1

Information-Based Trading and Autocorrelation in Individual Stock Returns

Xiangkang Yin and Jing Zhao

La Trobe University

Corresponding author, Department of Economics and Finance, La Trobe Business School, La Trobe University, Bundoora, Victoria 3086, Australia. Tel: 61-3-9479 3120, Email: j.zhao@latrobe.edu.au. The authors are grateful to Talis Putnins, the seminar participants at University of Bath, Shanghai University of Finance and Economics, Fudan University, Jiangxi University of Finance and Economics, University of Newcastle, and the 28th Australasian Finance & Banking Conference for their constructive comments. The research is supported by funding provided by the Australian Research Council Discovery Projects (DP140100113).

2

Information-Based Trading and Return Autocorrelation in Individual Stocks

ABSTRACT

Applying a recently developed approach, this paper estimates the arrival rates of orders driven by

private information and investor disagreement for each stock in a sample of NYSE-listed

companies. Autocorrelations of these arrival rates are determinants of their return autocorrelation.

Stock return tends to continue on consecutive days when privately-informed trading prevails,

leading to positive return autocorrelation. However, return tends to reverse itself on days with

continuous disagreement-driven trading, leading to more negative return autocorrelation.

Contrarian trading strategies conditional on measures of investor disagreement can yield

economically and statistically significant excess returns, after controlling for other determinants of

return autocorrelation.

JEL Classification: D82, G12, G14

Keywords: Information-based trading, return autocorrelation, private information, dispersion in

beliefs

3

Autocorrelation in short-horizon stock returns is well documented in the literature and

many studies find that autocorrelations in the returns of individual stocks are significant.1 The

evidence is fundamental to finance because it suggests predictability in stock returns and challenges

the efficient market hypothesis. Why are stock returns serially correlated and what factors affect

the correlation? This article addresses these questions from an information economics perspective.

The premise of this study is that speculative trading triggered by the arrival of superior private

information and order flows driven by investor disagreement in a stock are time-varying and

serially correlated.2 These correlations are profound factors in determining autocorrelation in stock

returns and its variation. More importantly, these two types of information-based trading play very

different roles in determining return autocorrelation. It is hypothesized that the propagation of

private information in consecutive trading days increases serial correlation in stock returns and

makes it positive, while dispersion in beliefs reduces the serial correlation and makes it more

negative. The paper focuses on separately and jointly testing these hypotheses using both panel

data and time-series data of individual stocks. It further demonstrates that the autocorrelation in

trading of a particular motive has statistically and economically profound impacts. That results in

the predictability of stock return to a certain extent and the profitability of investment strategies

exploiting this autocorrelation.

The link between privately-informed trading and return autocorrelation is intuitive. When

an investor receives a negative private signal of the future payoff of a stock, she/he sells the stock

and the price of the stock falls. However, this price usually only partially reflects the private

information and the low return in the current period can be followed by a low return in the next

1 See, for example, Jagadeesh (1990), Lehmann (1990), Conrad, Kaul and Nimalendran (1991), Copper (1999), Gervais, Kaniel and Mingelgrin (2001), Avramov, Chordia and Goyal (2006), and Hendershott and Seasholes (2014). 2 We use investor disagreement and dispersion in beliefs interchangeably in this paper to account for investor heterogeneity because they receive differential information and/or have varied interpretations for the same public information.

4

period as the negative private information is further spread and impounded into the price. This

intuition has been imbedded in theoretical analysis. For instance, Wang (1994), and Llorente,

Michaely, Saar and Wang (2002) propose models in which returns generated by privately-informed

trades tend to continue themselves. On the other hand, if some event, say the publication of a piece

of disputable news, triggers dispersion in investors’ beliefs about the future value of a stock,3 a

pessimistic investor is willing to lower the price to sell the stock. Since there is no substantial

change in the aggregate expectation of future stock payoff, a low return in the current period, as a

result of the lower price pushed by a pessimistic seller, is more likely to be followed by a high

return in the next period when an optimistic investor buys it. Thus, the disputable public news may

make the price alternate between up and down from one period to another and manifest return

reversal. The link between trades due to dispersion in beliefs and return autocorrelation has been

explored using various theoretical models,4 but it is Banerjee (2011) who first predicted that

increased disagreement should reduce return autocorrelation if investors have rational expectations.

Building on the theoretical foundations of prior studies, this paper analyzes return

autocorrelation by characterizing the information environment of a stock’s trading, i.e., its market

state. To this end, we identify every trading day’s information state for each stock in a sample of

NYSE (New York Stock Exchange) stocks. Then, nonparametric analysis and regression analysis

on pooled data and time series of individual stocks are conducted to estimate the contributions of

continuous privately-informed trading and disagreement-induced trading to the serial correlation

in individual stock returns. Consistent with theoretical predictions, it is found that returns on

consecutive days tend to continue when trading stemming from informed speculators prevails on

3 For instance, Kim and Verrecchia (1994) model market participants processing earnings announcements differently, resulting in more information heterogeneity at the time of an announcement. 4 Harris and Raviv (1993) show that stock returns in their model are negatively autocorrelated. Shalen (1993) demonstrates that dispersion about futures prices contributes to the positive correlation between consecutive absolute price changes.

5

these days, while returns tend to reverse if investors’ beliefs are highly heterogeneous on these

days. For instance, the first-order autocorrelation of close-to-close daily return is 0.089, on

average, if there is no continuous information-based trading. This autocorrelation is increased by

0.16 (i.e., becomes 0.071) on days with speculative trading on private information but reduced by

0.118 (i.e., becomes 0.207 on days with trading from disagreeing investors. In fact, it is further

demonstrated that return autocorrelation depends on the intensity of information-based trading in

a dynamic manner. Return continuation increases with the intensity of continuous privately-

informed trading, while return reversal increases with the intensity of continuous disagreement-

driven trading. These findings are robust and remain qualitatively similar when we take stock

illiquidity and turnover (Avramov, Chordia and Goyal, 2006), bid-ask bounce bias (Blume and

Stambaugh, 1983) and contemporaneous order imbalance (Chordia and Subrahmanyam, 2004) into

account in our analysis.

When firm size is taken into consideration, it appears that the effect magnitude of

information-based trading varies substantially. The connection between return autocorrelation and

continuous privately-informed trading is stronger for smaller firms because to them information

asymmetry is more likely to be a profound issue. On the other hand, the connection between return

autocorrelation and continuous disagreement-induced trading is stronger for larger firms, which

implies that sophisticated and confident investors who condition on prices and their own beliefs

are more likely to trade large stocks.

Since information-based trading, disagreement-driven trading in particular, is likely to be

highly persistent, this makes it possible to predict current return based on return and information-

based trading on the prior day. We develop contrarian and momentum trading strategies to examine

the statistical and economic significance of their profitability. When the contrarian strategy is

coupled with criteria for the lagged information state, it yields significantly positive returns and

6

outperforms both equal-weighted (EW) and value-weighted (VW) market portfolios by substantial

margins. For instance, the long-short zero-investment portfolio in our experiment generates

cumulative raw returns of 37% to 57% over a 2-year sample period. They surpass the EW market

portfolio by 3,095 to 4,973 basis points and the VW market portfolio by 1,989 to 3,713 basis points.

The effect of disagreement-driven trading complements the determinants of return autocorrelation

documented in the existing literature, as we find that our contrarian strategy is still highly profitable

after controlling for firm size and commonly known determinants such as stock illiquidity, trading

volume and liquidity provision.

This paper contributes to the literature in four related research areas. First, because very

different theoretical models can have similar implications for the time-series behavior of returns,

trading volume is often used as additional data to stock return data for the identification problem.

Regarding serial correlation in stock returns, the literature of volume-induced return reversal

include Campbell, Grossman and Wang (1993), Conrad, Hameed and Niden (1994), Sias and

Starks (1997), and Cooper (1999). However, the results are inconclusive. For instance, return

reversal decreases with trading volume for relatively small Nasdaq stocks as shown in Conrad,

Hameed and Niden (1994), but increase with trading volume for large NYSE stocks as shown in

Cooper (1999). Trading volume alone seems unable to fully reveal the hidden determinants of

return autocorrelation. Thus, we distinguish trades according to their trading motives and use

proxies of different trading activities to characterize the information environment of a stock market.

Our empirical evidence showing, firstly, the contrasting effects of the two types of information-

based trading on return autocorrelation, and secondly, the variation of these effects over firm size

may explain the inconclusiveness of prior findings. The importance of distinguishing trading

motives is also evidenced by further robustness tests which show these effects remain qualitatively

unchanged after controlling for trading turnovers.

7

Second, Llorente, Michaely, Saar and Wang (2002) empirically demonstrate that the

autocorrelation of a stock’s returns increases in the proxy of the stock’s information asymmetry

such as bid-ask spread. However, their focus is not on directly estimating the magnitude of

privately-informed trading’s effect on return autocorrelation or ascertaining whether privately-

informed trading makes this autocorrelation positive.5 On the other hand, there is growing

literature studying the impact of investor disagreement.6 Banerjee (2011) is the first empirical

analysis on the relationship between dispersion in beliefs and return autocorrelation. His empirical

findings provide limited support for the theoretical prediction of a negative relationship between

them.7 Our analysis differs from prior empirical studies as we directly regress daily return on

interaction terms of lagged return and daily measures of information-based trading. More

importantly, the same regression includes both effects of private information and dispersion in

beliefs, which not only quantitatively estimates these effects but also gauges their relative

significance.

Third, there has been widespread interest in short-run reversal strategies and their

profitability since the discovery of short-horizon return reversals by Jagadeesh (1990) and

Lehmann (1990).8 For instance, Lehmann (1990) demonstrates that contrarian strategies exploiting

the return reversals in individual stocks generate weekly abnormal returns of about 1.7%. Using

internal NYSE data, Hendershott and Seasholes (2014) form long-short portfolios that yield returns

5 Their conclusion is achieved by a two-stage analysis, where the first stage is a time-series analysis of an individual stock, in which future return is regressed on current return and the interaction between return and trading volume, and the second stage is a cross-sectional analysis, in which the coefficient of the interaction term estimated from the first stage is regressed against a proxy of information asymmetry of the stock. 6 See Harris and Raviv (1993), Shalen (1993), Banerjee and Kremer (2010), Carlin, Longstaff and Matoba (2014). 7 More specifically, these results show that difference in return autocorrelation between high- and low-disagreement stocks is statistically significant (insignificant) when disagreement is proxied by trading volume (dispersion in analyst forecasts). Nevertheless, trading volume can be driven by other factors in addition to dispersion in investors’ beliefs, such as change in risk aversion (Campbell, Grossman and Wang (1993)) and private information (Kyle (1985)). In prior studies, the empirical relationship between trading volume and return autocorrelation has been investigated but the findings are inconclusive. 8 Conrad, Hameed and Niden (1994), Ball, Kothari and Shanken (1995), Copper (1999), Avramov, Chordia and Goyal (2006), and Hendershott and Seasholes (2014).

8

around 13 to 20 basis points at a one-day horizon. Unlike previous studies, our contrarian trading

strategies explore the role of investor disagreement in determining return reversals. It shows that

the profitability of reversal strategies is statistically and economically significant, and increases

with trading intensity driven by dispersion in investors’ beliefs. Investor disagreement is

demonstrated to be a driver additional to well-known factors which can enhance the profitability

of the contrarian investment strategy.

A fourth related literature examines the impacts of trading activities on price efficiency.

The use of autocorrelation-based measures to test market efficiency dates back to early studies such

as Fama (1970), who argues that substantial return autocorrelation reflects deviation from random

walk pricing and is indicative of violations of an efficient market. Boehmer and Kelley (2009),

and Saffi and Siguardsson (2010) establish institutional trading activity and short selling as sources

of the improved short-horizon information environment. Our study complements the rich literature

on the role of information-based trading in equity markets. We directly link information-based

trading to return autocorrelation and find that stock prices on days with greater privately-informed

trading more strongly resemble a random walk. It therefore suggests that continuous trades from

privately-informed investors increase price efficiency. However, as consecutive trading driven by

investor disagreement lowers the return autocorrelation or leads to a greater absolute value of serial

correlation in returns, we infer such trading makes stock return deviate from a random walk further

and the market less efficient.

The remainder of this paper is organized as follows. Section I introduces the research

methodology and hypothesis development. Data and sample are described in Section II. Section

III demonstrates the opposite effects of privately-informed trading and disagreement-driven trading

on autocorrelation in individual stock returns. Section IV documents the profitability of

9

information-based trading strategies that exploit return autocorrelation. The concluding remarks

are provided in the last section.

I. Research Methodology and Hypothesis Development

Time-series behavior of stock returns, including the dynamics of return correlation, has

long been the research interests of academics and practitioners. Wang (1994) recognizes private

information as an important trading motive in addition to risk sharing. His theory shows that

returns generated by risk-sharing trades tend to reverse themselves while those generated by

informed trades tend to continue themselves. The theoretical model of Llorente, Michaely, Saar,

and Wang (2002) also separates hedging trades from informed trades and more sharply predicts

the dependence of a stock’s return continuation on the intensity of information asymmetry. These

models highlight the diffusion process of private information, which leads persistence of privately-

informed trading. Abstracting this diffusion and trading persistence, stock prices can be

martingales even with the presence of information asymmetry (Kyle (1982), and Glosten and

Milgrom (1985)).

Inspired by these theoretical works, our first hypothesis is that continuous privately-

informed trading is related to return autocorrelation and a greater measure of this type of trading

leads to a higher autocorrelation. Contrary to the existing literature, we directly measure daily

information asymmetry of a stock market by estimating expected flows of buy and sell orders rather

than use other proxies of information asymmetry such as bid-ask spread. We then isolate the effect

of privately-informed trading as a component determining the serial correlation in stock returns.

Another and perhaps more important aspect of our analysis is our measure of private information

taking into account the impact direction of information asymmetry. Theory and intuition tell us

that the fundamental reason for private information exerting a positive effect on return

10

autocorrelation is that the propagation of private information takes time so that the same piece of

private information can induce return to go up or down in two consecutive periods. Thus, the

hypothesis is tested through the continuation of privately-informed trading in the same direction.

Our second hypothesis conjectures that continuous trading stemming from disagreeing

investors has a negative effect on return autocorrelation and a greater measure of this type of trading

results in a lower autocorrelation. In a model of trading based on announcements of public

information, Harris and Raviv (1993) demonstrate that consecutive price changes exhibit negative

serial correlation. Banerjee (2011) predicts that investor disagreement is related negatively to

return autocorrelation if investors are rational and use price information to update their beliefs. The

intuition for the relationship between investor disagreement and return autocorrelation is

straightforward. If some information event triggers belief dispersion about the value of a stock,

optimists will initiate purchases and push the price up while pessimists will initiate sales and push

the price down. In this process, the price swings from one period to another, leading to a negative

serial correlation of stock return. The market reaches a new equilibrium through this interaction

process between optimistic and pessimistic investors. Disagreement is critical here because a piece

of non-controversial information will induce all investors to unanimously revise their valuation and

it will not be associated with abnormal trading (see, Llorente, Michaely, Saar and Wang (2002),

Banerjee and Kremer (2010)). Similar to the case concerning information asymmetry, we also

directly measure trading caused by investor disagreement at daily frequency and apply this measure

to separate the corresponding component that determines the serial correlation of returns. Since

the focus of this hypothesis is on the continuation of trading triggered by investor disagreement,

we test this hypothesis by separating consecutive days with this type of trading from other trading

days.

11

Properly and accurately measuring information-based trading is challenging and we here

adopt the Hidden Markov Model (HMM) approach of Yin and Zhao (2015). The most notable

advantage of this approach is its ability to produce time-varying measures of information-based

trading with satisfactory accuracy. Unlike static measures such as PIN (Easley, Kiefer, O’Hara

and Paperman (1996)) and PSOS (Duarte and Young (2009)), which remain constant over the

estimation window (ranging from a couple months to a year), these dynamic measures are at the

daily or even higher frequency. This dynamic nature is particularly valuable to the study of the

dynamic properties of stock returns such as autocorrelation in returns at the daily frequency.

Another outstanding feature of the HMM approach is its ability to capture not only highly positive

contemporaneous correlations between buy and sell order flows, but also the serial correlations of

buy orders and sell orders observed in transaction data.9 The autocorrelations in information-based

order flows are particularly valuable to this study as they are the determinants of the serial

correction of stock return. Through extensive Monte Carlo simulation experiments and real

transaction data analysis, Yin and Zhao (2015) demonstrate that the HMM is very effective in

identifying the market state of stock trading. In comparison with other prevailing models, the

HMM approach shows superior performance in estimating daily measures of information-based

trading as well as cumulative estimates over any time interval. It has also successfully been applied

to address issues related to earnings announcement and co-movement of stock returns. From a

technical point of view, the flexibility of HMM facilitates the estimation of model parameters and

9 Duarte and Young (2009) raise a concern regarding the original PIN model because of its failure to generate positive contemporaneous correlations between buy and sell order flows. Thus, dynamic measures based on the PIN model, such as those developed by Easley, Engle, O’Hara and Wu (2008), lack such contemporaneous correlations. Of course, static models of information-based trading exclude autocorrelation in buy or sell order flows because order arrival rates in these models are static over their estimation windows.

12

circumvents the computational overflow problem. This is a major technical difficulty when

estimating the PIN model and its variants for stocks with large daily trades.10

Consistent with our purpose of testing the two hypotheses, the HMM includes two motives

of information-based trading on a stock market: firstly, trades originating from speculative

investors who possess superior private information of the stock and take this informational

advantage to buy or sell the stock to maximize their profits/utility; and secondly, trades originating

from disagreeing investors because of their different interpretations of the same public information

or because they receive differential information of the stock. In addition, the model also includes

liquidity needs as the third trading motive, which is independent of information-based trading

motives. The information state of the market reflects whether private information events and/or

events triggering investor disagreement occur or not, and if they occur, how intense they are. Since

these events lead to different trading patterns, the state is uniquely associated with two random

variables: the numbers of buyer-initiated and seller-initiated order flows. More specifically, state

, is characterized by the expected values of these two random variables, ; and ; . The

different trading motives imply that both arrival rates of buy orders and sell orders under a

particular state have three components that 11

; ; ; ; , 1, 2, … , ,

; ; ; ; , 1, 2, … , , (1)

where ; and ; denote, respectively, the arrival rates of privately-informed buy and sell orders,

; and ; the arrival rates of disagreement-driven buy and sell orders, and ; and ; the

10 For stocks with large daily number of trades, direct computation of the likelihood function of the PIN model may result in a numerical overflow problem and make the convergence of the maximum likelihood estimation fail, see, for example, Easley, Engle, O’Hara and Wu (2008), Duarte and Young (2009), Easley, Hvidkjaer and O’Hara (2010), and Lin and Ke (2011). Using Expectation and Maximization Algorithm, the estimation of the HMM converges for all the sample stocks in this study. 11 We use the expected number of orders and the arrival rate of orders interchangeably throughout this paper because of the HMM approach’s Poisson assumption.

13

arrival rates of liquidity buy and sell orders. As (1) shows, the HMM allows for buy states and

sell states and they are determined and estimated through model estimation.

The random number of daily buy (sell) orders is modeled as the mixture of the random

numbers of buy (sell) orders of all states. Thus, trading day can be characterized by its probability

distribution over all states, . The very nature of the Markov model implies that any two

consecutive trading days are linked by a Markov chain so that Γ, where Γ is the transition

matrix of the Markov chain, whose element gives the probability of day 1 being on a particular

state conditional on day being on another (or the same) state. The parameters of the HMM

include the initial distribution of states , transition matrix Γ and order arrival rates λ ; and λ ;

( 1, 2, … , , 1, 2, … , . They can be estimated based on observed numbers of daily buy

orders and sell orders by maximizing the likelihood function (see Equation (A1) in Appendix A.1)

through Expectation and Maximization Algorithm. The details of the HMM and its estimation can

be found in Appendices A.1 and A.2 of this paper.

After the aggregate arrival rates of buy orders and sell orders under all states (i.e., λ ; and

λ ; have been estimated, we follow the HMM approach to apply the k-means clustering analysis

together with the jump method of Sugar and James (2003) to identify the three types of trading in

(1). We outline the basic idea of the estimation here and present its details in Appendix A.3. First,

we look into observed daily trade imbalances (the absolute value of net daily buys) and group them

into clusters according to their statistic properties. The clusters with a center strictly larger than

the center of the most frequent cluster are classified as ones involving privately-informed trading,

while those with a center smaller or equal to the center of the most frequent cluster are considered

without privately-informed trading. The rationale for this classification is the insight that

information asymmetry leads to one-sided trading and substantial trade imbalance (Kyle (1985),

Easley, Kiefer, O’Hara and Paperman (1996), Sarkar and Schwartz (2009)). The most frequent

14

cluster of trade imbalances is chosen to be the cutoff point, because it includes states which occur

most often and it is plausible to assume that most trading days do not have private information

dispersed on the market. Extensive simulation analysis has also demonstrated the validity of using

the most frequent cluster of trade imbalances for the cutoff point. For state , , we take the

absolute value of its expected number of net buys (i.e., λ ; λ ; ) as an out-of-sample

observation and assign it to the cluster whose center is the closest to it. If the assigned cluster

involves privately-informed trading, we estimate ; and ; by Equation (A3) in Appendix A.3.

If the assigned cluster does not involve privately-informed trading, ; and ; are equal to zero.

Similarly, we following the HMM approach and apply the k-means clustering analysis

together with the jump method to daily observations of balanced trades (i.e., the sum of buys and

sells minus the absolute value of net buys). Note that liquidity trading generates a normal level of

two-sided trades but an information event causing dispersion in investors’ beliefs results in a surge

in both buys and sells or a shock to balanced trading, as argued by Duarte and Young (2009) and

Sarkar and Schwartz (2009).12 Thus, clusters of balanced trades with a center strictly larger than

the center of the most frequent cluster are classified as ones that involve disagreement-induced

trading, while those with a center smaller or equal to the center of the most frequent cluster are the

ones not involving disagreement. Again, the cutoff point is the most frequent cluster of balanced

trades since information events that cause controversy among investors are not likely to occur at a

very high frequency. The choice of this cutoff point is strongly supported by simulation analysis.

For state , , the expected number of balanced trades λ ; λ ; λ ; λ ; is taken as an out-

of-sample observation and it is assigned to the cluster with the closest center to it. The state is

12 If investors reach a consensus on the public announcement of a stock, they should revise their valuations of the stock similarly so that the announcement does not lead to abnormal trading.

15

classified according to the cluster it is associated with and in turn order arrival rates ; and ;

can be estimated by Equation (A4) in Appendix A.3.

When arrival rates ; , ; , ; and ; are estimated, arrival rates of liquidity buy and

sell orders, ; and ; , can be backed out by (1). Since we know the probability distribution of

trading day over the state space through the estimation of the HMM, the arrival rate of the buy or

sell order of a particular type of trading on day is obtained by averaging the corresponding arrival

rates over the state space, weighted by the probabilities of the market states (see Equation (A5) in

Appendix A.3). Moreover, the aggregate arrival rates of buy and sell orders on day , ; and ; ,

can be derived by the sums of their three components that

; ; ; ; and ; ; ; ; . (2)

To facilitate cross-sectional comparison, we scale the components of information-based

trading in (2) by the total order arrival rate, ; ; . To a certain extent, the scaling also acts as

a controlling device for trading volume because the total number of order flows on day is a

random number with a mean of ; ; . Such scaling actually leads to some well-known

concepts in the literature. The scaled order arrival rate of privately-informed trading corresponds

to the developed by Easley, Kiefer, O’Hara, and Paperman (1996), while the scaled order

arrival rate of disagreement-driven trading corresponds to the S introduced by Duarte and

Young (2009). It should be noted, however, that the original and are constant over the

sample period and obtained through completely different methods. Therefore, following the

existing concepts in the literature and accounting for their dynamic nature we call the scaled

measures (dynamic probability of informed trading) and (dynamic probability of

symmetric order-flow shock) that

; ;

; ;, (3)

16

; ;

; ;. (4)

Note that 0 if privately-informed trading exists on day t. Hence, does not indicate

if the private signal is positive (inducing buyer-initiated orders) or negative (resulting in seller-

initiated orders). For empirical analysis, it is important to differentiate this trading direction. Thus,

we define (probability of net buys due to private information) as

; ;

; ;. (5)

II. Data and Sample Description

Our sample includes common stocks listed on the NYSE in the two-year period from

January 1, 2010 to December 31, 2011. From the Center of Research in Security Prices (CRSP),

we obtain data on daily return, the numbers of shares traded and the number of shares outstanding.

The following securities are eliminated from the sample since their trading characteristics might

differ from ordinary equities: certificates, American Depository Receipts, shares of beneficial

interest, units, companies incorporated outside the U.S., Americus Trust components, closed-end

funds, preferred stocks, and real estate investment trusts. To permit a reliable estimation of return

autocorrelation, we further require that stocks in the sample are traded on at least two-thirds of

days. After this screening process, there are 1,249 stocks in the sample.13 The transaction data

source is Thomason Reuter Tick History (TRTH). For each stock, transactions and quotes that

occur before and at the open are excluded, as well as those at and after the close. Quotes with a

zero bid or ask price, quotes for which the bid-ask spread is greater than 50% of the price, and

13 We have also applied a screen criterion to exclude penny stocks, which refers to a stock having a minimum share price over the sample period being less than one dollar. This screen leads to a reduction of 2.24% of the sample size. Nevertheless our findings are virtually unchanged when these penny stocks are excluded from the sample. Moreover, penny stocks usually have small market capitalization. Since we are also interested in size-stratified subsamples, we report our findings by including these penny stocks to create a more complete picture.

17

transactions with a zero price are also excluded to eliminate possible data errors. Data for

November 26, 2010 and November 25, 2011 are removed due to an early “day after thanksgiving”

closing. The Lee-Ready (1991) algorithm is applied to the transaction data to determine the daily

numbers of buys and sells.

Two daily return series are considered in this paper: open-to-close returns and close-to-

close returns. The close-to-close daily returns are obtained from CRSP, while the open-to-close

daily returns from the TRTH transaction database. The three measures of information-based

trading introduced in (3)-(5) are estimated by the HMM approach for each stock on each day.

Panel A of Table I summarizes the descriptive statistics of the entire sample and its three

subsamples stratified by average daily market capitalization over the sample period ( ). As

illustrated by column 2, the average daily of a stock over the sample period, , falls

with firm size. For example, the cross-sample mean of is 0.119 for the small stocks but

0.082 and 0.057 for the medium and large groups, respectively. This is consistent with the idea

that proxies for information asymmetry and smaller firms often have more private

information in their stock trading (Easley, Hvidkjaer and O’Hara (2002)). On the other hand,

is quite similar across the three size-based subsamples. Although one may expect the

availability of public information, such as media coverage or analysts’ following, to be related to

firm size, reflects the magnitude of belief heterogeneity in public news rather than simply

the occurrence of public news events themselves. The insignificant relationship between firm size

and is likely to indicate that traders have fewer disputes on public news of larger firms or

receive less differential information although their news events may occur more often. It is also

consistent with the empirical finding of Banerjee (2011) that none of the disagreement proxies are

strongly correlated with firm size. For our empirical analysis, a more important variable of

measuring information asymmetry than is the probability of net buys due to private

18

information. Column 4 reports the cross-sample descriptive statistics of its daily average over the

sample period ( ). The absolute mean values of for the entire sample and

the three subsamples are much smaller than their counterparts of .

INSERT TABLE I HERE

For each stock, we calculate the first-order autocorrelations (ACFs) of the two series of

daily returns and the three series of daily information-based trading measures. Their cross-

sectional summary statistics are presented in Panel B. As shown in columns 1 and 2, the median

autocorrelation is 0.040 in close-to-close returns ( ) and 0.024 in open-to-close returns

( ). It implies that autocorrelation in daily returns is negative for most sample stocks and

relatively weak in comparison with autocorrelations in information-based trading documented in

the last three columns. However, the serial correlations of and are 0.106 and 0.153,

small in comparison with the serial correlation of which is 0.460. This suggests that

investors’ private information is relatively shorter-lived than investor disagreement.

Daily trading activities due to private information are positively correlated through time for

most sample stocks, which is consistent with the theoretical setting of Chordia and Subrahmanyam

(2004) that traders split their orders over time to minimize their price impact. When examining the

three size-stratified subsamples, it is clear that private information in the small firms is longer-lived,

evidenced by its mean s of and values of 0.117 and 0.179, respectively. This

is probably because the greater information asymmetry in the small firms needs to be resolved by

longer time. In contrast, stocks in the large group on average have a larger of than

small and medium stocks.

19

III. The Opposite Effects of Continuous Privately-Informed Trading and Disagreement-

Driven Trading on Autocorrelation in Individual Stock Returns

Before rigorously testing the hypotheses developed in Section I, we sort trading days

according to their characteristics of information-based trading or stock return to examine their

validity. We first separate consecutive days with continuous trades driven by private information

in the same direction from other days; that is, select trading days according to ,

, 0. We also separate consecutive days without privately-informed trading at least on

one of the two days, i.e., , , 0. We then calculate the sample of each

stock’s return for these two different kinds of trading days to identify the impact of continuous

trading driven by private information. Rows (2) and (3) of Panel A in Table II indicate that the

mean s of and for the days satisfying , , 0 are 0.014 and

0.032, respectively, which are significant at the 1% level. On the other hand, for the days satisfying

, , 0 , the estimates of serial correlation are 0.093 and 0.062 ,

respectively, which are also significant at the 1% level. This evidence implies that private

information has a substantial impact on stock returns and it can change return autocorrelation from

significantly negative to significantly positive; that is, change stock return from reversal to

continuation. This change is profound, as shown in the lower part of Panel A that the difference

between the mean s over the two kinds of days, i.e., ACF(2)−ACF(3) is significant at the 1%

level.

INSERT TABLE II HERE

Alternatively, trading days can be distinguished based on whether there is consecutive

trading due to disagreement among investors. Rows (4) and (5) show that the mean s of

and on consecutive days with disagreement-driven trading are 0.180 and 0.150 ,

respectively. They are much more negative than those on days without consecutive disagreement-

20

driven trading, 0.024 and 0.019. The difference between the mean s of ( ) over

the two kinds of days, i.e., ACF(4)−ACF(5), is 0.156 0.130), significant at the 1% level. Thus,

as expected, continuous trading caused by dispersion in investors’ beliefs does substantially

enhance the negative serial correlation in individual stock returns.

We further jointly examine the effects of private information and investor disagreement on

return autocorrelation by considering four kinds of trading days in rows (6)-(9), which show

substantial in return autocorrelation. The comparison between rows (6) and (7) reveals the effect

of consecutive privately-informed trading when consecutive trading due to investor disagreement

does not appear. The comparison between rows (8) and (9) reveals the effect when disagreement

trading is effective on consecutive days. Once again, they demonstrate the significantly positive

impact of continuous privately-informed trading on serial correlation in returns, but the joint effects

of the two types of information-based trading lead to a negative correlation. On the other hand, the

comparisons between rows (8) and (6) and between rows (9) and (7), respectively, illustrate the

effect of investor disagreement on the consecutive trading days when privately-information trading

is effective and ineffective. The negative effect of belief dispersion is obvious and statistically

significant.

Panel B of Table II looks at the association of serial correlation in returns with information-

based trading from another angle, where trading days are sorted based on whether the consecutive

daily returns of a stock continue or reverse. Then we estimate the serial correlations of

and on these two kinds of days. Rows (II)-(III) demonstrate that autocorrelation in

( ) on days with close-to-close return continuation is much larger (smaller) than that on days

with return reversal. The difference is 0.168 ( 0.057), which is significant at the 1% level.

Similar results can be obtained for open-to-close return as shown in rows (IV)-(V) and the

following mean-difference test. Thus, return autocorrelation is positively connected with privately-

21

informed trading on consecutive days but negatively connected with disagreement driven trading

on consecutive days.

We further conduct a similar nonparametric analysis to the three size-stratified subsamples.

As shown in Panels C and D of Table II, the findings from Panel A and B qualitatively hold for the

three subsamples. A new observation is that the effect of private information on return

autocorrelation increases as stock size becomes smaller, as evidenced by the monotonicity of

, and in stock size shown in Panel C.

Moreover, the difference between the serial correlations of on days with return

continuation and days with return reversal falls as stock size increases, as shown by

in Panel D. Similar monotonicity can also be found in the difference between serial

correlations of .

In the subsections below, we confirm these observations through regression analysis.

Following Campbell, Grossman and Wang (1993), we study the dynamics of stock return

autocorrelation and the factors affecting the dynamics by estimating different forms of the

regression model:

, , , , , (6)

where , is stock i’s return on day t, the time-invariant individual effect,14 and , the error

term. In (6),autocorrelation is a linear function of factors , .15 We begin by reporting

the results based on panel data regressions for the entire sample and the three size-stratified

subsamples. We also conduct time-series regressions for individual stocks so that we can examine

more closely the individual stock level. Finally, we discuss the robustness of our empirical findings.

14 We report the results of panel regressions including firm fixed effects. If month fixed effects are also included, the results are qualitatively similar and the overall explanatory power of the regression model increases. 15 Vector , in (6) also includes variables of information-based trading on day . To simplify notations, subscript is omitted.

22

3.1 Panel regressions over the entire sample

We first estimate the fixed-effect panel regression model (6) with the specification that

, , , , ,, (7)

where dummy variable takes value 1 if the condition 0 is satisfied and zero otherwise.

As mentioned before, , , 0 implies that on the two consecutive trading

days private information leads trading activities in the same direction. Thus,

, , is used to capture the scenario that price on day 1 only partially

incorporates private information and there are continuous informed trades due to the same or

similar private signal on day t as well. Similarly, dummy variable , ,

captures

the continuation of investor disagreement on the market. Regression (6)-(7) resembles the feature

of Campbell, Grossman and Wang (1993), who regress return on the interaction term of lagged

return and trading volume to study the effect of trading volume on return autocorrelation.

The first hypothesis proposed in the previous section conjectures that return continuation is

larger on days with continuous privately-informed trades (i.e., 0 and the second hypothesis

argues that return reversal is larger on days with continuous disagreement-driven trades (i.e.,

0 ). To ensure the results’ robustness, the panel regressions are run for two return series, close-to-

close return and open-to-close return, and they are documented in Table III. Column 1 in Panel B

shows the testing results for close-to-close returns where only private information is taken into

account. The regression coefficient is significantly positive at the 1% level, thus supporting the

first hypothesis. In particular, return autocorrelation on days without consecutive informed trading

is 0.127 on average, but it increases by 0.157 on days with continuous informed trading. The

test on the lower part of the panel shows the difference 0.157 0.127 0.03 is statistically

significant at the 1% level. It suggests that continuously incorporating private information leads

to positive autocorrelation in stock returns. Comparing this return autocorrelation with the overall

23

return autocorrelation of 0.031 in Panel A, it shows that privately-informed trading not only

increases return autocorrelation but also shifts it from overall negative to positive. This positive

serial correlation in stock returns is consistent with what is reported in Table II through

nonparametric analysis. Results of open-to-close returns in column 4 of Panel B are also supportive

of the first hypothesis, as they exhibit even a greater degree of return continuation on days with

continuous informed trading.

INSERT TABLE III HERE

Furthermore, columns 2 and 5 in Panel B show the corresponding testing results for the

second hypothesis of the effects of dispersion in beliefs. The autocorrelation of close-to-close

returns reduces by 0.112 on days with continuous trades due to investor disagreement, and that of

open-to-close returns reduces by 0.089, which confirm the findings in Table II that serial

correlation in stock returns is more negative on trading days with consecutive disagreement-driven

trading. Both tests support the second hypothesis with an estimated coefficient significant at the

1% level.

We also jointly test the two hypotheses and report the results in columns 3 and 6 of Panel

B. The regression coefficients remain significant at the 1% level. In particular the autocorrelation

of close-to-close returns is 0.089 on days without consecutive information-based trading, and it

increases by 0.16 on days with continuous informed trading but reduces by 0.118 on days with

continuous disagreement trading. Open-to-close returns exhibit a similar pattern. Moreover, if we

compare estimates of in columns (3) and (1) or in columns (6) and (4), and compare estimates

of in columns (3) and (2) or in columns (6) and (5), we find that the changes in both estimates

themselves and their t-statistics are not substantial. This implies that the effects of private

information and belief heterogeneity on return autocorrelation are not considerably overlapped.

24

Overall, the regression results in Panel B show that return autocorrelation varies across days and

depends on the information environment of the market trading the stock.

Panel A of Table III reports the regression results with lagged return alone as the prime

explanatory variable (i.e., (7) is collapsed to , ). On the one hand, we find that the

return autocorrelation is negative on all trading days, ignoring the variation in a firm’s information

environment. On the other hand, lagged returns only explain 0.09% (0.03%) of the variance of

close-to-close (open-to-close) return series as indicated by in Panel A. Nevertheless, when we

distinguish days according to their associated types of information-based trading, lagged returns

explain 1.19% (1.10%) of the return variance, as shown in column 3 and 6 of Panel B, which is

improved over 12 (35) times compared to Panel A.

Although the results of regression model (6)-(7) demonstrate there is a strong association

between return autocorrelation of a stock and its information-based trading, we would like to

further investigate how this association changes with the intensity of information-based trading in

a dynamic manner. Thus, we introduce the interaction terms between lagged return and measures

of information-based trading into regressions and consider the following specification:

, , , , ,

, , , ,.

(8)

Columns 1 and 4 in Panel C of Table III show is significantly positive at the 1% level, supporting

the first hypothesis. That means return continuation increases with the intensity of continuous

privately-informed trading. Columns 2 and 4 in Panel C confirm that return reversal is enhanced

by the degree of continuous trading resulting from investor disagreement, since 0 and is

significant at the 1% level for both return series. When we jointly test the two hypotheses in

columns 3 and 6, the regression coefficients remain significant at the 1% level.

25

In the literature, return autocorrelation is used as a measure of price efficiency (see, for

example, Fama (1970), Boehmer and Kelley (2009), and Saffi and Siguardsson (2010)). Both

negative and positive return autocorrelations reflect deviation from random walk pricing and are

indicative of violations of the market efficiency hypothesis. Our results therefore link information-

based trading to price efficiency as well. For both two return series, the autocorrelation is negative

when there are no continuous information-based trades on two consecutive days as Panel B of

Table III shows. When there are continuous trades due to investor disagreement, return becomes

more negatively autocorrelated, but it becomes positively autocorrelated with a smaller absolute

value when there are continuous trades due to private information. These findings indicate that

privately-informed trading increases price efficiency. Prices on consecutive days with these trades

more closely track the fundamental value of the stock and more closely resemble a random walk.

However, consecutive trading triggered by investor disagreement increases the absolute value of

return autocorrelation so that return deviates more from a random walk.

3.2 Panel regressions over size-stratified subsamples

We estimate the fixed-effect panel regression model of (6) with specification (7) or (8) over

the three size-based subsamples and report the results in Table IV. Overall, the earlier results of

positive (negative) dynamic relationships between return autocorrelation and continuous privately-

informed trading (trading because of dispersion in beliefs) are generally robust and obtained for all

subsamples. However, the positive effect of private information is more substantial for small

stocks, while the negative effect of investor disagreement is more profound for large stocks. Let

us take column 3 of Panel B as an example. The estimate of declines from 0.192 to 0.143 and

then to 0.101 over the small, medium and large subsample, while the absolute value of estimate

increases from 0.110 to 0.123 and then to 0.126. The t-statistics of and display similar

monotonicity patterns.

26

INSERT TABLE IV HERE

Banerjee (2011) compares the predictions of rational expectations equilibrium models with

predictions made by difference of opinion models. He concludes that a negative relationship

between return autocorrelation and investor disagreement implies that investors condition on prices

in valuation and decision-making for investment. Thus, the regression coefficient in Panel C of

Table IV provides a proxy of investor sophistication in utilizing price information. Since is

negative and its absolute value increases in firm size as shown in columns 2 and 5, it implies that

sophisticated investors, who are rational and condition their trading on prices, are more likely to

trade in stocks that have larger market capitalization.

3.3 Time-series regression of individual stocks

To ensure the robustness of our findings, we also examine the daily time-series of each

individual stock over the sample period to more closely investigate the effects of information-based

trading at the individual stock level. Panel A of Table V reports the regressions of daily return on

the lagged return alone.16 In Panels B and C, we allow return autocorrelation to be time-varying

and depend on the information environment of stock trading as characterized by (7) and (8),

respectively. Apparently, the findings obtained from panel-data analysis hold. Taking column 3

of Penal B as an example, averaging over all sample stocks, autocorrelation in close-to-close

returns is 0.104 if there is no consecutive information-based trading. It increases by 0.165 on

days with continuous privately-informed trading and decreases by 0.133 on days with continuous

disagreement-drive trading. Moreover, the statistics of and reported in column 3 are very

close to their counterparts in column 1 and 2. This demonstrates that our findings are robust to

16 The fixed effect in (6) is replaced by a constant in time-series regressions of individual stocks but their estimates are not reported in Table V.

27

different model specifications. It also indicates that the contributions of the two types of

information-based trading to the return autocorrelation have little overlap.

INSERT TABLE V HERE

The standardized coefficient is a scale-invariant version of the regression coefficient, which

is calculated by multiplying the estimated regression coefficient by the ratio of the standard

deviation of the associated explanatory variable to the standard deviation of the dependent

variable.17 The squared standardized coefficient has been proposed as a metric for assessing the

relative importance of explanatory variables. The averages of standardized coefficients across the

sample stocks are documented in Table V. In general, return autocorrelation is more sensitive to

privately-informed trading in comparison to disagreement trading, as indicated by the larger

average standardized coefficient of . Let us take the full sample case in the third column of Panel

B for example again. On average, a one-standard deviation increase in previous day’s return

changes the contemporaneous return by 0.104 times of its standard deviation if these two days

have no consecutive information-based trading. Yet the contemporaneous return is changed by

0.104 0.115 0.011 standard deviation on days with consecutive privately-informed trades,

and by 0.104 0.073 0.177 standard deviation on days with consecutive disagreement

trades. These figures indicate that the impacts of information-based trading on stock returns are

not only statistically significant but also economically substantial.

We also group the results for the three size-based subsamples in Table V. The two

hypotheses are supported by all three subsamples. However, private information wields stronger

impacts on return autocorrelations of small firms while dispersion in beliefs has greater influence

on large firms.

17 Standardized coefficients are used to investigate the relative importance of explanatory variables, see for example Bushee and Noe (2000), Barton (2001), Baek, Bandopadhyaya and Du (2005), and Dieckmann and Plank (2012). A standardized coefficient of means that a one-standard-deviation change of the independent variable will lead to a  -standard-deviation change in the dependent variable.

28

3.4 Further tests and robustness checking

We conduct a number of further tests to ascertain the robustness of our results and

demonstrate that our findings are robust to various econometric specifications and alternative

definitions of stock return. 18 This subsection briefly reports the three major tests and their

conclusions. First, Avramov, Chordia and Goyal (2006) document that a strong relationship exists

between short-run return reversal and stock illiquidity even after controlling for trading volume.

Moreover, Duarte and Young (2009) are concerned that the PIN of Easley, Kiefer, O’Hara, and

Paperman (1996) may proxy for illiquidity of a stock. Many authors also investigate the effect of

trading volume on return autocorrelation albeit drawing quite different conclusions.19 In order to

examine whether our findings are driven by the time-varying stock illiquidity and stock trading

volume, we include a stock’s Amihud (2002) illiquidity measure and its turnover of the lagged day

as control variables in the panel regressions and individual stock regressions. In other words, we

regress model (6) by expanding specifications (7) and (8), respectively, to

, , , , ,

, , (9)

and

, , , , ,

, , , ,

, , ,

(10)

18 The results of robustness tests are available upon request. 19 See for example, Campbell. Grossman and Wang (1993), Conrad, Hameed and Niden (1994), Cooper (1999), and Avramov, Chordia and Goyal (2006).

29

where , and , are illiquidity and turnover of stock on day 1.20 The

regression results show that the effect of consecutive privately-informed (disagreement-driven)

trading on return autocorrelation remain significantly positive (negative) even after including the

effects of stock illiquidity and turnover. For instance, the estimates of and in (9) for close-

to-close return are 0.159 and −0.120, respectively, with corresponding t-statistics 19.806 and

−14.776. These figures are very close to their counterparts of 0.160 and −0.118 with t-statistics of

20.429 and −12.401 in Panel B of Table III, which are obtained from estimating specification (7)

that excluding , and , . On the other hand, these extended regressions

generate a significantly negative estimate of , which is consistent with Avramov, Chordia and

Goyal (2006). The estimate of is insignificant, which is consistent with the inconclusiveness of

previous studies’ findings on the effect of trading volume on return reversal.

Second, return computation may be subject to the well-known bid-ask bounce bias (Blume

and Stambaugh (1983)). To address this issue, we calculate daily returns based on the mid-point

of the quoted bid and ask prices corresponding to the first and last transaction of each day.

Although we find that negative autocorrelation in return series is reduced when mid-quote returns

are adopted, the effects of information-based trading are virtually unchanged.

Third, contemporaneous order imbalances are strongly and positively related to

contemporaneous returns (see for example, Kyle (1985), Glosten and Milgrom (1985), and Chordia

and Subrahmanyam (2004)). In order to examine whether our findings are driven by this

relationship, we include concurrent trade imbalance as a control variable in (6). Both panel

regressions and individual stock regressions indicate that the dynamic relationship between return

autocorrelation and information-based trading remains after controlling for current trade imbalance.

20 We follow Avramov, Chordia and Goyal (2006) and Campbell, Grossman and Wang (1993) to consider the levels of illiquidity and turnover in the analyses. However, the findings are robust if we instead follow Chordia Conrad, Hameed and Niden (1994) and Cooper (1999) and include the changes in illiquidity and turnover in the regressions.

30

IV. Predictive Evidence

Because market states are correlated through time, we investigate the ability of the measures

of information-based trading together with observed returns in predicting stock returns in the next

period. Our exercise is not searching for a powerful model of predicting stock return. Instead, we

intend to gauge the economic significance of the role that information-based trading plays in

determining serial correlation in individual stock returns.

4.1 Profitability of contrarian and momentum investment strategies

The trading strategies are based on the return forecast of tomorrow using the information

of today’s open-to-close return or together with today’s measures of information-based trading.

We first consider a contrarian trading strategy that buys (short sells) one share of a stock in the

sample at the opening ask (bid) and sells (covers) it at the closing bid (ask) if the previous day’s

open-to-close return of the stock is negative (positive). This strategy is implemented every day or

only on days with lagged (i.e., today’s) measure excesses a threshold. The rationale of this

trading strategy is using the negative serial correlation in stock returns to exploit the potential

profits. We calculate the average daily return for each invested stock and report its means across

the entire sample and the three subsamples in Panel A of Table VI. The result in the first row

indicates that such a contrarian strategy unconditional on lagged yields a significant daily

average return of 0.021% over the entire sample. However, the profitability of this strategy is

questionable because it yields a statistically insignificant return of 0.002% after adjustment for

the return of a value-weighted market portfolio.

INSERT TABLE VI HERE

Our earlier results show that returns are more likely to reverse themselves on days with

continuous disagreement trading, and this relationship becomes stronger when there is a larger

31

degree of investor disagreement. For this reason we implement this contrarian trading strategy

conditional on the magnitude of lagged .21 As shown in Panel A, the profitability of the

strategy monotonically increases in the magnitude of lagged . For instance, if we

implement the strategy with a threshold for the lagged greater than zero the average daily

raw return is 0.197%, which is a substantial increase relative to the implementation of the strategy

without considering disagreement-driven trading. This value monotonically increases to 0.246%

if the threshold of lagged is lifted to 0.2. By measuring a stock’s information environment

of trading, we can greatly enhance the profitability of the contrarian trading strategy through

smartly timing the return autocorrelation. This strategy is most effective for the small-firm

subsample and yields the highest average daily raw return of 0.324% if the implementation is

conditional on , 0.2. We also report the profits from the contrarian trading strategy

in terms of daily return adjusted by an equal-weighted (EW) or value-weighted (VW) market

portfolio in Panel A. Without examining information environment of trading, the contrarian

strategy fails to outperform the market portfolio consistently. However, when the trading strategy

is conditional on the lagged , it manages to robustly outperform both EW and VW market

portfolios. For instance, the contrarian strategy conditional on , 0.2 outperforms the

VW market portfolio by 21.6 base points per day if the universe of stock selection is the entire

sample. Moreover, it should be emphasized that all findings reported here and below have taken

the transaction costs of bid-ask spreads into account, as they are the consequence of buys at asks

and sells at bids.

We secondly consider a momentum trading strategy for each individual stock that buys

(short sells) a share at the opening ask (bid) and sells (covers) it at the closing bid (ask) if the

21 In robustness checks, we use lagged s estimated based on different estimation windows and obtain quantitatively similar results.

32

previous day’s open-to-close return is positive (negative). It is implemented unconditionally or

conditional on lagged . We calculate the average daily return from this strategy for each

invested stock and report its means across the entire sample and the three subsamples in Panel B

of Table VI. Apparently, no significantly positive returns in all the cases considered except in the

medium subsample.

The profitability of the contrarian trading strategy is driven by the persistence of

disagreement trading. In particular, the mean autocorrelation in over the entire sample

stocks is 0.460 as reported in Panel B of Table I. Thus, disagreement trading tends to continue

itself and lead to return reversal. However, the serial correlation in privately-informed trading is

relatively low and its mean over the entire sample stocks is 0.153. This indicates that private

information is short-lived and the privately-informed trading is less persistent. Moreover, without

information-based trading (i.e., with trading solely driven by liquidity needs), returns are

negatively autocorrelated. Subsequently the momentum trading strategy conditional on lagged

cannot consistently yield significant profits. As demonstrated in the regression analysis in

Subsection 3.1, trades from privately-informed investors increase price efficiency and prices on

consecutive days with these trades more strongly resembling a random walk, which is supported

by the non-profitability of the momentum trading found here. However, consecutive trading

triggered by investor disagreement drives returns deviating from a random walk, as further revealed

by the profitability of the contrarian trading strategy here.

4.2 Further examination of the contrarian trading strategy

Avramov, Choria and Goyal (2006) show that the profitability of a contrarian trading

strategy is linked to the portfolio’s liquidity because they find the largest potential profits occur in

low liquidity stocks. To examine whether the profitability of our contrarian strategies are driven

by the relationship between return reversal and stock illiquidity, we sort the sample stocks into

33

tertiles based on the stock’s illiquidity, which is proxied by the daily Amihud (2002) measure

averaged over the sample period. Panel A of Table VII documents the mean profits of contrarian

trading strategy across the three illiquidity-stratified subsamples. They confirm that the contrarian

trading strategy conditional on lagged yields a significantly positive return across the three

illiquidity-stratified subsamples. Regardless of the stock’s illiquidity, returns are more likely to

reverse themselves on days with more disagreement trading on the previous day, as evidenced by

the profitability that monotonically increases if the strategy is implemented with a larger lagged

. Nevertheless, the high illiquidity subsample yields the highest profitability, which is

consistent with Avramov, Choria and Goyal (2006).

INSERT TABLE VII HERE

In order to control for both firm size and illiquidity effects, we consider three size-stratified

subsamples and sort the stocks in each subsample into tertiles based on the stock’s illiquidity. Panel

B of Table VII reports the mean profits of contrarian trading strategy conditional on positive lagged

across the nine groups. They are all significantly positive at the 1% level.22 In sum,

implementing the contrarian investment strategy with consideration of lagged

systematically outperforms the implementation without consideration of lagged for all

three return measures reported in Table VII. It does not matter whether stock illiquidity and firm

size are further take into account or otherwise.

Liquidity providers are compensated by price concessions for their services (Kyle (1985),

and Grossman and Miller (1988)). When they unwind their net positions, the excess of price

concessions may intensify a negative autocorrelation in returns. So and Wang (2014) establish the

connection between variation in short-term return reversals and change in liquidity provision

22 The profits remain significantly positive across the nine groups, if we implement the contrarian trading strategy conditional on a larger value of lagged .

34

around earnings announcements. In order to examine whether the profitability of contrarian trading

strategy is driven by the time-variation in a stock’s illiquidity, we classify trading days of a stock

according to whether the change in illiquidity on the previous day is positive or not. Panel A of

Table VIII reports the mean profits of contrarian trading strategy from those two kinds of days,

respectively, where ∆ , , , denotes the change in illiquidity

measure on day 1. It illustrates that the contrarian trading strategy yields significant positive

returns on both two kinds of days if it is implemented with consideration of lagged . All

three return measures in the panel monotonically increase in lagged , no matter if the change

in lagged illiquidity is positive or negative.

Short sales are costly because of direct costs such as borrowing fees of stock loans and

indirect costs such as risk of a short position. Legal and institutional restrictions may also prohibit

some investors from selling short. These short-sale constraints can lead stocks to be overpriced

and in turn make short selling more profitable (Jones and Lamont (2002)). Avramov, Choria and

Goyal (2006) find that return reversals are mainly confined to the loser stocks (i.e., stocks that have

negative returns in the previous period). For this reason, Panel B of Table VIII separately reports

the mean profits of contrarian trading strategy from long positions and short positions. It

demonstrates that both long and short positions are significantly profitable although profits from

selling short are generally greater. In other words, the profitability of the contrarian trading strategy

does not solely rely on the short positions if lagged is taken into account.

INSERT TABLE VIII HERE

Finally, we consider a long-short contrarian portfolio, which longs the previous day’s losers

but shorts the previous day’s winners and has zero net investment, where a loser (winner) earns a

negative (positive) open-to-close return. The portfolio is constructed when the market opens with

stocks assigned to a long or short position with equal-weighting. The portfolio is held until the

35

market closes when it is liquidated. If we implement this strategy by selecting stocks from the

entire sample, the 2-year cumulative raw, EW-adjusted and VW-adjusted returns are −1.234%,

−5.516%, and −13.572%, respectively, as shown in Panel A in Table IX. However, if stock

selection is conditional on their lagged being positive these figures turn to 37.166%,

30.952% and 19.899%, respectively. As the panel documented, a more stringent conditioning

standard on lagged leads to an even greater return. Figure 1 displays the cumulative raw

returns from four portfolios, where stocks are selected without consideration of lagged or

requiring lagged greater than zero or 0.1 or 0.2. The EW-adjusted and VW-adjusted

counterparts are shown in Figures 2 and 3. The profitability of the long-short contrarian strategy

demonstrates that both the statistical and economic significance of the negative return

autocorrelation are enforced by disagreement-driven trading.

INSERT TABLE IX and FIGURES 1-3 HERE

To examine which stocks are more likely to be selected for the long-short contrarian

portfolios, we investigate the relationship between the characteristics of the stocks and the

frequency they are being chosen for the portfolio, in the scenario where the selection criterion

requires lagged to be positive. Specifically, for each stock in the sample we calculate the

frequency of trading days that the stock is selected for investment. We then rank stocks into

quintiles by this frequency from (stocks least often in the portfolio) to (stocks most often in

the portfolio). For each selected stock we calculate its average market capitalization, average daily

turnover, standard deviation of daily open-to-close returns, and average number of analysts

following it over the sample period. Panel B of Table IX reports the mean stock characteristics of

each quintile. On average, the stocks in ( are selected for investment for 10% (39.95%) of

trading days. Smaller firms are more likely to be selected, as evidenced by the average market

capitalization showing a general decline in the frequency of appearance in the portfolio, except for

36

the least often quintile. On the other hand, average trading turnover and number of analysts

following are concave in the frequency of stock selection, i.e., the extreme quintiles have smaller

turnovers and fewer analysts following, while average return standard deviation is convex in the

frequency of stock selection, i.e., stocks in the extreme quintiles are more volatile.

V. Concluding Remarks

This paper proposes an information-based trading explanation for the time-varying

autocorrelation in individual stock returns. Our empirical evidence shows that private information

and investor disagreement have opposite effects on return autocorrelation. Continuous privately-

informed trading is associated with return continuation, while continuous disagreement-driven

trading is associated with return reversal. Thus, the autocorrelation of a stock’s return reflects the

combined effects of these two types of information-based trading. A number of further tests

ascertain that the new findings are robust. These findings also suggest that privately-informed

trading is likely to increase price efficiency while trading due to dispersion in investors’ beliefs

may reduce price efficiency.

Moreover, the predictive evidence indicates that the relationship between current return and

lagged return can be strengthened considerably if lagged proxy for disagreement-driven trading is

accounted for. It is confirmed that the implementation of a contrarian trading strategy conditional

on lagged measures of disagreement-driven trading yields economically and statistically significant

excess returns.

37

Appendix: Hidden Markov Model Approach

A.1 Hidden Markov Model

Yin and Zhao (2015) adopt a Hidden Markov Model (HMM) to link the observed trading

data to the unobservable information state of the market of a risky asset. The hidden market state

reflects whether private information events and/or events triggering investor disagreement occur

or not, and if they occur, how intense they are. More specifically, the state is uniquely associated

with the distributions of the numbers of buyer-initiated order flows and seller-initiated order flows.

It can be characterized by the expected numbers of buy and sell orders over one unit of time period,

say one day. In turn, the random numbers of buy orders and sell orders on a particular trading day

follow the mixtures of state-dependent distributions across these market states. Therefore, each

trading day is associated with a probability distribution of it being at these market states and the

evolution of the daily state probability distribution portrays the trading process of the risky asset.

Formally, the HMM consists of two parts: firstly, a two-dimensional unobservable

stochastic process of state ≡ ; , ; : 1, … , , satisfying the Markov properties; and

secondly, a bivariate state-dependent trading process ≡ , : 1,⋯ , . In this model,

is the time horizon being considered, indicates the hidden state of market at time , and

and represent the observable time series of buyer-initiated and seller-initiated trades,

respectively. A two-dimensional state vector ≡ ; , ; is adopted for the convenience of

separately characterizing buy and sell states. In other words, varies over the two-dimensional

state space , ( 1, 2, … , and 1, 2, … , with a time-varying probability distribution,

where m and n are the ranges of the two components of hidden state. The Markov property of the

processes refers to the memoryless property that the distribution of depends on only the first-

lagged state and the distribution of depends only on the current state , i.e.,

Pr Pr | and Pr , Pr | ,

38

where ≡ , , … , and ≡ , , … , . Although the Markov property implies

that conditioning on the history of the process up to time t is equivalent to conditioning only on the

most recent value of , a dependence structure exists in the evolution of hidden states, which can

be described by the transition matrix of the Markov chain:

Γ

γ , ; , γ , ; ,γ , ; , γ , ; ,

⋯γ , ; , γ , ; ,γ , ; , γ , ; ,

⋮ ⋱ ⋮γ , ; , γ , ; ,γ , ; , γ , ; ,

⋯γ , ; , γ , ; ,γ , ; , γ , ; ,

,

where

γ , ; , ≡ Pr ; , ; ; , ;

is the probability that the state is , at time 1 conditional on it being , at time . The

unconditional probability of the hidden state being in state , at time t, , ; ≡ Pr ;

, ; , is a key variable of any HMM. Denoting these probabilities by the row vector

≡ , ; , , ; , … , , ; , … , , ; , … , , ; , , ; ,

one can deduce the distribution of states at time 1 from its distribution at time t by Γ.

In the literature it is usually assumed that buy and sell order flows arrive at the market over

a time period according to a bivariate independent Poisson process (Easley, Hvidkjaer and O’Hara

(2002), Duarte and Young (2009), and Roşu (2009)). Yin and Zhao (2015) make a less restrictive

assumption by assuming that for any state , buy and sell order flows arrive at the market

according to a bivariate independent Poisson process.23 Because the independence of buy and sell

order arrivals is assumed under a given state, the HMM can accommodate a contemporaneous

correlation between the numbers of buy and sell orders and serial correlations in buy orders and in

sell orders through the interaction between and the evolution of states. Thus, if on trading day

23 The Poisson distribution is a one-parameter distribution with its mean (arrival rate) equal to variance. We will use order arrival rate and the expected number of orders interchangeably.

39

the market is at state , , the probability of observing buy orders and sell orders,

Pr | ; , ; , is equal to , where

;λ ;

!and ;

λ ;

!,

and λ ; and λ ; in the above expressions are the arrival rates of buys and sells, respectively, when

buy state is i and sell state is j. The marginal distribution of observing , on day t can

be calculated by

Pr Pr | ; , ; Pr ; , ; ,

where -diagonal matrix is defined by

≡0

⋱0

and ≡1⋮1

.

The parameters of the model include the initial distribution of states , transition matrix Γ and

order arrival rates λ ; and λ ; ( 1, 2, … , , 1, 2, … , . They can be estimated by

maximizing the following likelihood function through Expectation and Maximization Algorithm:

Γ Γ ⋯Γ . (A1)

The numbers of buy and sell states, m and n, are determined in model selection according to an

information criterion, such as Akaike Information Criterion (AIC) or Bayesian Information

Criterion (BIC).

A.2 Estimation of the HMM

Denote η the vector of forward probabilities, whose 1 -th element is

η , ; Pr , ; , ; η , ; γ , ; , , .

40

It can be rewritten in a matrix form of η η Γ with η . Therefore, the

likelihood function in (A1) can be computed recursively in terms of the forward probabilities as

follows:

Γ Γ Γ ⋯Γ .

The Baum-Welch algorithm (see Baum, Petrie, Soules and Weiss (1970)) exploits the fact

that the Complete-Data Log-Likelihood (CDLL) can be directly applied to maximization even if

the likelihood of the observed data cannot be applied. In the current case, the hidden states are

regarded as missing data while the CDLL is the log-likelihood of parameter set based on

observed time series of buy and sell order flows and the unobservable time series of states, i.e.,

log Pr , | , where is a time series realization of state variable

with t ranging from 1 to T. To apply the Expectation and Maximization (EM) algorithm, define

ζ Γ ζ as the vector of backward probabilities for 1, 2, … , 1 with ζ ′,

where the 1 -th element of is

ζ , ; Pr , , … , | ; , ; .

Further, let , ; and , ; , ; be zero-one variables that

, ; 1ifandonlyif ; , ; ,

, ; , ; 1ifandonlyif ; , ; , ; , ; .

With this notation, the CDLL of the HMM is given by

log Pr ,

, ; log , ; , ; , ; log , ; , , ; log ,

,,

.

The EM algorithm for estimating the HMM involves the following two steps:

E Step: Compute the conditional expectations of the missing data, given the observations

and the current estimate of . Specifically, conditional expectations of , ; and

, ; , ; are estimated by:

41

, ; Pr ; , ; ,η , ; ζ . ;

|,

, ; , ; Pr ; , ; , ; , ; | ,η , ; , ; , , ζ , ;

|.

M Step: Maximize the CDLL, where the missing data are replaced by their conditional

expectations, to determine the estimate of . Thus, all , ; and , ; , ; in CDLL are

replaced by their conditional means , ; and , ; , ; , and CDLL is maximized with respect

to , Γ, and λ ; and λ ; . The solution to the maximization problem consists of

, ; , ; , , ; ,∑ , ; , ;

∑ ∑ ∑ , ; , ;,

,

∑ ∑ , ;∑ ∑ , ;

and ,∑ ∑ , ;∑ ∑ , ;

.

The above E and M steps are repeated many times until some convergence criterion has been

satisfied, for instance the improvement in the CDLL is less than 10 . This EM algorithm provides

three sets of parameter estimates: , Γ, and λ ; and λ ; . Once and Γ are estimated, can be

obtained through applying Γ .

Applying Bayes’ rule, the posterior distribution of states can be calculated by

Pr ; , ;

η , ; ζ , ;|

, ; . (A2)

When implementing the EM algorithm for estimating the HMM, it is relatively convenient

to find plausible starting values for the initial distribution of states and the transition matrix. One

strategy is to assign a uniform starting value to all elements of the initial state distribution and the

transition matrix. If the number of states is , we assign ′ and Γ , where is the

matrix of size with all elements equal to 1. In order to improve the convergence speed, we

run an -means clustering on the observed buys and sells and then use the centers of the clusters

as the initial starting values of the state-dependent order arrival rates.

42

A.3 Estimation of the Arrival Rates of Order Flows

After obtaining λ ; , λ ; and Pr ; , ; in the process of

estimating the HMM, Yin and Zhao’s (2015) HMM approach adopts k-means clustering analysis

and the jump method of Sugar and James (2003) to identify the three types of trading for each

market state , :

; ; ; ; and ; ; ; ; .

where ; and ; are arrival rates of privately-informed buy and sell orders when the market is

at state , , ; and ; are disagreement-driven buy and sell order arrival rates, and ; and ;

are the arrival rates of liquidity buy and sell orders. The identification involves two steps.

Step One: Partitioning hidden states to determine the arrival rates of privately-informed

buys and sells, i.e., ; and ; .

Private information leads to one-sided trading and substantial trade imbalance (Kyle (1985),

Easley, Kiefer, O’Hara and Paperman (1996), Sarkar and Schwartz (2009)). Following this insight,

k-means clustering analysis is applied to observed trading imbalances over the whole estimation

window, i.e., | | for 1, 2, … , , and the number of clusters is determined by the jump

method of Sugar and James (2003). If there is only one cluster, the observed trading imbalances

are similar and there is no significant evidence for the existence of private information during the

period. Therefore, ; ; 0 for all hidden states. The rationale behind such a claim is the

common “wisdom” that trading due to liquidity needs or disagreement among investors is two-

sided and only generates small trade imbalances, while privately-informed trading is often

associated with a substantial trade imbalance. If there is privately-informed trading, the daily trade

imbalances, on average, cannot be consistent and similar over time for the whole sample period.

If the clustering analysis indicates that there are multiple clusters with different centers of

trade imbalances, the clusters with their centers no larger than that of the most frequent cluster are

43

identified as the clusters without privately-informed trading but those with a strictly larger center

are classified as clusters associated with privately-informed trading. The rationale of the

classification once again is that private information induces profound trading imbalances. The

most frequent cluster is chosen to be the cutoff point because it includes states which occur most

often; and it is plausible to assume that the most frequently appearing states involve no private

information. After all, liquidity trading occurs most frequently and privately-informed trading is

less frequent. The simulation by Yin and Zhao (2015) and our sample data show that the most

frequent cluster in trade imbalances always turns out to be the cluster with the smallest center.

Extensive simulation also demonstrates the validity of adopting the most frequent cluster of trade

imbalance for the cutoff point. After clusters of trade imbalances have been determined,

λ ; λ ; are treated as an out-of-sample observation and it is assigned to the cluster whose

center is the closest to it. If λ ; λ ; belongs to a cluster without privately-informed trading,

state , is considered to involve no private-information, i.e., ; ; 0, and is used to

denote the set of such states. If state , does not belong to , it contains private information.

Then,

; λ ; λ ; λ ; # λ ; # and ; 0 if λ ; λ ; ,

; 0 and ; λ ; λ ; λ ; # λ ; # if λ ; λ ; , (A3)

where #, # is a matching state of state , , which is a state in with balanced trades being the

closest to the balanced trades of state , .24 The matching state is used to proxy the small trade

imbalance caused by liquidity and/or disagreement trading in state , .

Step Two: Classifying hidden states to determine the arrival rates of buy and sell orders

driven by disagreement among investors, i.e., ; and ; .

24 Mathematically, #, # ∗, ∗ ∈ λ ; λ ; λ ; λ ; λ ; ∗ λ ; ∗ λ ; ∗ λ ; ∗ .

44

Liquidity trading exists on each trading day, which is two-sided in the sense that their

average numbers of buys and sells are not considerably different. As argued by Duarte and Young

(2009), and Sarkar and Schwartz (2009), if an information event causes dispersion in investors’

beliefs, both buys and sells should increase substantially, leading to a shock to balanced trading.

Thus, a k-means clustering analysis is applied to the observations of balanced trades

| | 1, 2, … , to separate states with investor disagreement from states without

disagreement, and the number of clusters is determined by the jump method of Sugar and James

(2003). If only one cluster exists, it implies that no substantial disagreement among investors and

all two-sided orders are generated by liquidity traders. Therefore, ; ; 0 for all , .

If more than one cluster is detected, following the similar rationale and method of cluster

classification in Step One, the clusters with their centers strictly larger than that of the most frequent

cluster of balanced trades are considered as the ones that are associated with investor disagreement

but clusters with a smaller center are classified as not involving disagreement-driven trading. That

is, disagreement causes sizable balanced trades while liquidity trading, although existing on each

trading day, leads to a relatively small amount of trades. The most frequent cluster is selected for

cutoff point for the reason that liquidity trading is assumed to occur every day and more frequently

than disagreement-driven trading. This selection of the cutoff point is also strongly supported by

extensive simulation analysis. For state , , the expected number of balanced trades λ ; λ ;

λ ; λ ; is taken as an out-of-sample observation and it is assigned to the cluster whose center

is closest to it. Denote the set consisting of the states in the clusters with disagreement and the

remaining states constitute set . If state , belongs to , its arrival rates of disagreement-driven

buys and sells are set to zero, i.e., ; ; 0. If state , belongs to , its expected buys

and sells triggered by investor disagreement are, respectively,

45

; λ ; ; max∗, ∗ ∈ ∩

λ ; ∗ and ; λ ; ; max∗, ∗ ∈ ∩

λ ; ∗ , (A4)

where ; and ; are obtained in the first step. The last terms in the above equations proxy the

expected numbers of liquidity buys and sells ; , and ; , respectively. Set includes both

liquidity trading and privately-informed trading while set includes both liquidity trading and

disagreement-induced trading. Their intersection, i.e., set ∩ , includes states that involve only

liquidity trading. The largest arrival rates of buy and sell orders in ∩ is used to subtract

liquidity order arrival rates from the aggregate buy and sell order arrival rates, to ensure that the

arrival rates of buy and sell orders driven by investor disagreement are not exaggerated.

The arrival rates of liquidity buy orders and sell orders in state , can be derived by

; ; ; ; and ; ; ; ; .

Moreover, the arrival rates of different types of trades on trading day t can be estimated by

; ;,

Pr ; , ; , ; ;,

Pr ; , ; ,

; ;,

Pr ; , ; , ; ;,

Pr ; , ; ,

; ;,

Pr ; , ; , ; ;,

Pr ; , ; ,

(A5)

where the conditional probability of the hidden state, Pr ; , ; , is available after

the estimation of the HMM, as shown by (A2).

46

REFERENCES Amihud, Yakov, 2002. Illiquidity and stock returns: cross-section and time-series effects, Journal

of Financial Markets 5, 31–56. Avramov, Doron, Tarun Chordia and Amit Goyal, 2006. Liquidity and autocorrelations in

individual stock returns, Journal of Finance 61, 2365–2394. Baek, In-Mee, Arindam Bandopadhyaya, and Chan Du, 2005. Determinants of market-assessed

sovereign risk: Economic fundamentals or market risk appetite? Journal of International Money and Finance 24, 533–548.

Ball, Ray, S. P., Kothari, and Jay Shanken, 1995. Problems in measuring portfolio performance: an application to contrarian investment strategies, Journal of Financial Economics 38, 79–107.

Banerjee, Snehal, 2011. Learning from prices and the dispersion in beliefs, Review of Financial Studies 24, 3025–3068.

Banerjee, Snehal, and Ilan Kremer, 2010. Disagreement and learning: dynamic patterns of trade, Journal of Finance 65, 1269–1302.

Barton, Jan, 2001. Does the use of financial derivatives affect earnings management decisions? The Accounting Review 76, 1, 1–26.

Baum, Leonard E., Ted Petrie, George Soules, and Norman Weiss, 1970. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains, Annals of Mathematical Statistics 41, 164–171.

Blume, Marshall E., and Robert F. Stambaugh, 1983. Biases in computed returns: An application to the size effect, Journal of Financial Economics 12, 387–404.

Boehmer, Ekkehart, and Eric K. Kelley, 2009. Institutional investors and the informational efficiency of prices, Review of Financial Studies 22, 3563–3594.

Bushee, Brian J., and Christopher F. Noe, 2000. Corporate disclosure practices, institutional investors, and stock return volatility, Journal of Accounting Research 38, 171–202.

Campbell, John Y., Sanford J. Grossman, and Jiang Wang, 1993. Trading volume and serial correlation in stock returns, Quarterly Journal of Economics 108, 905–939.

Carlin, Bruce I., Francis A. Longstaff, and Kyle Matoba, 2014. Disagreement and asset prices, Journal of Financial Economics 114, 226–238.

Chordia, Tarun, and Avanidhar Subrahmanyam, 2004. Order imbalance and individual stock return: Theory and evidence, Journal of Financial Economics 72, 485–518.

Conrad, Jennifer, Allaudeen Hameed, and Cathy Niden, 1994. Volume and autocovariances in short-horizon individual security returns, Journal of Finance 49, 1305–1329.

Conrad, Jennifer, Gautam Kaul, and M. Nimalendran, 1991. Components of short-horizon individual security returns, Journal of Financial Economics 29, 365-384.

Cooper, Michael, 1999. Filter rules based on price and volume in individual security overreaction, Review of Financial Studies 12, 901–35.

Dieckmann, Stephan, and Thomas Plank, 2012. Default risk of advanced economies: An empirical analysis of credit default swaps during the financial crisis, Review of Finance 16, 903–934.

Duarte, Jefferson, and Lance Young, 2009. Why is PIN priced? Journal of Financial Economics 91, 119–138.

Easley, David, Robert F. Engle, Maureen O’Hara, and Liuren Wu, 2008. Time-varying arrival rates of informed and uninformed trades, Journal of Financial Econometrics 6, 171–207.

Easley, David, Soeren Hvidkjaer, and Maureen O’Hara, 2002. Is information risk a determinant of asset returns? Journal of Finance 57, 2185–2221.

Easley, David, Soeren Hvidkjaer, and Maureen O’Hara, 2010. Factoring information into returns, Journal of Financial and Quantitative Analysis 45, 293–309.

47

Easley, David, Nicholas M. Kiefer, Maureen O’Hara, and Joseph B. Paperman, 1996. Liquidity, information, and infrequently traded stocks, Journal of Finance 51, 1405-1436.

Fama, Eugene F., 1970. Efficient capital markets: A review of theory and empirical work, Journal of Finance 25, 383–417.

Gervais, Simon, Ron Kaniel, and Dan H. Mingelgrin, 2001. The high-volume return premium, Journal of Finance 56, 877–919.

Glosten, Lawrence R., and Paul R. Milgrom, 1985. Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, Journal of Financial Economics 14, 71–100.

Grossman, Sanford J., and Merton H. Miller, 1988. Liquidity and market structure, Journal of Finance 43, 617–633.

Harris, Milton, and Artur Raviv, 1993. Differences of opinion make a horse race, Review of Financial Studies 6, 473-506.

Hendershott, Terrence, and Mark S. Seasholes, 2014. Liquidity provision and stock return predictability, Journal of Banking & Finance 45, 140–151.

Jagadeesh, Narasimhan, 1990. Evidence of predictable behavior of security returns, Journal of Finance 45, 881–898.

Jones, Charles M., and Owen A. Lamont, 2002. Short-sale constraints and stock returns, Journal of Financial Economics 66, 207–239.

Kim, Oliver, and Robert E. Verrecchia, 1994. Market liquidity and volume around earnings announcements, Journal of Accounting and Economics 17, 41-67.

Kyle, Albert S., 1985. Continuous auctions and insider trading, Econometrica 53, 1315–1326. Lee, Charles M.C., and Mark J. Ready, 1991. Inferring trade direction from intraday data, Journal

of Finance 46, 733–747. Lehmann, Bruce N., 1990. Fads, martingales, and market efficiency, Quarterly Journal of

Economics 105, 1–28. Lin, Hsiou-Wei W., and Wen-Chyan Ke, 2011. A computing bias in estimating the probability of

informed trading, Journal of Financial Market 14, 625–640. Llorente, Guillermo, Roni Michaely, Gideon Saar, and Jiang Wang, 2002. Dynamic volume-return

relation of individual stocks, Review of Financial Studies 15, 1005–1047. Newey, Whitney K., and Kenneth D. West, 1994. Automatic lag selection in covariance matrix

estimation, Review of Economic Studies 61, 631–653. Roşu, Ioanid, 2009. A dynamic model of the limit order book, Review of Financial Studies 22,

4601–4641. Saffi, Pedro A.C., and Kari Siguardsson, 2010. Price efficiency and short selling, Review of

Financial Studies 24, 821–852. Sarkar, Asani, and Robert A. Schwartz, 2009. Market sidedness: Insights into motives for trade

initiation, Journal of Finance 64, 375–423. Shalen, Catherine T., 1993. Volume, volatility and the dispersion of beliefs, Review of Financial

Studies 6, 405–434. Sias, Richard W., and Laura T. Starks, 1997. Return autocorrelation and institutional investors,

Journal of Financial Economics 46, 103–131. So, Eric C., and Sean Wang, 2014. News-driven return reversals: Liquidity provision ahead of

earnings announcements, Journal of Financial Economics 114, 20–35. Sugar, Catherine A., and Gareth M. James, 2003. Finding the number of clusters in a dataset,

Journal of the American Statistical Association 98, 750–763. Wang, Jiang, 1994. A model of competitive stock trading volume, Journal of Political Economy

102, 127–168.

48

Yin, Xiangkang, and Jing Zhao, 2015. A hidden Markov model approach to information-based trading: Theory and applications, Journal of Applied Econometrics 30, 1210–1234.

49

Table I Summary statistics of the sample characteristics

The sample in the table includes 1,249 common stocks that traded on NYSE between January 1, 2010 and December 31, 2011. Panel A reports the summary statistics of the sample characteristics, where , ,

and are respectively the averages of daily market capitalization, , and of a stock over the sample period. In Panel B, denotes daily close-to-close return while is daily open-to-close return. Panel B presents the cross-sectional summary statistics of autocorrelations in these two return time series and three time series of information-based trading measures for the entire sample and the three size-based subsamples.

Panel A: Characteristics of the sample stocks

inmil.$ Entire sample Mean 7908.36 0.086 0.085 ‐0.004Median 1960.80 0.079 0.076 ‐0.002Std.Dev. 22120.16 0.042 0.045 0.017Minimum 12.00 0.005 0.001 ‐0.170Maximum 355227.33 0.409 0.268 0.076Small stock Mean 477.10 0.119 0.082 ‐0.011Median 471.41 0.111 0.073 ‐0.007Std.Dev. 273.03 0.044 0.048 0.024Medium stock Mean 2094.80 0.082 0.087 ‐0.001Median 1968.20 0.077 0.077 0.000Std.Dev. 739.88 0.027 0.045 0.012Large stock Mean 21171.03 0.057 0.087 ‐0.001Median 9795.59 0.054 0.078 0.000Std.Dev. 34714.96 0.027 0.044 0.011Panel B: First-order autocorrelation (ACF) in daily time series

Entire sample Mean ‐0.043 ‐0.027 0.106 0.460 0.153Median ‐0.040 ‐0.024 0.092 0.467 0.150Std.Dev. 0.071 0.072 0.090 0.115 0.080Minimum ‐0.314 ‐0.227 ‐0.106 ‐0.004 ‐0.078Maximum 0.156 0.175 0.517 0.798 0.508Small stock Mean ‐0.053 ‐0.039 0.117 0.410 0.179Median ‐0.051 ‐0.032 0.104 0.413 0.179Std.Dev. 0.078 0.074 0.096 0.130 0.073Medium stock Mean ‐0.030 ‐0.012 0.095 0.463 0.151Median ‐0.028 ‐0.011 0.087 0.470 0.146Std.Dev. 0.071 0.071 0.081 0.098 0.076Large stock Mean ‐0.046 ‐0.030 0.106 0.507 0.129Median ‐0.043 ‐0.026 0.091 0.515 0.119Std.Dev. 0.061 0.068 0.090 0.092 0.084

50

Table II Serial correlation in stock returns and measures of information-based trading

In Panel A, the first row reports the means of first-order autocorrelations ( s) in two return series over the whole sample period. For stock i, consecutive trading days 1 and are selected according to the information-based trading measures, and , on these days. Then the s of the two return series on these trading days are calculated. The cross-sectional means of these s are reported in rows (2)-(9). The lower segment of the panel shows the difference between s presented in the corresponding row. For instance, denotes the difference between return s of days satisfying the conditions in rows (2) and (3). Panel B reports the s of

and when trading days are sorted according to consecutive returns. Panels C and D are the counterparts of Panels A and B, respectively, for the three size-based subsamples. Symbols ***, ** and * indicate significance at the 1%, 5%, 10% level, respectively.

Panel A: Serial correlation in stock returns on sorted days

(1) All days ‐0.043*** ‐0.027***(2) Days with , , 0 0.014*** 0.032***(3) Days with , , 0 ‐0.093*** ‐0.062***(4) Days with , , 0 ‐0.180*** ‐0.150***(5) Days with , , 0 ‐0.024*** ‐0.019***(6) Days with , , 0 and , , 0 0.068*** 0.069***(7) Days with , , 0 and , , 0 ‐0.073*** ‐0.042***(8) Days with , , 0 and , , 0 ‐0.128*** ‐0.080***(9) Days with , , 0 and , , 0 ‐0.172*** ‐0.184***Difference between ACFs over two kinds of days:

0.107*** 0.094*** ‐0.156*** ‐0.130*** 0.142*** 0.111*** 0.043** 0.104*** ‐0.197*** ‐0.149*** ‐0.100*** ‐0.142***

Panel B: Serial correlation in information-based trading measures on sorted days

(I) All days 0.153*** 0.460***(II) Days with , , 0 0.229*** 0.439***(III) Days with , , 0 0.061*** 0.496***(IV) Days with , , 0 0.234*** 0.439***(V) Days with , , 0 0.063*** ‐0.488***Difference between ACFs over two kinds of days:

0.168*** ‐0.057*** 0.171*** ‐0.050***

51

Panel C: Serial correlation in stock returns on sorted days for the three size-based subsamples

Close-to-close return Open-to-close return

Small stock Medium

stock Large stock Small stock

Medium stock

Large stock

‐0.053*** ‐0.030*** ‐0.046*** ‐0.039*** ‐0.012*** ‐0.030*** 0.031*** 0.028*** ‐0.018** 0.047*** 0.056*** ‐0.008 ‐0.132*** ‐0.070*** ‐0.080*** ‐0.081*** ‐0.039*** ‐0.067*** ‐0.178*** ‐0.175*** ‐0.189*** ‐0.149*** ‐0.133*** ‐0.167*** ‐0.043*** ‐0.009** ‐0.021*** ‐0.034*** ‐0.011** ‐0.013*** 0.070*** 0.081*** 0.054*** 0.067*** 0.089*** 0.053*** ‐0.109*** ‐0.049*** ‐0.061*** ‐0.074*** ‐0.016 ‐0.036*** ‐0.086*** ‐0.117*** ‐0.182*** ‐0.034** ‐0.053*** ‐0.153***) ‐0.185*** ‐0.165*** ‐0.167*** ‐0.184*** ‐0.203*** ‐0.166***

ACF(2)−ACF(3) 0.163*** 0.098*** 0.062*** 0.129*** 0.096*** 0.059***ACF(4)−ACF(5) ‐0.135*** ‐0.166*** ‐0.168*** ‐0.114*** ‐0.122*** ‐0.154***ACF(6)−ACF(7) 0.179*** 0.130*** 0.115*** 0.141*** 0.104*** 0.089***ACF(8)−ACF(9) 0.099** 0.048 ‐0.015 0.150*** 0.151*** 0.014ACF(8)−ACF(6) ‐0.156*** ‐0.198*** ‐0.236*** ‐0.101** ‐0.141*** ‐0.205***ACF(9)−ACF(7) ‐0.076** ‐0.116*** ‐0.106*** ‐0.110** ‐0.188*** ‐0.130***Panel D: Serial correlation in information-based trading measures on sorted days for the three size-based subsamples

Small stock Medium

stock Large stock Small stock

Medium stock

Large stock

0.179*** 0.151*** 0.129*** 0.410*** 0.463*** 0.507*** 0.266*** 0.228*** 0.193*** 0.419*** 0.430*** 0.468*** 0.057*** 0.062*** 0.064*** 0.448*** 0.491*** 0.548*** 0.263*** 0.245*** 0.193*** 0.400*** 0.443*** 0.473***

0.067*** 0.053*** 0.069*** 0.434*** 0.484*** 0.547***ACF(II)−ACF(III) 0.209*** 0.166*** 0.129*** ‐0.030*** ‐0.061*** ‐0.080***ACF(IV)−ACF(V) 0.197*** 0.192*** 0.124*** ‐0.034*** ‐0.042*** ‐0.074***

52

Table III Panel regression analysis of return autocorrelation

This table contains the results of panel data regressions that examine the dynamic relationship between return autocorrelation and measures of information-based trading for the entire sample stocks. In each regression, the dependent variable is daily return of stock i on day , , , and the explanatory variables are listed in the table. We consider two return series, close-to-close return and open-to-close return. All regressions include firm fixed effects (not reported). In Panel A, lagged return alone serves as the main explanatory variable. In Panel B, dummy variables are used to distinguish days with continuous information-based trading from other days, where takes value 1 if the condition is satisfied and 0 otherwise. P-values of hypothesis tests on regression coefficients are also reported. Regressions in Panel C allow return autocorrelation to change with the intensity of information-based trading. T-statistics, computed based on the standard errors clustered by firm, are shown in parentheses. Symbols ***, ** and * indicate significance at the 1%, 5%, 10% level, respectively.

Panel A: Time-invariant serial correlation in daily returns

Explanatory variable (coefficient) Close-to-close return Open-to-close return

, ‐0.031*** ‐0.017***  ‐10.268 ‐6.466 Adj. 0.09% 0.03% 0.09% 0.03%

Panel B: Autocorrelation in daily returns conditional on the existence of information-based trading on consecutive days

Explanatory variable (coefficient) Close-to-close return Open-to-close return

1 2 3 4 5 6

, ‐0.127*** ‐0.012*** ‐0.089*** ‐0.125*** ‐0.011** ‐0.097***  ‐20.970 ‐2.812 ‐14.668 ‐21.733 ‐2.484 ‐16.935

, , , 0.157*** 0.160*** 0.176*** 0.178***

20.168 20.429 22.363 22.745, , ,

‐0.112*** ‐0.118*** ‐0.089*** ‐0.093*** ‐12.195 ‐12.401 ‐9.010 ‐9.342Adj. 0.87% 0.54% 1.18% 0.92% 0.31% 1.10% 0.87% 0.54% 1.19% 0.92% 0.31% 1.10%

Hypothesis testing on regression coefficients 0.030 0.072 0.051 0.081

[p-value] 0.000 0.000 0.000 0.000 ‐0.125 ‐0.207 ‐0.100 ‐0.190

[p-value] 0.000 0.000 0.000 0.000Panel C: Autocorrelation in daily returns conditional on the intensity of information-based trading on consecutive days

Explanatory variable (coefficient) Close-to-close return Open-to-close return

1 2 3 4 5 6, ‐0.070*** ‐0.019*** ‐0.040*** ‐0.060*** ‐0.019*** ‐0.043***

  ‐14.048 ‐4.665 ‐8.115 ‐12.032 ‐4.301 ‐8.051, ,

, , ,

2.753***4.754

2.571***4.613

2.479***5.009

2.389***4.931

, ,

, , ,

‐0.309***‐9.746

‐0.285***‐8.754

‐0.213***‐5.783

‐0.183***‐4.879

Adj. 0.67% 0.61% 0.96% 0.66% 0.29% 0.77% 0.67% 0.61% 0.96% 0.67% 0.29% 0.77%

53

Table IV Size-stratified results of panel regressions of return autocorrelation

This table contains the results of panel data regressions that examine the dynamic relationship between return autocorrelation and measures of information-based trading for the three size-based subsamples. Dependent variable is , , which is close-to-close return or open-to-close return. All regressions include firm fixed effects (not reported). Panel A considers lagged return as the only main explanatory variable. Panel B uses dummy variables to distinguish days with continuous information-based trading from other days. P-values of hypothesis tests on regression coefficients are also reported. Panel C allows return autocorrelation to change with the degree of information-based trading. T-statistics, computed based on the standard errors clustered by firm, are shown in parentheses. Symbols ***, ** and * indicate significance at the 1%, 5%, 10% level, respectively.

Panel A: Time-invariant serial autocorrelation in daily returns

Explanatory variable (coefficient) Close-to-close return Open-to-close return

Small stockMedium

stock Large stock Small stock

Medium stock

Large stock

, ‐0.037*** ‐0.017*** ‐0.035*** ‐0.024*** 0.000 ‐0.021***  ‐7.502 ‐3.394 ‐10.186 ‐5.831 0.056 ‐5.530

Adj. 0.13% 0.03% 0.12% 0.05% 0.00% 0.05% 0.13% 0.03% 0.12% 0.05% 0.00% 0.05%

Panel B: Autocorrelation in daily returns conditional on the existence of information-based trading on consecutive days

Explanatory variable (coefficient) Close-to-close return Open-to-close return

1 2 3 4 5 6 Small stock

, ‐0.151*** ‐0.017** ‐0.113*** ‐0.140*** ‐0.013* ‐0.114***  ‐12.514 ‐3.900 ‐10.601 ‐15.841 ‐1.917 ‐12.850

, , , 0.190*** 0.192*** 0.199*** 0.201***

15.679 16.444 17.663 17.845, , ,

‐0.107*** ‐0.110*** ‐0.085*** ‐0.091*** ‐3.801 ‐4.491 ‐5.321 ‐5.583

Adj. 1.21% 0.56% 1.48% 1.14% 0.30% 1.31% 1.21% 0.56% 1.49% 1.14% 0.30% 1.31%

Hypothesis testing on regression coefficients 0.039 0.079 0.059 0.087

[p-value] 0.000 0.000 0.000 0.000 ‐0.125 ‐0.223 ‐0.098 ‐0.205

[p-value] 0.000 0.000 0.000 0.000Medium stock

, ‐0.107*** ‐0.001 ‐0.069*** ‐0.113*** ‐0.003 ‐0.086***  ‐13.318 ‐0.786 ‐8.894 ‐12.397 ‐0.550 ‐9.563

, , , 0.138*** 0.143*** 0.173*** 0.174***

11.974 12.917 13.717 13.896, , ,

‐0.116*** ‐0.123*** ‐0.089*** ‐0.090*** ‐12.899 ‐12.291 ‐8.683 ‐8.857

Adj. 0.63% 0.45% 0.96% 0.84% 0.26% 1.01% 0.63% 0.46% 0.97% 0.84% 0.26% 1.01%

Hypothesis testing on regression coefficient 0.031 0.074 0.060 0.087

[p-value] 0.000 0.000 0.000 0.000b b ‐0.117 ‐0.192 ‐0.092 ‐0.176[p-value] 0.000 0.000 0.000 0.000

54

Close-to-close return Open-to-close return Explanatory variable (coefficient) 1 2 3 4 5 6 Large stock

, ‐0.100*** ‐0.017*** ‐0.060*** ‐0.089*** ‐0.015*** ‐0.057***  ‐10.919 ‐3.936 ‐6.499 ‐10.531 ‐2.635 ‐6.341

, , , 0.096*** 0.101*** 0.092*** 0.095***

5.854 6.203 7.429 7.693, , ,

‐0.122*** ‐0.126*** ‐0.103*** ‐0.107*** ‐8.104 ‐8.498 ‐8.445 ‐8.645

Adj. 0.56% 0.66% 0.91% 0.44% 0.46% 0.69% 0.56% 0.66% 0.91% 0.44% 0.46% 0.69%

Hypothesis testing on regression coefficient ‐0.004 0.040 0.002 0.038

[p-value] 0.619 0.000 0.778 0.000b b ‐0.139 ‐0.186 ‐0.119 ‐0.164[p-value] 0.000 0.000 0.000 0.000

Panel C: Autocorrelation in daily returns conditional on the intensity of information-based trading on consecutive days

Close-to-close return Open-to-close return Explanatory variable (coefficient) 1 2 3 4 5 6 Small stock

, ‐0.081*** ‐0.028*** ‐0.057*** ‐0.067*** ‐0.022*** ‐0.053***  ‐9.476 ‐3.844 ‐6.763 ‐9.078 ‐3.207 ‐6.639

, ,

, , ,

2.676***3.953

2.540***3.860

2.317***4.594

2.250***4.530

, ,

, , ,

‐0.258***‐4.697

‐0.225***‐3.984

‐0.185***‐3.305

‐0.148***‐2.595

Adj. 0.90% 0.56% 1.10% 0.83% 0.27% 0.91% 0.91% 0.56% 1.10% 0.83% 0.27% 0.91%

Medium stock , ‐0.055*** ‐0.007 ‐0.024*** ‐0.055*** ‐0.014** ‐0.040***

  ‐9.804 ‐1.346 ‐3.857 ‐6.324 ‐2.378 ‐4.527, ,

, , ,

3.266***4.123

3.002***4.041

4.828***3.517

4.680***3.479

, ,

, , ,

‐0.306***‐9.745

‐0.286***‐9.018

‐0.193***‐4.779

‐0.160***‐3.928

Adj. 0.44% 0.51% 0.75% 0.65% 0.21% 0.73% 0.44% 0.51% 0.75% 0.66% 0.21% 0.74%

Large stock , ‐0.068*** ‐0.009* ‐0.020*** ‐0.056*** ‐0.014** ‐0.022***

  ‐12.667 ‐1.897 ‐3.499 ‐8.327 ‐2.255 ‐3.086, ,

, , ,

4.119***4.149

3.627***3.999

3.056**2.271

2.727**2.171

, ,

, , ,

‐0.528***‐13.677

‐0.515***‐13.205

‐0.385***‐8.785

‐0.374***‐8.437

Adj. 0.50% 1.09% 1.22% 0.36% 0.65% 0.75% 0.50% 1.09% 1.22% 0.36% 0.65% 0.76%

55

Table V Time series regressions of return autocorrelation for individual stocks

Panel A of the table consists of time series regressions of a stock’s return against its lagged return, , while Panels B and C are comprised of time series regressions of the following models, respectively,

,

. Regression coefficients and Newey-West (1987) adjusted t-statistics (in parentheses) are averaged over the entire sample or over one of the three subsamples. The table also displays the percentage of sample stocks with the regression coefficient being positive coeff. 0 , significantly positive or negative at the 10% level (coeff.* >0 or coeff.* <0), average adjusted , and average standardized coefficient of the individual regressions across the entire sample and the three size-based subsamples.

Panel A: Time-invariant serial autocorrelation in daily returns

Close-to-close return Open-to-close return

Entire sample

Small stock

Medium stock

Large stock

Entire sample

Small stock

Medium stock

Large stock

Averagecoeff. ‐0.043 ‐0.053 ‐0.029 ‐0.046 ‐0.027 ‐0.038 ‐0.012 ‐0.029

Averaget‐stat. ‐0.688 ‐0.901 ‐0.433 ‐0.729 ‐0.453 ‐0.689 ‐0.170 ‐

0.500% coeff. 0 27.78% 25.18% 33.41% 24.76% 36.51% 31.41% 44.71% 33.41%% coeff.* 0 2.64% 3.60% 3.61% 0.72% 5.60% 4.32% 8.65% 3.85%% coeff.* 0 20.82% 31.65% 13.94% 16.83% 18.41% 24.46% 11.54% 19.23%Averageadj. 0.49% 0.69% 0.39% 0.38% 0.39% 0.50% 0.32% 0.35%Average 0.69% 0.89% 0.59% 0.58% 0.59% 0.70% 0.52% 0.55%Panel B: Autocorrelation in daily returns conditional on the existence of information-based trading on consecutive days Close-to-close return Open-to-close return 1 2 3 4 5 6 Entire sample Averagecoeff. ‐0.140 ‐0.024 ‐0.104 ‐0.134 ‐0.019 ‐0.104 Averaget‐statistics ‐1.848 ‐0.439 ‐1.411 ‐1.763 ‐0.349 ‐1.374% coeff. 0 8.44% 39.02% 15.95% 8.44% 39.02% 15.95%% coeff.* 0 0.36% 5.00% 1.43% 0.75% 4.50% 1.31%% coeff.* 0 55.36% 16.07% 41.25% 52.91% 14.82% 41.84% Averagestandardizedcoeff. ‐0.140 ‐0.024 ‐0.104 ‐0.134 ‐0.019 ‐0.104Averagecoeff. 0.156 0.165 0.167 0.174 Averaget‐statistics 1.451 1.560 1.535 1.592% coeff. 0 87.05% 88.37% 87.05% 88.37%% coeff.* 0 43.21% 46.79% 45.40% 48.41%% coeff.* 0 1.43% 1.61% 0.75% 0.75% Averagestandardizedcoeff. 0.108 0.115 0.117 0.122Averagecoeff. ‐0.129 ‐0.133 ‐0.119 ‐0.123 Averaget‐statistics ‐0.957 ‐0.997 ‐0.902 ‐0.941% coeff. 0 22.89% 20.83% 22.89% 20.83%% coeff.* 0 0.18% 0.18% 0.19% 0.38%% coeff.* 0 19.11% 23.21% 20.45% 23.45% Averagestandardizedcoeff. ‐0.071 ‐0.073 ‐0.059 ‐0.061Averageadj. 1.64% 1.09% 2.08% 1.57% 0.98% 2.00%Average 2.03% 1.49% 2.67% 1.97% 1.38% 2.60%

56

Close-to-close return Open-to-close return 1 2 3 4 5 6Small stock Averagecoeff. ‐0.183 ‐0.043 ‐0.155 ‐0.167 ‐0.035 ‐0.146 Averaget‐statistics ‐2.357 ‐0.758 ‐2.003 ‐2.162 ‐0.615 ‐1.883% coeff. 0 3.74% 30.48% 7.49% 5.62% 35.39% 7.30%% coeff.* 0 0.00% 3.74% 0.53% 0.56% 2.25% 0.56%% coeff.* 0 66.84% 24.60% 57.75% 63.48% 22.47% 54.49% Averagestandardizedcoeff. ‐0.183 ‐0.043 ‐0.155 ‐0.167 ‐0.035 ‐0.146Averagecoeff. 0.216 0.222 0.216 0.222 Averaget‐statistics 1.973 2.075 2.000 2.040% coeff. 0 96.26% 95.72% 96.63% 96.07%% coeff.* 0 59.89% 63.10% 58.99% 60.11%% coeff.* 0 0.53% 0.00% 0.56% 0.00% Averagestandardizedcoeff. 0.152 0.158 0.153 0.157Averagecoeff. ‐0.100 ‐0.104 ‐0.081 ‐0.087 Averaget‐statistics ‐0.756 ‐0.789 ‐0.650 ‐0.695% coeff. 0 20.86% 24.06% 29.78% 26.40%% coeff.* 0 0.53% 0.53% 0.56% 1.12%% coeff.* 0 17.65% 20.32% 13.48% 16.29% Averagestandardizedcoeff.   ‐0.054 ‐0.055 ‐0.041 ‐0.044Averageadj. 2.33% 1.21% 2.68% 2.03% 0.92% 2.32%Average 2.72% 1.61% 3.27% 2.43% 1.32% 2.91%Medium stock Averagecoeff. ‐0.126 ‐0.008 ‐0.087 ‐0.136 ‐0.011 ‐0.107 Averaget‐statistics ‐1.619 ‐0.132 ‐1.176 ‐1.751 ‐0.195 ‐1.394% coeff. 0 11.29% 47.31% 20.97% 9.04% 42.37% 15.82%% coeff.* 0 0.54% 6.99% 1.61% 1.13% 7.91% 2.82%% coeff.* 0 46.24% 9.68% 31.72% 54.24% 10.73% 45.20% Averagestandardizedcoeff. ‐0.126 ‐0.008 ‐0.087 ‐0.136 ‐0.011 ‐0.107Averagecoeff. 0.154 0.165 0.193 0.197 Averaget‐statistics 1.413 1.544 1.711 1.758% coeff. 0 81.18% 84.95% 89.27% 88.70%% coeff.* 0 41.94% 44.09% 51.41% 55.37%% coeff.* 0 1.08% 2.15% 0.56% 0.56% Averagestandardizedcoeff. 0.107 0.115 0.134 0.138Averagecoeff. ‐0.137 ‐0.143 ‐0.106 ‐0.109 Averaget‐statistics ‐1.044 ‐1.086 ‐0.782 ‐0.810% coeff. 0 10.22% 11.29% 22.03% 19.77%% coeff.* 0 0.00% 0.00% 0.00% 0.00%% coeff.* 0 19.89% 24.73% 19.21% 18.64% Averagestandardizedcoeff.   ‐0.076 ‐0.079 ‐0.057 ‐0.058Averageadj. 1.41% 0.94% 1.89% 1.67% 0.81% 2.04%Average 1.80% 1.34% 2.48% 2.07% 1.21% 2.63%

57

Close‐to‐closereturn Open‐to‐closereturn 1 2 3 4 5 6Large stock Averagecoeff. ‐0.112 ‐0.020 ‐0.069 ‐0.099 ‐0.013 ‐0.060 Averaget‐statistics ‐1.567 ‐0.425 ‐1.053 ‐1.375 ‐0.236 ‐0.845% coeff. 0 10.70% 34.22% 22.99% 10.67% 39.33% 24.72%% coeff.* 0 0.53% 4.28% 2.14% 0.56% 3.37% 0.56%% coeff.* 0 52.94% 13.90% 34.22% 41.01% 11.24% 25.84% Averagestandardizedcoeff. ‐0.112 ‐0.020 ‐0.069 ‐0.099 ‐0.013 ‐0.060Averagecoeff. 0.098 0.108 0.093 0.103 Averaget‐statistics 0.966 1.061 0.893 0.980% coeff. 0 76.47% 78.07% 75.28% 80.34%% coeff.* 0 27.81% 33.16% 25.84% 29.78%% coeff.* 0 2.67% 2.67% 1.12% 1.69% Averagestandardizedcoeff. 0.066 0.074 0.063 0.070Averagecoeff. ‐0.150 ‐0.153 ‐0.169 ‐0.173 Averaget‐statistics ‐1.071 ‐1.115 ‐1.275 ‐1.318% coeff. 0 8.02% 6.95% 16.85% 16.29%% coeff.* 0 0.00% 0.00% 0.00% 0.00%% coeff.* 0 19.79% 24.60% 28.65% 35.39% Averagestandardizedcoeff.   ‐0.082 ‐0.085 ‐0.079 ‐0.081Averageadj. 1.17% 1.12% 1.67% 1.01% 1.20% 1.65%Average 1.57% 1.52% 2.26% 1.40% 1.60% 2.24%Panel C: Serial autocorrelation in daily returns conditional on the intensity of information-based trading on consecutive days

Close-to-close return Open-to-close return 1 2 3 4 5 6Entire sample Averagecoeff. ‐0.107 ‐0.014 ‐0.060 ‐0.098 ‐0.017 ‐0.066 Averaget‐statistics ‐1.635 ‐0.273 ‐1.056 ‐1.602 ‐0.332 ‐1.149% coeff. 0 7.68% 43.21% 22.14% 9.19% 40.90% 18.39%% coeff.* 0 0.18% 6.43% 2.68% 0.75% 5.25% 2.25%% coeff.* 0 45.71% 15.00% 32.86% 47.28% 15.20% 34.33% Averagestandardizedcoeff. ‐0.107 ‐0.014 ‐0.060 ‐0.098 ‐0.017 ‐0.066Averagecoeff. 13.857 12.595 13.604 12.633 Averaget‐statistics 2.159 2.046 2.252 2.145% coeff. 0 94.46% 92.32% 97.19% 96.25%% coeff.* 0 65.89% 60.89% 67.35% 64.17%% coeff.* 0 0.54% 0.71% 0.94% 0.94% Averagestandardizedcoeff. 0.100 0.092 0.107 0.101Averagecoeff. ‐0.744 ‐0.677 ‐0.655 ‐0.584 Averaget‐statistics ‐1.603 ‐1.443 ‐1.280 ‐1.106% coeff. 0 9.29% 15.36% 20.08% 24.20%% coeff.* 0 0.00% 0.00% 0.38% 0.38%% coeff.* 0 42.86% 39.46% 35.65% 31.14% Averagestandardizedcoeff.   ‐0.100 ‐0.090 ‐0.073 ‐0.064Averageadj. 1.62% 1.63% 2.43% 1.56% 1.20% 2.11%Average 2.01% 2.02% 3.02% 1.96% 1.60% 2.70%

58

Close-to-close return Open-to-close return 1 2 3 4 5 6Small stock Averagecoeff. ‐0.129 ‐0.039 ‐0.098 ‐0.112 ‐0.035 ‐0.095 Averaget‐statistics ‐2.008 ‐0.735 ‐1.698 ‐1.885 ‐0.683 ‐1.632% coeff. 0 5.35% 31.55% 12.30% 6.18% 33.71% 10.11%% coeff.* 0 0.00% 4.81% 0.53% 0.00% 2.25% 1.12%% coeff.* 0 59.36% 27.81% 50.80% 53.37% 25.84% 47.19% Averagestandardizedcoeff. ‐0.129 ‐0.039 ‐0.099 ‐0.112 ‐0.035 ‐0.095Averagecoeff. 8.456 8.107 7.149 7.062 Averaget‐statistics 2.521 2.479 2.552 2.525% coeff. 0 99.47% 98.40% 98.88% 98.31%% coeff.* 0 80.21% 77.01% 75.84% 74.72%% coeff.* 0 0.00% 0.00% 0.56% 0.56% Averagestandardizedcoeff. 0.125 0.121 0.124 0.123Averagecoeff. ‐0.654 ‐0.573 ‐0.555 ‐0.466 Averaget‐statistics ‐1.141 ‐0.973 ‐0.880 ‐0.668% coeff. 0 13.90% 22.46% 28.65% 37.64%% coeff.* 0 0.00% 0.00% 0.56% 1.12%% coeff.* 0 26.74% 24.06% 23.03% 17.98% Averagestandardizedcoeff.   ‐0.072 ‐0.060 ‐0.049 ‐0.038Averageadj. 2.28% 1.45% 2.71% 1.97% 1.06% 2.31%Average 2.67% 1.85% 3.30% 2.36% 1.46% 2.91%Medium stock Averagecoeff. ‐0.095 ‐0.001 ‐0.046 ‐0.092 ‐0.008 ‐0.061 Averaget‐statistics ‐1.394 ‐0.015 ‐0.815 ‐1.474 ‐0.171 ‐1.083% coeff. 0 10.22% 51.61% 26.34% 10.73% 44.63% 18.64%% coeff.* 0 0.54% 9.14% 4.30% 2.26% 7.91% 2.82%% coeff.* 0 36.56% 8.06% 26.88% 44.63% 12.99% 32.77% Averagestandardizedcoeff. ‐0.095 ‐0.001 ‐0.046 ‐0.092 ‐0.008 ‐0.061Averagecoeff. 11.492 10.368 12.385 11.630 Averaget‐statistics 2.125 1.999 2.336 2.212% coeff. 0 93.55% 92.47% 97.74% 97.18%% coeff.* 0 64.52% 58.06% 71.19% 67.23%% coeff.* 0 0.54% 1.08% 0.56% 0.56% Averagestandardizedcoeff. 0.098 0.089 0.115 0.108Averagecoeff. ‐0.699 ‐0.629 ‐0.616 ‐0.532 Averaget‐statistics ‐1.622 ‐1.461 ‐1.222 ‐1.039% coeff. 0 10.75% 17.20% 19.77% 21.47%% coeff.* 0 0.00% 0.00% 0.56% 0.00%% coeff.* 0 42.47% 40.32% 33.90% 31.07% Averagestandardizedcoeff.   ‐0.100 ‐0.089 ‐0.072 ‐0.061Averageadj. 1.40% 1.41% 2.15% 1.59% 1.03% 2.05%Average 1.80% 1.80% 2.74% 1.99% 1.43% 2.64%

59

Close-to-close return Open-to-close return 1 2 3 4 5 6Large stock Averagecoeff. ‐0.098 ‐0.003 ‐0.034 ‐0.089 ‐0.008 ‐0.041 Averaget‐statistics ‐1.502 ‐0.066 ‐0.654 ‐1.446 ‐0.142 ‐0.730% coeff. 0 7.49% 46.52% 27.81% 10.67% 44.38% 26.40%% coeff.* 0 0.00% 5.35% 3.21% 0.00% 5.62% 2.81%% coeff.* 0 41.18% 9.09% 20.86% 43.82% 6.74% 23.03% Averagestandardizedcoeff. ‐0.098 ‐0.003 ‐0.034 ‐0.089 ‐0.008 ‐0.041Averagecoeff. 21.612 19.297 21.271 19.200 Averaget‐statistics 1.831 1.661 1.869 1.699% coeff. 0 90.37% 86.10% 94.94% 93.26%% coeff.* 0 52.94% 47.59% 55.06% 50.56%% coeff.* 0 1.07% 1.07% 1.69% 1.69% Averagestandardizedcoeff. 0.076 0.066 0.081 0.072Averagecoeff. ‐0.877 ‐0.829 ‐0.792 ‐0.754 Averaget‐statistics ‐2.045 ‐1.895 ‐1.738 ‐1.611% coeff. 0 3.21% 6.42% 11.80% 13.48%% coeff.* 0 0.00% 0.00% 0.00% 0.00%% coeff.* 0 59.36% 54.01% 50.00% 44.38% Averagestandardizedcoeff.   ‐0.128 ‐0.120 ‐0.099 ‐0.092Averageadj. 1.18% 2.02% 2.43% 1.13% 1.52% 1.97%Average 1.57% 2.42% 3.02% 1.52% 1.91% 2.57%

60

Table VI Profits from contrarian and momentum strategies

The contrarian trading strategy buys (short sells) one share of a stock at the opening ask (bid) and sells (covers) at the closing bid (ask) if the previous day’s open-to-close return of the stock is negative (positive). The strategy is also implemented conditional on the previous day’s . Conversely, the momentum trading strategy buys (short sells) one share of a stock at the opening ask (bid) and sells (covers) at the closing bid (ask) if the previous day’s open-to-close return of the stock is positive (negative). It is also implemented conditional on the previous day’s . Profits are measured by the daily raw returns, or returns adjusted by an equal-weighted (EW) or a value-weighted (VW) market portfolio averaged over the sample period. Panel A (B) reports the mean profits of contrarian (momentum) strategy across the entire sample stocks and the three subsamples, with the t-statistics reported in parentheses. Symbols ***, ** and * indicate significance at the 1%, 5%, 10% level, respectively.

Panel A: Profits from contrarian trading strategy

Implementation condition Entire sample Small stock Medium stock Large stock Daily raw return Unconditionalon 0.021%*** 0.025%* 0.011% 0.029%***  3.867 1.796 1.425 5.367

, 0 0.197%*** 0.190%*** 0.134%*** 0.159%***  12.092 5.240 6.861 8.609

, 0.05 0.199%*** 0.262%*** 0.165%*** 0.169%***  11.949 6.500 7.612 8.813

, 0.1 0.204%*** 0.266%*** 0.170%*** 0.174%***  11.929 6.498 7.504 8.696

, 0.15 0.222%*** 0.298%*** 0.185%*** 0.182%***  12.347 6.896 7.780 8.802

, 0.2 0.246%*** 0.324%*** 0.201%*** 0.212%*** 11.797 6.513 7.780 7.911 DailyEW‐adjustedreturnUnconditionalon 0.020%*** 0.025%* 0.009% 0.026%*** 3.581 1.803 1.165 4.812

, 0 0.169%*** 0.197%*** 0.145%*** 0.165%*** 10.714 5.195 7.253 8.248

, 0.05 0.204%*** 0.270%*** 0.168%*** 0.174%*** 11.602 6.307 7.477 8.530

, 0.1 0.208%*** 0.270%*** 0.174%*** 0.179%*** 11.5820 6.208 7.499 8.536

, 0.15 0.225%*** 0.305%*** 0.189%*** 0.180%*** 11.895 6.691 7.637 8.243

, 0.2 0.249%*** 0.338%*** 0.201%*** 0.206%*** 10.858 6.278 7.632 6.339 DailyVW‐adjustedreturnUnconditionalon ‐0.002% 0.003% ‐0.013%* 0.005% ‐0.278 0.211 ‐1.738 0.960

, 0 0.138%*** 0.172%*** 0.108%*** 0.133%*** 8.954 4.626 5.541 7.023

, 0.05 0.171%*** 0.240%*** 0.132%*** 0.142%*** 9.984 5.737 5.985 7.294

, 0.1 0.176%*** 0.241%*** 0.138%*** 0.147%*** 10.013 5.669 6.043 7.322

, 0.15 0.193%*** 0.275%*** 0.154%*** 0.149%*** 10.412 6.148 6.338 7.086

, 0.2 0.216%*** 0.307%*** 0.166%*** 0.174%*** 9.713 5.814 6.419 5.715

61

Panel B: Profits from momentum trading strategy

Implementation condition Entire sample Small stock Medium stock Large stock DailyrawreturnUnconditionalon ‐0.021%*** ‐0.025%* ‐0.011% ‐0.029%***  ‐3.867 ‐1.796 ‐1.425 ‐5.367

, 0 ‐0.010% ‐0.026% 0.019%* ‐0.023%***  ‐1.247 ‐1.300 1.733 ‐2.899

, 0.025 ‐0.003% ‐0.031% 0.036%*** ‐0.014%  ‐0.313 ‐1.275 2.965 ‐1.453

, 0.05 0.003% ‐0.032% 0.043%*** ‐0.001%  0.329 ‐1.275 3.187 ‐0.071

, 0.075 0.012% ‐0.015% 0.057%*** ‐0.007%  0.982 ‐0.539 3.821 ‐0.4560

, 0.10 0.028%* ‐0.011% 0.070%*** 0.026% 1.834 ‐0.343 3.590 0.905 DailyEW‐adjustedreturnUnconditionalon ‐0.023%*** ‐0.025%* ‐0.013%* ‐0.032%*** ‐4.122 ‐1.779 ‐1.667 ‐5.834

, 0 ‐0.009% ‐0.022% 0.023%* ‐0.028%***  ‐1.000 ‐1.024 1.937 ‐2.917

, 0.025 ‐0.002% ‐0.023% 0.038%*** ‐0.020%*  ‐0.153 ‐0.865 2.733 ‐1.851

, 0.05 0.007% ‐0.027% 0.044%*** 0.003%  0.593 ‐0.995 2.975 0.216

, 0.075 0.014% ‐0.004% 0.051%*** ‐0.007%  1.018 ‐0.148 3.090 ‐0.323

, 0.10 0.024% ‐0.007% 0.068%*** 0.009% 1.372 ‐0.208 3.486 0.261 DailyVW‐adjustedreturnUnconditionalon ‐0.044%*** ‐0.047%*** ‐0.034%*** ‐0.052%*** ‐7.957 ‐3.364 ‐4.513 ‐9.709

, 0 ‐0.024%*** ‐0.039%* 0.005% ‐0.039%***  ‐2.882 ‐1.902 0.473 ‐4.449

, 0.025 ‐0.016% ‐0.040% 0.021% ‐0.030%***  ‐1.6010 ‐1.596 1.634 ‐2.946

, 0.05 ‐0.008% ‐0.043% 0.028%** ‐0.009%  ‐0.719 ‐1.641 1.979 ‐0.701

, 0.075 ‐0.001% ‐0.022% 0.038%** ‐0.019%  ‐0.055 ‐0.755 2.451 ‐1.025

, 0.10 0.011% ‐0.023% 0.056%*** 0.000% 0.672 ‐0.714 2.939 ‐0.014

62

Table VII Effect of stock illiquidity on profitability of contrarian trading strategies

The contrarian trading strategy, that buys (short sells) one share of a stock at the opening ask (bid) and sells (covers) at the closing bid (ask) if the previous day’s open-to-close return of the stock is negative (positive), is implemented unconditionally or conditional on the previous day’s . For each stock, profits are measured by the average daily raw, EW-adjusted or VW-adjusted return and illiquidity is proxied by the average daily Amihud (2002) measure over the sample period. Panel A reports the mean profits of contrarian strategy for three illiquidity-stratified subsamples. In Panel B, stocks in each size-stratified subsample are sorted into tertiles based on illiquidity. The panel reports the mean profits of contrarian strategy conditional on , 0for the nine groups. T-statistics are reported in parentheses. Symbols ***, ** and * indicate significance at the 1%, 5%, 10% level, respectively. Panel A: Profits from contrarian trading strategy for illiquidity-stratified subsamples

Implementation condition Low

illiquidity stock Medium

illiquidity stock High

illiquidity stock DailyrawreturnUnconditionalon 0.027%* 0.007% 0.031%***  1.929 0.971 5.236

, 0 0.160%*** 0.143%*** 0.180%***  8.297 5.893 5.456

, 0.05 0.174%*** 0.173%*** 0.250%***  8.573 6.797 6.627

, 0.1 0.180%*** 0.176%*** 0.254%***  8.253 6.833 6.643

, 0.15 0.188%*** 0.188%*** 0.289%***  8.468 7.010 7.091

, 0.2 0.219%*** 0.200%*** 0.318%*** 7.790 7.078 6.656 DailyEW‐adjustedreturnUnconditionalon 0.027%* 0.005% 0.028%*** 1.920 0.766 4.663

, 0 0.168%*** 0.155%*** 0.184%*** 8.203 6.322 5.265

, 0.05 0.174%*** 0.179%*** 0.258%*** 8.274 7.010 6.322

, 0.1 0.181%*** 0.183%*** 0.260%*** 8.040 7.130 6.265

, 0.15 0.183%*** 0.193%*** 0.298%*** 7.992 7.112 6.814

, 0.2 0.212%*** 0.202%*** 0.331%*** 6.322 7.228 6.305 DailyVW‐adjustedreturnUnconditionalon 0.004% ‐0.016%** 0.007% 0.324 ‐2.301 1.173

, 0 0.136%*** 0.118%*** 0.160%*** 6.928 4.870 4.684

, 0.05 0.144%*** 0.142%*** 0.228%*** 7.063 5.611 5.745

, 0.1 0.150%*** 0.146%*** 0.231%*** 6.900 5.729 5.714

, 0.15 0.153%*** 0.157%*** 0.268%*** 6.852 5.867 6.264

, 0.2 0.181%*** 0.167%*** 0.300%*** 5.712 6.013 5.851

63

Panel B: Profits from contrarian trading strategy conditional on positive lagged DPSOS for subsamples double sorted by firm size and stock illiquidity

Low

illiquidity stockMedium

illiquidity stockHigh

illiquidity stock DailyrawreturnSmallStock 0.207%*** 0.147%*** 0.216%***  3.336 3.077 2.835 Mediumstock 0.134%*** 0.131%*** 0.137%***  3.656 3.884 4.349 Largestock 0.157%*** 0.158%*** 0.162%***  4.540 4.945 5.458 DailyEW‐adjustedreturnSmallStock 0.189%*** 0.151%*** 0.251%*** 3.153 2.801 3.106 Mediumstock 0.146%*** 0.153%*** 0.136%*** 4.093 4.401 4.001 Largestock 0.155%*** 0.167%*** 0.174%*** 4.422 4.654 5.162 DailyVW‐adjustedreturnSmallStock 0.164%*** 0.126%** 0.228%*** 2.726 2.431 2.872 Mediumstock 0.108%*** 0.113%*** 0.104%*** 3.085 3.333 3.129 Largestock 0.126%*** 0.137%*** 0.138%*** 3.708 4.186 4.227

64

Table VIII Decomposition of the profits of the contrarian trading strategy

The contrarian trading strategy, which buys (short sells) one share of a stock at the opening ask (bid) and sells (covers) at the closing bid (ask) if the previous day’s open-to-close return of the stock is negative (positive), is implemented unconditionally or conditional on the previous day’s . For each stock, profits are measured by the average daily raw, EW-adjusted or VW-adjusted return over the sample period. In Panel A, ∆ , is stock i’s change in Amihud’s (2002) illiquidity measure on day . The panel separately reports the mean profits of the contrarian investment strategy for lagged ∆ being negative and positive. Panel B decomposes the profits of contrarian trading strategy into two components, one from long positions and the other from short positions and separately reports their means. T-statistics are reported in parentheses. Symbols ***, ** and * indicate significance at the 1%, 5%, 10% level, respectively.

Panel A: Daily profits of contrarian trading strategy on days with negative and positive ∆ ,

Rawreturn EW‐adjustedreturn VW‐adjustedreturn

negative∆ ,

positive∆ ,

negative∆ ,

positive∆ ,

negative∆ ,

positive∆ ,

Unconditionalon 0.002% 0.044%*** 0.054%*** ‐0.008% 0.028%*** ‐0.024%*** 0.293 5.570 7.692 ‐0.935 4.117 ‐3.046

, 0 0.102%*** 0.223%*** 0.246%*** 0.084%*** 0.206%*** 0.063%*** 5.370 9.418 12.143 3.351 10.486 2.579

, 0.05 0.135%*** 0.272%*** 0.288%*** 0.116%*** 0.245%*** 0.095%*** 6.290 10.435 12.593 4.186 11.036 3.535

, 0.1 0.139%*** 0.279%*** 0.294%*** 0.120%*** 0.251%*** 0.100%*** 6.285 10.592 12.531 4.249 11.011 3.650

, 0.15 0.149%*** 0.305%*** 0.313%*** 0.132%*** 0.269%*** 0.113%*** 6.442 11.270 12.547 4.473 11.137 3.958

, 0.2 0.158%*** 0.341%*** 0.331%*** 0.157%*** 0.286%*** 0.138%*** 5.888 12.194 11.345 5.033 10.109 4.619

Panel B: Daily profits from the long and short positions of contrarian trading strategy

Rawreturn EW‐adjustedreturn VW‐adjustedreturn Long Short Long Short Long ShortUnconditionalon 0.042%*** 0.012%* 0.023%*** 0.026%*** ‐0.017%** 0.023%*** 5.274 1.714 3.185 3.280 ‐2.335 3.079

, 0 0.169%*** 0.171%*** 0.101%*** 0.255%*** 0.037%** 0.262%*** 8.127 7.961 5.482 9.797 1.998 10.582

, 0.05 0.198%*** 0.216%*** 0.114%*** 0.312%*** 0.049%** 0.317%*** 8.902 8.588 5.826 10.428 2.505 11.133

, 0.1 0.207%*** 0.217%*** 0.120%*** 0.316%*** 0.055%* 0.321%*** 9.262 8.642 6.111 10.540 2.815 11.235

, 0.15 0.224%*** 0.235%*** 0.132%*** 0.335%*** 0.066%* 0.342%*** 9.907 8.761 6.653 10.538 3.361 11.231

, 0.2 0.250%*** 0.262%*** 0.148%*** 0.370%*** 0.083%*** 0.376%*** 9.108 9.408 6.007 10.838 3.364 11.619

65

Table IX Profits from long-short contrarian portfolios and the characteristics of stocks selected by a portfolio

At the beginning of each trading day, a stock with a positive (negative) lagged open-to-close return and a lagged satisfying the specified condition is assigned to a short (long) position with an equal weight to a portfolio. The

portfolio is held until the market closes and is then liquidated. Daily returns of the individual stocks are calculated on the first and last quotes of the actual trades. All buys take place at the ask and all sells at the bid. The universe of stock selection is the entire sample stocks or one of the three subsamples. Panel A reports the cumulative raw returns, EW-adjusted returns, and VW-adjusted returns of six such long-short portfolios over the 2-year sample period. Panel B documents the characteristics of stocks in the long-short contrarian portfolio with , 0. In the panel,

sample stocks are sorted into quintiles according to the percentage of days they are selected by the portfolio, where Q1Q5 represents the stocks least (most) frequently appearing in the portfolio. For each quintile, the cross-sectional

mean of average daily market capitalization ( ), average daily turnover ( ), the standard deviation of daily open-to-close returns ( ), and average number of analysts following ( ) are reported in columns 2 to 5, respectively.

Panel A: Cumulative profits from a long-short portfolio formed based on lagged return and

Implementation condition Raw return EW-adjusted return VW-adjusted return Withoutconsidering ‐1.234% ‐5.516% ‐13.572%

, 0 37.166% 30.952% 19.889%, 0.05 43.179% 36.462% 25.002%, 0.1 44.386% 37.623% 26.069%, 0.15 52.678% 45.509% 33.285%, 0.2 57.085% 49.726% 37.126%

Panel B: Characteristics of stocks in a long-short portfolio formed based on lagged return and lagged DPSOS being positive Fraction of days (in mil. $) Q1 LeastOften 10.00% 9018 0.96% 2.45% 9.745Q2 16.04% 13575 1.08% 2.04% 11.265Q3 21.15% 9522 1.06% 2.03% 12.089Q4 26.34% 5240 1.07% 2.05% 11.327Q5 MostOften 39.95% 2156 0.64% 2.20% 6.642

66

Figure 1. Cumulative raw returns for long-short portfolios over the 2-year sample period. This figure displays cumulative raw returns from long-short contrarian portfolios formed by conditioning on lagged returns and lagged measures of . Each day, stocks with a positive (negative) lagged open-to-close return are assigned to the short (long) position with equal-weighting when market opens, and it is held till the market closes when the portfolio is liquidated. In Panel A, the portfolio is formed based on lagged return only, without considering the stocks’ , .

In Panels B, C and D, additional condition of stock selections that , 0 , , 0.1 and

, 0.2, respectively, are applied.

67

Figure 2. Cumulative EW-adjusted returns for long-short portfolios over the 2-year sample period. This figure displays cumulative returns from long-short contrarian portfolios formed by conditioning on lagged returns and lagged measures of , where the return is adjusted by return on the equal-weighted (EW) market portfolio. Each day, stocks with a positive (negative) lagged open-to-close return are assigned to the short (long) position with equal-weighting when market opens, and it is held till market closes when the portfolio is liquidated. In Panel A, the portfolio is formed based on legged return only. In Panels B, C and D, additional conditions of stock selection that ,

0, , 0.1 and , 0.2, respectively, are applied.

68

Figure 3. Cumulative VW-adjusted returns for long-short portfolios over the 2-year sample period. This figure displays cumulative returns from long-short contrarian portfolios formed by conditioning on lagged returns and lagged measures of , where the return is adjusted by return on the value-weighted (VW) market portfolio. Each day, stocks with a positive (negative) lagged open-to-close return are assigned to the short (long) position with equal-weighting when market opens, and it is held till market closes when the portfolio is liquidated. In Panel A, the portfolio is formed based on lagged return only. In Panel B, C and D, additional conditions of stock selection that ,

0, , 0.1 and , 0.2, respectively, are applied.