel B demonstrate that dollar trading volumes on information days are significantly larger than the volumes on non-information days. As in Panel A, this conclusion is robust to the level of trading activity and the use of parametric vs. nonparametric tests. Taken together, the results in Table 3’s Panels A and B strongly confirm H1 and suggest that the level of informed trading is considerably higher than suggested by previous theoretical models. These models typically require a large quantity of uninformed or liquidity volume relative to informed volume in order to provide camouflage for trading on asymmetric information. Next, we test hypotheses H2 and H3 by calculating mean and median serial correlations for trading volume and order imbalance, as well as contemporaneous correlations between volume and price variance. Parametric and nonparametric tests are then used to determine whether correlations differ across the two information structures. Measures for trading volume, order imbalance, price variance, and absolute price change are compiled at 5-min intervals on all trading days for each firm. Trading volume is the total number of shares traded in a 5-min interval. Order imbalance is the difference between the number of shares in buyer-initiated trades and the number of shares in seller-initiated trades divided by the total number of shares traded in an interval. Price variance is measured as the variance of transaction prices Žrecorded at 30-s intervals. over the 5-min period. 9

We also test H1 using share volumes instead of the dollar volumes reported in Panel B of Table 3. The share volume results are equivalent to those reported in Panels A and B.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

435

Table 4 shows that the mean Žmedian. serial correlation in trading volume is 0.1599 Ž0.1175. for non-information days and 0.3220 Ž0.2828. for information days. The information day autocorrelations are significantly larger than the non-information day values at the 0.0001 level based on a parametric t-test and nonparametric sign test. Clearly, the presence of informed trading induces higher levels of positive autocorrelation, an empirical finding that confirms Wang’s Ž1998. proposition 3 and our H2. The mean Žmedian. serial correlation in order imbalance is 0.1469 Ž0.1834. for non-information days and 0.2262 Ž0.2078. for information days. The information day autocorrelations are significantly larger than the non-information day values at the 0.0001 level based on the same parametric and nonparametric tests. Therefore, both volume and order imbalance autocorrelations reveal patterns that are consistent with those predicted in H2. The mean Žmedian. contemporaneous correlation between volume and price variance is 0.0904 Ž0.0586. for non-information days and 0.1627 Ž0.1434. for information days. Differences in contemporaneous correlations are statistically significant at the 0.0001 level using the paired t-test and sign test. This is an interesting result because it demonstrates that the relation between volume and price variance is a function of trading type Že.g., informed vs. uninformed.. The findings not only confirm that there exists an overall positive relationship between volume and price variance, but also show that the relation is strengthened because of informed trading. The last column shows that equivalent results are obtained by using absolute price changes instead of price variance. These contemporaneous relationships confirm Wang’s Ž1998. proposition 4 and provide support for H3. The remainder of Table 4 gives more detailed information for the mean and median results. The number of firms and observations is reported for each category, and overall correlation results are partitioned into subgroups based on positive vs. negative correlations and significant vs. insignificant Žat the 0.05 level. magnitudes. The information days have higher percentages of positive correlations that are significant at the 0.05 level for all four categories. Volume serial correlations on information days are significantly positive for 85.6% of the firms compared to 70.0% of the serial correlations on non-information days. Similar results are obtained for order imbalance serial correlations Ž96.7% vs. 84.7%., volume–price variance contemporaneous correlations Ž68.1% vs. 41.5%., and volume–absolute price change contemporaneous correlations Ž84.8% vs. 70.2%.. It should also be noted that the percentage of firms with statistically significant and negative correlations is either zero or trivially small for all four correlation categories. 4.4. Regression analysis 4.4.1. AI and HPB estimation The test results from hypotheses H1, H2, and H3 provide empirical evidence that supports the Wang Ž1998. model. However, some of this evidence is also

436

Table 5 Estimates of the liquidity and heterogeneity parameters in the price innovation process

All 5-min intervals Ž10:05–12:30 and 14:30–15:55. used in estimation 2nd interval Ž10:05–10:10. 3rd interval Ž10:10–10:15. 4th interval Ž10:15–10:20. 5th interval Ž10:20–10:25. 6th interval Ž10:25–10:30. 7th interval Ž10:30–10:35. 8th interval Ž10:35–10:40. 9th interval Ž10:40–10:45. 10th interval Ž10:45–10:50. 11th interval Ž10:50–10:55. 12th interval Ž10:55–11:00. 13th interval Ž11:00–11.05. 14th interval Ž11:05–11:10. 15th interval Ž11:10–11:15. 16th interval Ž11:15–11:20. 17th interval Ž11:20–11:25. 18th interval Ž11:25–11:30. 19th interval Ž11:30–11:35. 20th interval Ž11:35–11:40.

Liquidity parameter, l t

Percentage of firms with estimate of liquidity parameter positiversignificant, etc., at 0.05 level

Heterogeneity parameter, g t

Percentage of firms with estimate of heterogeneity parameter positiversignificant, etc., at 0.05 level

Mean Žmedian.

Ž1.

Ž2.

Ž3.

Ž4.

Mean Žmedian.

Ž1.

Ž2.

Ž3.

Ž4.

375

0.00386 Ž0.00361.

99%

1%

0%

0%

y0.00025 Žy0.00001.

21%

26%

30%

23%

374 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375

0.00555 Ž0.00411. 0.00509 Ž0.00429. 0.00531 Ž0.00451. 0.00550 Ž0.00436. 0.00501 Ž0.00415. 0.00486 Ž0.00421. 0.00453 Ž0.00386. 0.00457 Ž0.00383. 0.00420 Ž0.00370. 0.00423 Ž0.00354. 0.00392 Ž0.00349. 0.00397 Ž0.00328. 0.00375 Ž0.00329. 0.00365 Ž0.00307. 0.00364 Ž0.00309. 0.00377 Ž0.00327. 0.00369 Ž0.00305. 0.00335 Ž0.00299. 0.00362 Ž0.00313.

48% 58% 67% 68% 67% 72% 69% 74% 70% 73% 70% 72% 69% 70% 72% 70% 71% 68% 70%

42% 37% 29% 30% 30% 27% 29% 25% 30% 27% 29% 26% 31% 28% 27% 29% 29% 30% 29%

6% 2% 3% 1% 2% 1% 1% 1% 0% 1% 1% 1% 1% 1% 1% 0% 0% 1% 1%

4% 3% 0% 0% 1% 0% 0% 0% 0% 0% 0% 0% 0% 1% 0% 0% 0% 1% 0%

0.00062 Ž0.00026. y0.00079 Žy0.00008. y0.00053 Žy0.00011. y0.00011 Žy0.00006. y0.00034 Žy0.00008. y0.00022 Žy0.00001. y0.00047 Žy0.00003. y0.00045 Ž0.00002. y0.00050 Žy0.00005. y0.00021 Žy0.00008. y0.00017 Žy0.00001. y0.00042 Žy0.00004. y0.00043 Žy0.00004. y0.00019 Žy0.00004. y0.00004 Žy0.00007. y0.00054 Žy0.00001. 0.00023 Ž0.00002. y0.00064 Žy0.00002. y0.00014 Žy0.00002.

8% 5% 7% 6% 4% 4% 5% 6% 6% 6% 6% 4% 4% 5% 6% 6% 5% 5% 6%

49% 39% 35% 41% 38% 46% 41% 46% 41% 41% 42% 42% 42% 41% 38% 42% 47% 41% 43%

34% 46% 49% 45% 50% 43% 49% 45% 47% 46% 46% 46% 48% 47% 46% 46% 41% 46% 44%

9% 9% 10% 9% 8% 7% 6% 3% 7% 7% 6% 8% 6% 7% 10% 7% 7% 7% 7%

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Number of firms with valid estimates

375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375

0.00348 Ž0.00305. 0.00353 Ž0.00312. 0.00356 Ž0.00312. 0.00335 Ž0.00284. 0.00342 Ž0.00287. 0.00320 Ž0.00271. 0.00337 Ž0.00281. 0.00347 Ž0.00321. 0.00325 Ž0.00282. 0.00356 Ž0.00307. 0.00415 Ž0.00368. 0.00370 Ž0.00329. 0.00369 Ž0.00339. 0.00382 Ž0.00318. 0.00365 Ž0.00305. 0.00366 Ž0.00314. 0.00346 Ž0.00298. 0.00349 Ž0.00300. 0.00350 Ž0.00291. 0.00357 Ž0.00312. 0.00341 Ž0.00310. 0.00345 Ž0.00306. 0.00348 Ž0.00311. 0.00339 Ž0.00290. 0.00365 Ž0.00321. 0.00370 Ž0.00335. 0.00460 Ž0.00392.

69% 67% 68% 66% 66% 65% 66% 68% 65% 68% 67% 72% 72% 73% 70% 69% 70% 71% 70% 72% 70% 70% 72% 69% 74% 72% 66%

30% 32% 31% 33% 33% 34% 32% 30% 32% 31% 33% 28% 26% 27% 30% 30% 29% 27% 29% 28% 29% 29% 27% 30% 26% 28% 33%

1% 1% 1% 1% 1% 1% 1% 1% 2% 0% 0% 0% 1% 0% 1% 0% 1% 1% 1% 0% 0% 1% 0% 1% 0% 0% 1%

1% 0% 0% 1% 1% 1% 1% 1% 1% 1% 0% 0% 0% 0% 0% 0% 0% 1% 0% 0% 1% 0% 0% 0% 0% 0% 0%

y0.00036 Ž0.00000. y0.00009 Žy0.00002. y0.00015 Ž0.00000 y0.00016 Ž0.00000. y0.00052 Ž0.00000. y0.00038 Žy0.00005. y0.00012 Ž0.00004. 0.00000 Ž0.00003. y0.00061 Ž0.00000. y0.00020 Ž0.00005. y0.00024 Ž0.00013. 0.00024 Ž0.00002. y0.00043 Žy0.00007. y0.00057 Žy0.00014. y0.00015 Žy0.00003. y0.00056 Žy0.00002. y0.00025 Žy0.00002. y0.00023 Žy0.00007. y0.00023 Žy0.00010. y0.00042 Žy0.00005. y0.00038 Žy0.00002. 0.00011 Žy0.00003. y0.00008 Žy0.00003. y0.00002 Žy0.00003. 0.00004 Ž0.00005. 0.00006 Ž0.00003. 0.00020 Ž0.00055.

3% 5% 6% 6% 6% 7% 6% 7% 6% 7% 13% 9% 5% 6% 6% 8% 8% 6% 7% 4% 5% 5% 4% 7% 7% 7% 27%

47% 44% 43% 44% 43% 38% 46% 46% 44% 47% 46% 43% 40% 36% 42% 39% 41% 37% 37% 43% 41% 41% 43% 41% 46% 46% 43%

41% 44% 45% 42% 45% 48% 40% 39% 44% 39% 33% 42% 45% 47% 46% 43% 43% 47% 47% 40% 45% 45% 43% 44% 41% 41% 23%

9% 7% 7% 7% 6% 7% 8% 8% 6% 7% 9% 6% 9% 12% 6% 10% 8% 10% 9% 13% 9% 9% 10% 8% 6% 5% 7%

The liquidity and heterogeneity parameters, l t and g t , are estimated on a firm-by-firm basis across all trading days for all 5-min intervals as well as for each 5-min interval using D p˜t sg t p˜ty1 q l t D y˜t , where D p˜t is price innovation and D y˜t is order imbalance in interval t ŽWang, 1998, Eq. 8, p. 326.. The first daily 5-min interval Ži.e., 10:00–10:05. is not included as one-period lagged values are needed in deriving the estimates. This table shows the means and medians of the estimated parameters across all sample firms and the percentages of firms with estimated parameters that are Ž1. positive and significant, Ž2. positive and not significant, Ž3. negative and not significant, and Ž4. negative and significant, all at the 0.05 level.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

21st interval Ž11:40–11:45. 22nd interval Ž11:45–11:50. 23rd interval Ž11:50–11:55. 24th interval Ž11:55–12:00. 25th interval Ž12:00–12:05. 26th interval Ž12:05–12:10. 27th interval Ž12:10–12:15. 28th interval Ž12:15–12:20. 29th interval Ž12:20–12:25. 30th interval Ž12:25–12:30. 31st interval Ž14:30–14:35. 32nd interval Ž14:35–14:40. 33rd interval Ž14:40–14:45. 34th interval Ž14:45–14:50. 35th interval Ž14:50–14:55. 36th interval Ž14:55–15:00. 37th interval Ž15:00–15:05. 38th interval Ž15:05–15:10. 39th interval Ž15:10–15:15. 40th interval Ž15:15–15:20. 41st interval Ž15:20–15:25. 42nd interval Ž15:25–15:30. 43rd interval Ž15:30–15:35. 44th interval Ž15:35–15:40. 45th interval Ž15:40–15:45. 46th interval Ž15:45–15:50. 47th interval Ž15:50–15:55.

437

438

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

consistent with alternative theoretical models and previously reported empirical results. Serially correlated volume, for example, is produced by the models of Foster and Viswanathan Ž1993. and Harris and Raviv Ž1993.. Harris Ž1987. and Campbell et al. Ž1993. present empirical evidence consistent with positive volume autocorrelations. Contemporaneous correlation between volume and price variance is obtained by Kim and Verricchia Ž1991., and documented by Karpoff Ž1987.. In order to disentangle some of these results and to further isolate Wang’s Ž1998. predictions, we partition volume into its three proposed components and test for significance in each component ŽH4.. We also investigate the inter-temporal patterns of trading on asymmetric information ŽH6. and heterogeneous beliefs ŽH7. and compare these results to theoretical values. Before testing H4, H6, and H7, we first estimate the parameters of Wang’s Ž1998. Eq. Ž8.. We then use the parameter estimates of the heterogeneity parameter, g , and liquidity parameter, l, to construct the full asymmetric information ŽAI. and heterogeneous prior beliefs ŽHPB. measures. According to H5, the heterogeneity parameter, g , should be nonzero in the following regression model: D p˜t s g t p˜ty1 q l t D y˜t q ´ t ,

Ž 10 .

where D p˜t is price innovation and D y˜t is order imbalance in interval t. We estimate the regression model ŽEq. Ž10.. on a firm-by-firm basis across 5-min intervals, as well as on a firm-by-firm basis for each 5-min interval. Reported t-statistics are adjusted for heteroscedasticity and autocorrelation using the Newey and West Ž1987. procedure. Table 5 presents the means and medians of the estimated parameters across all sample firms and the percentages of firms with estimated parameters that are Ž1. positive and significant, Ž2. positive and not significant, Ž3. negative and not significant, and Ž4. negative and significant, all at the 0.05 level.10 Across all time intervals, the liquidity coefficients are significantly positive Žnegative. for 99% Ž0%. of our sample firms. The estimated liquidity parameters are also highly significant and positive for all 5-min intervals throughout the trading day. Fig. 1 provides a graph of the median inter-temporal liquidity estimates. The graph displays a general U-shape with higher values at the open and close of trading. The heterogeneity parameter estimates are significantly positive Žnegative. for 21% Ž23%. of our sample firms across all time intervals. This evidence clearly

10 The results in Tables 5, 7 and 8 are based on firm-by-firm regressions. This method ignores the possibility of substantial cross-correlations, and does not yield a joint test across regression coefficients. To address these issues, we apply the Chordia et al., Ž2000. method to test for cross-equation dependence. We find that mean Žmedian. cross-correlations are small Ži.e., between 0.0157 and 0.0344. but more significant than suggested by chance. We use the cross-correlation estimates to conduct a joint test across all coefficients of interest, following Christie Ž1982.. In every case, the results confirm the conclusions reported herein.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454 439

Fig. 1. Intra-day pattern of the median estimate of the liquidity parameter.

440 P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Fig. 2. Intra-day pattern of the median estimate of the heterogeneity parameter.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

441

supports the claim of H5 that g should be nonzero. The estimated g-coefficients at 5-min intervals are generally less significant than the overall case, but still more significant than by chance. We note that the highest level of significance Ž27%. occurs in the 47th interval at the close of the market, as anticipated by Wang’s Ž1998. model. Fig. 2 provides a graph of the median inter-temporal heterogeneity estimates. Although the graph shows a large increase in heterogeneity coefficients at the close of trading, there is also evidence of larger coefficients at the open of trading both in the morning and afternoon sessions.11 Using the parameter estimates from Eq. Ž10., we construct the full measures for AI and HPB as follows: AI t s  1 y Ž StrSty1 . rl2t 4 Ž Sty1 y St . ,

Ž 11 .

where l t is the liquidity parameter, St is the error variance of price at the end of an interval t, and Ž St -1 y St . is the contemporaneous price variance over the interval t; and: HPB t s Ž g t2rl 2t . Ž S 0 y Sty1 . ,

Ž 12 .

where g t is the heterogeneity parameter, l t is the liquidity parameter, and Ž S 0 y Sty1 . is the cumulative price variance from the beginning of the day up to the end of the previous interval t y 1. The 5-min interval, firm-by-firm estimates of AI and HPB are summarized in Panel A of Table 6. The initial results Žin columns three and four. contain some rather extreme values, particularly for the mean AI measures. We, therefore, windsorize the data by eliminating AI and HPB measures in the top and bottom one percentile of the respective distributions. The last two columns present the windsorized results. We note that the AI measures are substantially larger than the HPB measures for every 5-min interval across the trading day. Based on these findings, it appears that trading volume is affected by asymmetric information much more than by heterogeneous prior beliefs. Positive levels of HPB throughout the trading day, however, are consistent with H4. In Panel B, we test for equality of means Žmedians. across all 5-min intervals for both the AI and HPB measures. The results reveal highly significant inter-temporal variation for each measure. This finding is consistent with H7 since HPB is expected to remain at low levels throughout the trading day, and then peak at the close of trading. Furthermore, the largest HPB magnitude Žfor the windsorized 11 Our Figs. 1 and 2 correspond to Wang’s Ž1998. Figs. 3 and 4, respectively. Although comparisons of simulated data with real trading data are tenuous at best, we observe that our figures display considerably more inter-temporal variation than Wang’s simulations. In particular, the simulations do not adequately capture actual open-of-trading behavior. And finally, we note that our Figs. 1 and 2 are most consistent with the case of K s 0.50, implying the prevalence of underconfident informed traders. Although not reported herein, graphs of HPB trading intensity and inter-temporal price volatility Žcorresponding to Wang’s Figs. 2 and 6, respectively. are also consistent with the case of K s 0.50.

442

Table 6 Measures of trading on asymmetric information and heterogeneous prior beliefs

2nd interval Ž10:05–10:10. 3rd interval Ž10:10–10:15. 4th interval Ž10:15–10:20. 5th interval Ž10:20–10:25. 6th interval Ž10:25–10:30. 7th interval Ž10:30–10:35. 8th interval Ž10:35–10:40. 9th interval Ž10:40–10:45. 10th interval Ž10:45–10:50. 11th interval Ž10:50–10:55. 12th interval Ž10:55–11:00. 13th interval Ž11:00–11:05. 14th interval Ž11:05–11:10. 15th interval Ž11:10–11:15. 16th interval Ž11:15–11:20. 17th interval Ž11:20–11:25. 18th interval Ž11:25–11:30. 19th interval Ž11:30–11:35.

Before windsorizing both measures to eliminate observations at the highest and lowest 1% levels

After windsorizing both measures to eliminate observations at the highest and lowest 1% levels

Number of firms included

Number of firms included

332 368 374 375 374 375 375 375 375 375 375 375 375 374 375 375 375 373

Trading on asymmetric information ŽAI t .

Trading on heterogeneous prior beliefs ŽHPB t .

Trading on asymmetric information ŽAI t .

Trading on heterogeneous prior beliefs ŽHPB t .

Mean Žmedian. 11.07782 Ž0.01921. 0.89214 Ž0.01816. 1.25459 Ž0.01567. 1.18191 Ž0.01662. 0.50846 Ž0.01182. 0.93490 Ž0.00907. 0.77626 Ž0.00938. 31.57081 Ž0.01000. 0.50604 Ž0.00880. 6.42817 Ž0.01038. 143.90100 Ž0.01025. 1.57078 Ž0.00913. 0.85460 Ž0.00950. 32.05655 Ž0.00931. 1.39815 Ž0.01004. 0.66788 Ž0.00755. 0.62105 Ž0.00900. 1.96536 Ž0.00875.

Mean Žmedian.

Mean Žmedian.

Mean Žmedian.

0.15464 Ž0.00005. 0.00181 Ž0.00004. 0.03880 Ž0.00004. 0.00081 Ž0.00004. 0.00069 Ž0.00005. 0.00039 Ž0.00005. 0.00127 Ž0.00004. 0.00420 Ž0.00005. 0.00121 Ž0.00006. 0.00073 Ž0.00007. 2.53280 Ž0.00007. 0.00906 Ž0.00009. 0.00134 Ž0.00009. 0.00710 Ž0.00010. 0.00325 Ž0.00011. 0.00101 Ž0.00009. 0.00083 Ž0.00009. 0.01778 Ž0.00011.

0.25654 Ž0.01734. 0.19709 Ž0.01609. 0.18710 Ž0.01368. 0.15559 Ž0.01481. 0.15086 Ž0.01091. 0.14169 Ž0.00905. 0.14935 Ž0.00934. 0.12524 Ž0.00914. 0.12454 Ž0.00880. 0.14752 Ž0.00980. 0.14090 Ž0.00962. 0.13939 Ž0.00859. 0.16486 Ž0.00866. 0.14471 Ž0.00852. 0.14040 Ž0.00848. 0.15827 Ž0.00727. 0.13420 Ž0.00843. 0.12790 Ž0.00816.

0.00034 Ž0.00004. 0.00032 Ž0.00004. 0.00029 Ž0.00004. 0.00031 Ž0.00004. 0.00025 Ž0.00005. 0.00023 Ž0.00005. 0.00027 Ž0.00004. 0.00023 Ž0.00005. 0.00031 Ž0.00006. 0.00032 Ž0.00007. 0.00034 Ž0.00007. 0.00037 Ž0.00009. 0.00042 Ž0.00009. 0.00040 Ž0.00009. 0.00042 Ž0.00011. 0.00035 Ž0.00009. 0.00041 Ž0.00009. 0.00041 Ž0.00010.

328 365 372 375 374 375 375 375 375 374 375 374 375 374 375 373 375 372

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Panel A: Measures of trading on asymmetric information and heterogeneous prior beliefs

375 373 375 375 373 374 372 373 371 372 372 374 375 374 375 375 374 375 372 375 375 373 375 375 375 375 375 375

1.00166 Ž0.00788. 0.91982 Ž0.01033. 0.51956 Ž0.00833. 1.79993 Ž0.00974. 0.90256 Ž0.00973. 1.22240 Ž0.00955. 0.49702 Ž0.01241. 1.71984 Ž0.00988. 0.85116 Ž0.00881. 0.59525 Ž0.00984. 0.80336 Ž0.01189. 2.23689 Ž0.01107. 0.43486 Ž0.00877. 1.47831 Ž0.01048. 0.83146 Ž0.01028. 0.97014 Ž0.01134. 56.49796 Ž0.00779. 0.51538 Ž0.00935. 0.65537 Ž0.00992. 1.63411 Ž0.01113. 0.64768 Ž0.00923. 0.69557 Ž0.00829. 1.89529 Ž0.00898 . 2.59852 Ž0.01066. 1.33167 Ž0.01037. 0.51927 Ž0.01016. 0.91564 Ž0.01456. 19.39076 Ž0.02936.

0.00346 Ž0.00012. 0.00317 Ž0.00009. 0.00212 Ž0.00014. 0.00283 Ž0.00015. 0.00142 Ž0.00016. 0.01225 Ž0.00015. 0.00105 Ž0.00014. 0.01214 Ž0.00015. 0.00299 Ž0.00012. 0.00651 Ž0.00014. 0.00316 Ž0.00021. 0.00240 Ž0.00022. 0.00309 Ž0.00013. 0.00207 Ž0.00011. 0.00126 Ž0.00019. 0.09282 Ž0.00015. 0.16651 Ž0.00018. 0.00321 Ž0.00017. 0.00386 Ž0.00022. 0.00149 Ž0.00022. 0.00107 Ž0.00020. 0.00148 Ž0.00019. 0.00563 Ž0.00017. 0.00305 Ž0.00020. 0.01181 Ž0.00020. 0.00269 Ž0.00021. 0.00394 Ž0.00016. 0.02127 Ž0.00052.

375 371 374 375 373 373 371 372 370 370 371 374 373 373 375 374 373 374 370 375 375 373 374 374 373 374 375 370

0.12052 Ž0.00673. 0.14275 Ž0.00876. 0.11058 Ž0.00790. 0.12559 Ž0.00951. 0.13593 Ž0.00955. 0.13651 Ž0.00836. 0.13006 Ž0.01136. 0.14424 Ž0.00880. 0.15646 Ž0.00806. 0.13772 Ž0.00870. 0.16143 Ž0.01027. 0.13335 Ž0.00946. 0.12672 Ž0.00850. 0.13369 Ž0.01034. 0.12485 Ž0.00953. 0.13875 Ž0.00995. 0.14042 Ž0.00739. 0.13691 Ž0.00857. 0.13237 Ž0.00867. 0.11818 Ž0.00964. 0.12513 Ž0.00841. 0.14984 Ž0.00822. 0.15735 Ž0.00829. 0.14898 Ž0.00990. 0.16530 Ž0.00917. 0.14012 Ž0.00935. 0.16361 Ž0.01394. 0.27192 Ž0.02262.

0.00045 Ž0.00011. 0.00049 Ž0.00009. 0.00050 Ž0.00014. 0.00049 Ž0.00015. 0.00051 Ž0.00016. 0.00057 Ž0.00014. 0.00052 Ž0.00013. 0.00060 Ž0.00014. 0.00054 Ž0.00012. 0.00057 Ž0.00013. 0.00068 Ž0.00021. 0.00077 Ž0.00021. 0.00050 Ž0.00013. 0.00053 Ž0.00011. 0.00056 Ž0.00019. 0.00056 Ž0.00015. 0.00065 Ž0.00017. 0.00059 Ž0.00017. 0.00065 Ž0.00021. 0.00071 Ž0.00022. 0.00067 Ž0.00020. 0.00071 Ž0.00019. 0.00073 Ž0.00016. 0.00072 Ž0.00020. 0.00069 Ž0.00019. 0.00065 Ž0.00021. 0.00058 Ž0.00016. 0.00133 Ž0.00052. (continued on next page)

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

20th interval Ž11:35–11:40. 21st interval Ž11:40–11:45. 22nd interval Ž11:45–11:50. 23rd interval Ž11:50–11:55. 24th interval Ž11:55–12:00. 25th interval Ž12:00–12:05. 26th interval Ž12:05–12:10. 27th interval Ž12:10–12:15. 28th interval Ž12:15–12:20. 29th interval Ž12:20–12:25. 30th interval Ž12:25–12:30. 31st interval Ž14:30–14:35. 32nd interval Ž14:35–14:40. 33rd interval Ž14:40–14:45. 34th interval Ž14:45–14:50. 35th interval Ž14:50–14:55. 36th interval Ž14:55–15:00. 37th interval Ž15:00–15:05. 38th interval Ž15:05–15:10. 39th interval Ž15:10–15:15. 40th interval Ž15:15–15:20. 41st interval Ž15:20–15:25. 42nd interval Ž15:25–15:30. 43rd interval Ž15:30–15:35. 44th interval Ž15:35–15:40. 45th interval Ž15:40–15:45. 46th interval Ž15:45–15:50. 47th interval Ž15:50–15:55.

443

444

Table 6 Ž continued . Trading on asymmetric information ŽAI t .

Trading on heterogeneous prior beliefs ŽHPB t .

F-statistic w p-valuex in repeated measures parametric ANOVA: Test for equality of means across intervals F-statistic w p-valuex in Friedman’s two-way nonparametric ANOVA: Test for equality of medians across intervals

7.54 w0.0001x

18.89 w0.0001x

10.49 w0.0001x

28.83 w0.0001x

Panel C: Serial correlations of the AI t and HPB t measures over the 46 intervals

Trading on asymmetric information ŽAI t .

Trading on heterogeneous prior beliefs ŽHPB t .

Mean Žmedian. value of correlation Mean Žmedian. value of p-value on the significance of correlation Mean Žmedian. number of observations used in calculation of correlation Total number of firm-days included Percentage of firm-days with serial correlation significant at 0.05 level

0.1430 Ž0.1014. 0.5884 Ž0.6459. 29 Ž31. 28,349 5.4%

0.1615 Ž0.1109. 0.5552 Ž0.6027. 29 Ž31. 28,599 7.5%

AI t is a proxy for informed trading on asymmetric information and is measured by w1yŽ St r Sty1 .xr l2t 4Ž Sty1 y St ., where l t is the liquidity parameter, St is the error variance of price at the end of an interval t, and Ž Sty 1 y St . is the contemporaneous price variance over the interval ŽWang, 1998, p. 329.. HPB t is a proxy for informed trading on heterogeneous prior beliefs and is measured by Žg t2 r l2t .Ž S 0 y St y 1 ., where g t is the heterogeneity parameter, l t is the liquidity parameter, and Ž S 0 y St y 1 . is the cumulative price variance from the beginning of the day up to the end of previous interval t y1 ŽWang, 1998, p. 329.. The contemporaneous price variances, Ž Sty 1 y St ., are computed for all 5-min intervals based on transaction prices recorded 30 s apart. The error variance of price St at the end of day is estimated by the variance of closing prices on all information days over the sample period. The error variances of price during the day are then derived accordingly using the end-of-day St and the contemporaneous price variances. The liquidity and heterogeneity parameters, l t and g t , are estimated on a firm-by-firm basis across all trading days for each interval using D p˜t sg t p˜ty1 q l t D y˜t , where D p˜t is price innovation and D y˜t is order imbalance in interval t ŽWang, 1998, Eq. 8, p. 326.. The first daily 5-min interval Ži.e., 10:00–10:05. is not included as one-period lagged values are needed in deriving the estimates. Panel A shows, in 5-min intervals, the mean Žmedian. measures of AI t and HPB t before and after a windsorization procedure that eliminates extreme values at the highest and lowest 1% levels among all observations. Panel B shows the results of testing for equality of the mean Žmedian. measures of AI t and HPB t across the 46 intra-day intervals. Panel C shows the summary results of serial correlations of the AI t and HPB t measures over the 46 intra-day intervals.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Panel B: Test for equality of mean median measures across the 46 intervals

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445

data. occurs precisely in the last trading interval, adding additional support for H7. Significant variation for the AI measure, however, leads to the rejection of H6. According to Wang’s Ž1998. model, trading from asymmetric information should be flat throughout the day. In the next subsection, we measure the impact of AI and HPB on traded volumes. The results show that while the AI measure varies significantly within the day, its impact on traded volume remains relatively flat. In Panel C, we calculate serial correlations for AI and HPB at 5-min intervals over the trading day. The mean Žmedian. serial correlations for AI and HPB are 0.1430 Ž0.1014. and 0.1615 Ž0.1109., respectively. The results show that trading on asymmetric information and heterogeneous prior beliefs are not strongly correlated at 5-min intervals. The percentage of firm-days with significant autocorrelations at the 5% level is 5.4% for AI and 7.5% for HPB. 4.4.2. AI and HPB impact on Õolume As discussed above in Section 2, liquidity or uninformed trading is not strategic in nature and evolves randomly through time. According to Wang’s Ž1998. specification, trading based on asymmetric information is captured by Eq. Ž11., and trading based on heterogeneous prior beliefs is represented by Eq. Ž12.. In this subsection, we analyze these proposed relationships by testing each components’ impact on traded volume Žas opposed to analyzing the magnitudes of AI and HPB measures.. Previous subsections provide evidence that is consistent with the existence of separate AI and HPB motivations for trading. In this subsection, we test the ability of each component to generate trading volume. We estimate the following regression model: VOL t s a q b 1 INFOt q b 2 INFO )AI t q b 3 INFO ) HPB t 5

q

Ý wmVOL tym q ´ t ,

Ž 13 .

ms1

where VOL t is the total trading volume over 5-min interval t, and AI t and HPB t are defined above. INFOt is a dummy variable that takes on the value of one if the trading day is classified as an information day Žsee Section 3.3. and zero otherwise. INFO )AI t and INFO ) HPB t represent interaction terms that measure the incremental effect of asymmetric information and heterogeneous prior beliefs, respectively, on trading volume during information days. Because VOL t exhibits considerable autocorrelation, we include five lagged values as independent variables.12 All non-dummy variables are transformed by taking natural logarithms. 12 In addition to the lag specification in model Ž13., we estimate models with 10, 15, 20, and 25 lagged values for volume. The significance of AI and HPB-related coefficients are unchanged by altering the number of lagged values.

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P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

The sample size is reduced to 375 companies for regressions Ž13. and Ž14. since we require a minimum of 200 5-min intervals with active trading Ži.e., 200 observations. over the 330-day sample period. We expect trading volume on information days to exceed trading volume on non-information days Ži.e., b 1 ) 0.. According to Wang’s Ž1998. Eqs. Ž23., Ž24., and Ž25., volume on information days can come from three sources; the informed trader, market maker, and liquidity trader. The b 1 coefficient is expected to capture trading volume arriving from the market maker, as well as any additional liquidity trader-initiated order flow.13 A significant b 2 Ž b 3 . coefficient is evidence that trading on asymmetric information Žheterogeneous beliefs. appreciably impacts traded volume. Table 7 provides a summary of the firm-by-firm regression results using the 1% windsorized data, as described in Section 4.4.1. Reported t-statistics are adjusted for heteroscedasticity and autocorrelation using the Newey and West Ž1987. procedure. Similar to previous tables, we present a breakdown of positive and negative coefficients, and significant and insignificant results Žat the 0.05 level.. As hypothesized, mean Žmedian. b 2 and b 3 coefficients are consistently significant at the 0.05 level. INFO has a mean Žmedian. coefficient of 0.804 Ž0.811.. It takes a positive value for 95.2% Ži.e., 84.5% plus 10.7%. of the firms and is significantly positive for 84.5% of the firms. This result confirms that volume is higher on information days. INFO )AI has a mean Žmedian. coefficient of 0.064 Ž0.060., is positive for 99.7% of the firms, and significantly positive for 97.6% of the firms. This finding demonstrates that a significant amount of additional trading volume is generated by asymmetric information. INFO ) HPB has a mean Žmedian. coefficient of 0.006 Ž0.004., is positive for 58.4% of the firms, and significantly positive for 22.4% of the firms. This finding is particularly important because it shows that heterogeneous prior beliefs lead to significant volume increases even after controlling for liquidity and asymmetric information trading. Overall, these results are very supportive of Wang’s Ž1998. claim that both asymmetric information and heterogeneous prior beliefs generate higher trading volume. More specifically, the INFO ) HPB results provide additional evidence in support of H4. Next, we refine our analysis in order to capture the inter-temporal impact of AI and HPB measures on trading volumes. According to the model Žand H6., asymmetric information trading will be smoothed over the day as the informed trader attempts to conceal his private signal. Heterogeneous beliefs trading is expected to have little impact on volume until the final few intervals of the trading

13 It should be noted, however, that higher levels of uninformed trading on information days relative to non-information days violates one of the simplifying assumptions of the EKOP Ž1996. model Ži.e., a constant value for ´ .. We thank an anonymous referee for clarifying this point.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

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period when it is predicted to increase ŽH7.. We test these predictions, with respect to H6 and H7, using the following regression specification: 8

8

VOL t s a q Ý f iTIME i ,t q Ý d j AI ) TIME j,t is2

js1

8

q Ý u k HPB ) TIME k ,t q ks1

5

Ý wmVOL tym q ´ t ,

Ž 14 .

ms1

where VOL t , AI t , and HPB t are defined as above and TIME t is a dummy variable for the different intervals of the day. The time periods 1, 2, . . . , 8 correspond to 10:05–10:30, 10:30–11:00, 11:00–11:30, 11:30–12:00, 12:00–12:30, 14:30– 15:00, 15:00–15:30, and 15:30–15:55. The first 5-min interval Ž10:00–10:05. of the day is dropped in the regression analysis since HPB requires a lagged value. We use 30-min Ždummy. intervals in order to isolate the volume effects predicted by Wang Ž1998. and our H7. As before, the interaction variables, AI ) TIME and HPB ) TIME, are designed to measure the marginal volume effects of AI and HPB trading for each of the eight time intervals. Panel A of Table 8 summarizes the regression results from estimating Eq. Ž14. on a firm-by-firm basis. As seen by analyzing the TIME coefficients Ži.e., f 2 , f 3 , . . . , f 8 ., the overall inter-temporal volume pattern is U-shaped, as documented in previous studies. The inter-temporal pattern of volume impact from asymmetric information ŽAI ) TIME. is relatively smooth throughout the trading day and does not show any consistent trending tendency. The minumum and maximum mean Žmedian. d-values are 0.058 Ž0.054. and 0.065 Ž0.061., respectively. The intertemporal pattern of volume impact from heterogeneous beliefs ŽHPB ) TIME., however, shows a significant increase in the last trading interval. In fact, both the mean and median u-values reach their daily maximums during the last interval of trading. In addition to the upward movement toward the end of trading, there is less overall uniformity in HPB’s impact on trading volume with minimum and maximum mean Žmedian. u-values of y0.043 Žy0.042. and 0.022 Ž0.021., respectively. This evidence suggests that trading on heterogeneous prior beliefs leads to significant volume increases only at the close of trading. Panels B and C provide additional results with respect to hypotheses H6 and H7. According to H6, there should be insignificant variation across the daily time intervals. The repeated measures ANOVA test compares d coefficients across the time intervals Ži.e., d 1 s d 2 s d 3 s . . . s d 8 .. The F-statistic of 1.37 Žpanel B. is not significant at conventional levels, thereby confirming the hypothesis that the volume impact of trading on asymmetric information is relatively uniform during the day. According to H7, there should be a volume surge at the end of the day due to heterogeneous beliefs trading. Panel C provides paired t-test results from comparing u-values at the end of the day Ži.e., u 8 . to the mean u-value from the 7 . Ž . rest of the day Ži.e., 1r7S ks 1 u k . The t-statistic of 8.34 p-values 0.0001 shows

448

Variable

Intercept INFO INFO )AI INFO ) HPB VOL t y 1 VOL t y 2 VOL t y 3 VOL t y 4 VOL t y 5

a b1 b2 b3 w1 w2 w3 w4 w5

Mean Žmedian. estimated coefficient of the variable

Number Žpercent. firms with coefficient ) 0 and t-statistic significant at 0.05 level

Number Žpercent. of firms with coefficient ) 0 and t-statistic not significant at 0.05 level

Number Žpercent. of firms with coefficient- 0 and and t-statistic not significant at 0.05 level

Number Žpercent. of firms with coefficient- 0 and t-statistic significant at 0.05 level

2.753 Ž2.792. 0.804 Ž0.811. 0.064 Ž0.060. 0.006 Ž0.004. 0.150 Ž0.142. 0.096 Ž0.097. 0.072 Ž0.071. 0.061 Ž0.063. 0.057 Ž0.057.

375 Ž100.0%. 317 Ž84.5%. 366 Ž97.6%. 84 Ž22.4%. 370 Ž98.7%. 351 Ž93.6%. 326 Ž86.9%. 315 Ž84.0%. 292 Ž77.9%.

0 Ž0.0%. 40 Ž10.7%. 8 Ž2.1%. 135 Ž36.0%. 5 Ž1.3%. 23 Ž6.1%. 46 Ž12.3%. 55 Ž14.7%. 77 Ž20.5%.

0 Ž0.0%. 14 Ž3.7%. 1 Ž0.3%. 123 Ž32.8%. 0 Ž0.0%. 1 Ž0.3%. 3 Ž0.8%. 5 Ž1.3%. 6 Ž1.6%.

0 Ž0.0%. 4 Ž1.1%. 0 Ž0.0%. 33 Ž8.9%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%.

Total number of regressions Mean Žmedian. number of observations Mean Žmedian. adjusted R 2 Mean Žmedian. F-statistic Number Žpercentage. of firms with F-statistic significant at the 0.05 level

375 3463 Ž2521. 0.2612 Ž0.2409. 196.86 Ž106.85. 375 Ž100.0%.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Table 7 Firm-by-firm time series regression of trading volume on measures of trading on asymmetric information and trading on heterogeneous prior beliefs

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

449

that the end-of-day HPB impact on volume is significantly more intense than throughout the rest of the day. These results are supportive of H6 and H7, in particular, and Wang’s Ž1998. model in general. In summary, we have shown that the HPB parameter ŽEq. 10. is nonzero, and that the HPB measure ŽEq. 12. is positive and increasing at the end of trading ŽSection 4.4.1, Tables 5 and 6.. We have also shown that the HPB measure has a significant impact on traded volume, particularly at the end of the day ŽSection 4.4.2, Tables 7 and 8.. These findings are especially important because they demonstrate that trading on heterogeneous beliefs can influence overall trading volume levels, thereby reducing the need to rely solely on liquidity trading as camouflage. We believe that the results reported herein are the first to document this phenomenon.

Notes to Table 7: 5

VOL t s a q b 1 INFOt q b 2 INFO )AI t q b 3 INFO ) HPB t q

Ý wmVOL tym q ´ t , ms 1

where VOL t is trading volume measured by the number of shares Žin thousands. traded during a 5-min interval t. INFOt is a dummy variable, which is assigned a value of one if interval t is from a day classified as an information day and zero otherwise. INFO )AI t is an interaction variable between INFOt and AI t measuring the extent of trading on asymmetric information on an information day. INFO ) HPB t is an interaction variable between INFOt and HPB t measuring the extent of trading on heterogeneous prior beliefs on an information day. AI t is a proxy for informed trading on asymmetric information and is measured by w1yŽ St r Sty1 .xr l2t 4Ž Sty1 y St ., where l t is the liquidity parameter, St is the error variance of price at the end of an interval t, and Ž Sty1 y St . is the contemporaneous price variance over the interval ŽWang, 1998, p. 329.. HPB t is a proxy for informed trading on heterogeneous prior beliefs and is measured by Žg t2 r l2t .Ž S 0 y Sty1 ., where g t is the heterogeneity parameter, l t is the liquidity parameter, and Ž S 0 y Sty1 . is the cumulative price variance from the beginning of the day up to the end of the previous interval t y1 ŽWang, 1998, p. 329.. The contemporaneous price variances, Ž Sty 1 y St ., are computed for all 5-min intervals based on transaction prices recorded 30 s apart. The error variance of price St at the end of the day is estimated by the variance of closing prices on all information days over the sample period. The error variances of price during the day are then derived accordingly using the end-of-day St and the contemporaneous price variances. The liquidity and heterogeneity parameters, l t and g t , are estimated on a firm-by-firm basis across all trading days for each interval using D p˜t sg t p˜ty1 q l t D y˜t , where D p˜t is price innovation and D y˜t is order imbalance in interval t ŽWang, 1998, Eq. 8, p. 326.. The first daily 5-min interval Ži.e., 10:00–10:05. is not included in the regression as lagged values are needed in deriving AI t and HPB t . Measures of AI t and HPB t are windsorized to eliminate extreme values at the highest and lowest 1% levels among all observations. Lagged volume measures, VOL t y m , ms1, . . . , 5 are also added to control for serial correlations in trading volume. All non-dummy variables, VOL t , VOL t y m , AI t , and HPB t are transformed by taking natural logarithms. Companies with trades in less than 200 5-min intervals throughout the sample period are eliminated from the analyses. The regression is run on a firm-by-firm basis on a final sample of 375 companies. All statistics in the individual firm-by-firm regressions are adjusted for arbitrary heteroskedasticity and serial correlation using the Newey and West Ž1987. and Hansen’s Ž1982. generalized method of moments ŽGMM. procedures. Results of all t-tests are presented on the basis of two-tail significance level.

Intercept TIME 2 TIME 3 TIME 4 TIME 5 TIME 6 TIME 7 TIME 8 AI ) TIME 1 AI ) TIME 2 AI ) TIME 3 AI ) TIME 4 AI ) TIME 5 AI ) TIME 6 AI ) TIME 7 AI ) TIME 8 HPB ) TIME 1 HPB ) TIME 2 HPB ) TIME 3 HPB ) TIME 4 HPB ) TIME 5

a f2 f3 f4 f5 f6 f7 f8 d1 d2 d3 d4 d5 d6 d7 d8 u1 u2 u3 u4 u5

10:30–11:00 11:00–1130 11:30–12:00 12:00–12:30 14:30–15:00 15:00–15:30 15:30–15:55 10:05–10:30 10:30–11:00 11:00–11:30 11:30–12:00 12:00–12:30 14:30–15:00 15:00–15:30 15:30–15:55 10:05–10:30 10:30–11:00 11:00–11:30 11:30–12:00 12:00–12:30

Mean Žmedian. estimated coefficient of the variable

Number Žpercent. of firms with coefficient ) 0 and t-statistic significant at 0.05 level

Number Žpercent. of firms with coefficient ) 0 and t-statistic not significant at 0.05 level

Number Žpercent. of firms with coefficient- 0 and t-statistic not significant aat 0.05 level

Number Žpercent. of firms with coefficient- 0 and t-statistic significant at 0.05 level

2.892 Ž2.905. 0.365 Ž0.397. 0.184 Ž0.176. 0.228 Ž0.219. 0.177 Ž0.171. 0.494 Ž0.515. 0.282 Ž0.253. 0.796 Ž0.735. 0.063 Ž0.058. 0.065 Ž0.061. 0.062 Ž0.058. 0.060 Ž0.055. 0.060 Ž0.054. 0.061 Ž0.055. 0.058 Ž0.054. 0.058 Ž0.054. y0.043 Žy0.042. y0.001 Ž0.002. y0.017 Žy0.013. y0.013 Žy0.010. y0.018 Žy0.017.

375 Ž100.0%. 71 Ž18.9%. 47 Ž12.5%. 46 Ž12.3%. 48 Ž12.8%. 122 Ž32.5%. 64 Ž17.1%. 177 Ž47.2%. 255 Ž68.0%. 297 Ž79.2%. 278 Ž74.1%. 265 Ž70.7%. 265 Ž70.7%. 277 Ž73.9%. 282 Ž75.2%. 262 Ž69.9%. 9 Ž2.4%. 13 Ž3.5%. 10 Ž2.7%. 14 Ž3.7%. 10 Ž2.7%.

0 Ž0.0%. 207 Ž55.2%. 176 Ž46.9%. 195 Ž52.0%. 175 Ž46.7%. 171 Ž45.6%. 212 Ž56.5%. 142 Ž37.9%. 97 Ž25.9%. 69 Ž18.4%. 89 Ž23.7%. 103 Ž27.5%. 91 Ž24.3%. 88 Ž23.5%. 79 Ž21.1%. 100 Ž26.7%. 58 Ž15.5%. 180 Ž48.0%. 137 Ž36.5%. 144 Ž38.4%. 121 Ž32.3%.

0 Ž0.0%. 87 Ž23.2%. 142 Ž37.9%. 123 Ž32.8%. 136 Ž36.3%. 70 Ž18.7%. 87 Ž23.2%. 48 Ž12.8%. 20 Ž5.3%. 8 Ž2.1%. 8 Ž2.1%. 6 Ž1.6%. 19 Ž5.1%. 10 Ž2.7%. 14 Ž3.7%. 13 Ž3.5%. 141 Ž37.6%. 153 Ž40.8%. 190 Ž50.7%. 187 Ž49.9%. 217 Ž57.9%.

0 Ž0.0%. 10 Ž2.7%. 10 Ž2.7%. 11 Ž2.9%. 16 Ž4.3%. 12 Ž3.2%. 12 Ž3.2%. 8 Ž2.1%. 3 Ž0.8%. 1 Ž0.3%. 0 Ž0.0%. 1 Ž0.3%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%. 167 Ž44.5%. 29 Ž7.7%. 38 Ž10.1%. 30 Ž8.0%. 27 Ž7.2%.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

Part A: Summary results of firm-by-firm time series regressions Variable

450

Table 8 Firm-by-firm time series regression of information ŽINFO. days trading volume on intra-day variables of trading on asymmetric information and trading on heterogeneous prior beliefs

u6 u7 u8 w1 w2 w3 w4 w5

14:30–15:00 15:00–15:30 15:30–15:55

y0.001 Ž0.002. y0.014 Žy0.013. 0.022 Ž0.021. 0.161 Ž0.151. 0.098 Ž0.098. 0.076 Ž0.077. 0.066 Ž0.066. 0.065 Ž0.064.

42 Ž11.2%. 6 Ž1.6%. 79 Ž21.1%. 363 Ž96.8%. 333 Ž88.8%. 311 Ž82.9%. 298 Ž79.5%. 280 Ž74.7%.

Total number of regressions Mean Žmedian. number of observations Mean Žmedian. adjusted R 2 Mean Žmedian. F-statistic Number Žpercentage. of firms with F-statistic significant at the 0.05 leve

152 Ž40.5%. 142 Ž37.9%. 148 Ž39.5%. 12 Ž3.2%. 40 Ž10.7%. 59 Ž15.7%. 70 Ž18.7%. 89 Ž23.7%.

138 Ž36.8%. 193 Ž51.5%. 123 Ž32.8%. 0 Ž0.0%. 2 Ž0.5%. 5 Ž1.3%. 7 Ž1.9%. 6 Ž1.6%.

375 2292 Ž1681. 0.2633 Ž0.2459. 38.92 Ž20.21. 375 Ž100.0%.

Panel B: Test for firm-by-firm differences among the g coefficients during the day F-statistic w p-valuex in repeated measures analysis of variance ŽANOVA.: H0 : d 1 s d 2 s d 3 s . . . s d 8

1.37 w0.2133x

Panel C: Test for firm-by-firm difference between the end-of-day u coefficient and the average u coefficient from the rest of the day T-statistic w p-valuex in paired t-test: 7 H0 : u 8 s1r7S ks1 uk

8.34 w0.0001x

43 Ž11.5%. 34 Ž9.1%. 25 Ž6.7%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%. 0 Ž0.0%.

P. Brockman, D.Y. Chungr Journal of Empirical Finance 7 (2000) 417–454

HPB ) TIME 6 HPB ) TIME 7 HPB ) TIME 8 VOL t y 1 VOL t y 2 VOL t y 3 VOL t y 4 VOL t y 5

451

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452

5. Conclusion The purpose of this study is to analyze inter-temporal trading patterns attributable to informed trading, and to distinguish between asymmetric information and heterogeneous prior beliefs. Although liquidity and asymmetric information motives for trading are well established in the literature, there is much less consensus about the role played by heterogeneous beliefs. We overcome previous limitations by combining the testable hypotheses of Wang Ž1998. with the estimation procedures used in Easley et al. Ž1996.. If trading on heterogeneous prior beliefs describes actual order flows, then this motive could be a source of

Notes to Table 8: 8

VOL t s a q

8

8

5

Ý fiTIME i ,t q Ý d j AI ) TIME j,t q Ý u k HPB ) TIME k ,t q Ý wmVOL tym q ´ t ,

is 2

js 1

ks1

ms1

where VOL t is trading volume measured by the number of shares Žin thousands. traded during a 5-min interval t. AI t is a proxy for informed trading on asymmetric information and is measured by w1yŽ St r Sty1 .xr l2t 4Ž Sty1 y St ., where l t is the liquidity parameter, St is the error variance of price at the end of an interval t, and Ž Sty 1 y St . is the contemporaneous price variance over the interval ŽWang, 1998, p. 329.. HPB t is a proxy for informed trading on heterogeneous prior beliefs and is measured by Žg t2 r l2t .Ž S 0 y Styl ., where g t is the heterogeneity parameter, l t is the liquidity parameter, and Ž S 0 y Sty1 . is the cumulative price variance from the beginning of the day up to the end of the previous interval t y1 ŽWang, 1998, p. 329.. The contemporaneous price variances, Ž Sty 1 y St ., are computed for all 5-min intervals based on transaction prices recorded 30 s apart. The error variances of price St at the end of a day is estimated by the variance of closing prices on all information days over the sample period. The error variances of price during the day are then derived accordingly using the end-of-day St and the contemporaneous price variances. The liquidity and heterogeneity parameters, l t and g t , are estimated on a firm-by-firm basis across all trading days for each interval using D p˜t sg t p˜ty1 q l t D y˜t , where D p˜t is price innovation and D y˜t is order imbalance in interval t ŽWang, 1998, Eq. 8, p. 326.. The first daily 5-min interval Ži.e., 10:00–10:05. is not included in the regression as lagged values are required for deriving AI t and HPB t . Measures of AI t and HPB t are windsorized to eliminate extreme values at the highest and lowest 1% levels among all observations. TIME i,t are dummy variables identifying the time period to which interval t belongs, with is1, . . . , 8, representing 10:05–10:30, 10:30–11:00, 11:00–11:30, 11:30–12:00, 12:00–12:30, 14:30–15:00, 15:00–15:55 respectively. AI ) TIME j, t are interaction variables between AI t and TIME j, t , js1, . . . , 8, and HPB ) TIME k,t are interaction variables between HPB t and TIME k,t , k s1, . . . , 8. TIME 1, t Ži.e., for 10:05–10:30. is not included in the regression model to avoid perfect collinearity among the set of dummy variables. Lagged volume measures, VOL t y m , ms1, . . . , 5, are also added to control for serial correlations in trading volume. All non-dummy variables, VOL t , VOL t y m , AI t , and HPB t , are transformed by taking natural logarithms. Companies with trades in less than 200 5-min intervals throughout the sample period are eliminated from the analyses. The regression is run on a firm-by-firm basis on a final sample of 375 companies. All t-statistics in the individual firm-by-firm regressions are adjusted for arbitrary heteroskedasticity and serial correlation using the Newey and West Ž1987. and Hansen’s Ž1982. generalized method of moments ŽGMM. procedures. Results of all t-tests are presented on the basis of two-tail sgnificance level.

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considerable trading volume and may be responsible for previously documented trading patterns. In particular, Wang Ž1998. proposes that the increase in volume at the end of the trading day is a consequence of trading on heterogeneous prior beliefs. In this study, we estimate the EKOP model parameters and calculate posterior probabilities of informed and uninformed trading. The posterior probability results demonstrate that informed vs. uninformed classifications are quite robust to the 80% cut-off rule suggested in Easley et al. Ž1998a.. After isolating informed trading from uninformed trading, we test seven hypotheses related to Wang’s Ž1998. heterogeneous beliefs model. Volume and order imbalance autocorrelations, as well as volume and price variance cross correlations, are consistent with the presence of asymmetric information and heterogeneous prior beliefs. We show that the heterogeneous prior beliefs parameter is significantly different from zero, and estimate the full AI and HPB measures. Although the HPB measure behaves in a manner consistent with Wang’s Ž1998. model, the AI measure displays considerably more inter-temporal variation than expected Ži.e., H6 is rejected.. Firm-by-firm regression analysis provides additional support for the existence of trading on heterogeneous prior beliefs. We show that the impact of asymmetric information on volume is relatively smooth throughout the day while trading on heterogeneous beliefs displays a relative increase at the end of the day. The results are quite consistent with the patterns predicted by Wang Ž1998.. Our empirical findings contribute to the literature in the following three ways. First, there is currently little if any empirical evidence of trading on heterogeneous prior beliefs, and yet several recent theoretical models propose the existence of such motives. The findings confirm that trading on heterogeneous beliefs does indeed lead to volume increases. Second, we specifically test the validity of Wang’s Ž1998. model. In addition to proposing the existence of heterogeneous trading, his model describes its inter-temporal behavior. The findings show considerable similarity to the hypothesized patterns of asymmetric information and heterogeneous beliefs. Third, we demonstrate the usefulness of the EKOP model in testing theoretical trading models. The use of posterior probabilities to distinguish between information and non-information days provides a credible method for testing various Žtheoretical. market microstructure models. Future research can add to the results reported herein by applying the EKOP procedure to other such models.

Acknowledgements The authors wish to thank Geert Bekaest Žthe editor. and the two anonymous referees for their helpful comments. The work described in this paper was supported by a research grant from the Hong Kong Polytechnic University ŽProject No. A-PA91..

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