Heterogeneous Prior Beliefs, Differential Interpretation and
the Consensus Effect of Quarterly Earnings Signals and Trading Volume
Rowland K. Atiase,
Alex Dontoh, and
Michael J. Gift*
May 2011
Correspondence to:
Rowland K. Atiase, Department of Accounting, CBA 4M.202,
McCombs School of Business, The University of Texas at Austin,
Austin, Texas 78712-1172;
Phone: (512) 471-5841, FAX: (512) 471-3904;
e-mail: [email protected]
JEL classification: D84, G14, M41, G10
Keywords:
Empirical Capital Markets; Heterogeneous Prior Beliefs; Differential Interpretation;
The Consensus Effect; Quarterly Earnings Signals; Trading Volume Reaction
*
Atiase is with the McCombs School of Business, the University of Texas at Austin; Dontoh is
with the Stern School of Business, New York University; and Gift is with the Faculty of Business
Administration, University of Macau. We would like to thank Bipin Ajinkya, Ed Cannon, Somchai
Supattarakul and Senyo Tse for many helpful discussions, and Hayford Addai, Florence Atiase,
Harold Cartey, Mike Crawley, Joe Dowd, Robert Freeman, Alicia Jackson, Bill Kinney, Pierre
Liang, Pam Losefsky, Paul Newman, participants at the 2010 American Accounting Association
Annual Meeting, San Francisco, CA, participants at the 18th Conference on Pacific Basin Finance,
Economics, Accounting, Business and Management, Beijing, China, and workshop participants at
the University of Florida, the Hong Kong University of Science and Technology, the International
Symposium on Forecasting, University of South Florida, for their helpful comments. The authors
gratefully acknowledge the contribution of I/B/E/S International Inc. for providing earnings per share
forecast data, available through the Institutional Brokers Estimate System. These data have been
provided as part of a broad academic program to encourage earnings expectation research. The first
author acknowledges financial support provided by the Department of Accounting and the McCombs
School of Business at the University of Texas at Austin, and the Ernst and Young Foundation.
Heterogeneous Prior Beliefs, Differential Interpretation and
the Consensus Effect of Quarterly Earnings Signals and Trading Volume
ABSTRACT
Models of financial economists including Karpoff (1986), Varian (1989), Holthausen and
Verrecchia (1990), and Dontoh and Ronen (1993) have demonstrated that there are three distinct
fundamental determinants of trading volume reaction to new information releases: first, the
extent of differences in investors’ prior beliefs; second, differences in their interpretations of the
information; and third, the level of consensus that the information release induces among them.
Although these effects are well-understood theoretically, empirical studies that investigate
trading volume reaction to the arrival of new information have tended to combine these three
fundamental determinants, thereby masking their distinct incremental effects on trade. In this
paper we examine all three potential sources of trade in response to information: heterogeneous prior
beliefs, differential interpretation, and the consensus effect of the news. We find that all three of
these effects have a distinct incremental impact on trading volume, thereby corroborating the
theoretical models of financial economists.
JEL classification: D84, G14, M41, G10
Keywords:
Empirical Capital Markets; Heterogeneous Prior Beliefs; Differential Interpretation;
The Consensus Effect; Quarterly Earnings Signals; Trading Volume Reaction
Heterogeneous Prior Beliefs, Differential Interpretation and
the Consensus Effect of Quarterly Earnings Signals and Trading Volume
1. Introduction
Models of financial economists (Karpoff (1986), Varian (1989), Holthausen and Verrecchia
(1990, hereafter HV), and Dontoh and Ronen (1993, hereafter DR)) show that an information event
can stimulate trade for three fundamental reasons. The first reason is that differences in investors’
prior beliefs cause them to take positions that must be unwound in light of new information. A
second reason is that investors can interpret new information differently, thus revising their prior
beliefs differentially and motivating a re-shuffling of assets to new owners. Third, given
heterogeneous prior beliefs, a given magnitude of differential interpretation can result in a decrease or
an increase in consensus, thereby inducing more or less trade, i.e., the consensus effect (discussed
below). Although these effects are well-understood theoretically, empirical researchers have tended to
combine these three distinct origins of trading response to news, thereby masking their incremental
effects on trade. For example, some researchers claim that the size of trading response reflects the
amount of information that the event conveys to investors. But the size of trading response may
reflect not only the amount of information but also the extent of heterogeneity in investors’ prior
beliefs, how differently they interpret the information, and the extent of consensus induced by the
information. Empirically, however, it is not clear whether trading in response to news generally
reflects any one of these explanations or some combination of them. Indeed, if a subset of these
variables dominates and the others are empirically unimportant, then that would suggest that trade
occurs primarily due to the dominant variables. On the other hand, if evidence shows that all three
reasons provide significant explanation for trading, then that will suggest that the three reasons for
trade that appear in the theoretical literature (heterogeneous prior beliefs, differential interpretations,
and the consensus effect) are all empirically important, and therefore empirical trading volume models
that exclude or fail to control for any of these determinants are misspecified with biased estimated
coefficients.
2
This paper seeks to sort out empirically these three distinct motives for trade. Investors’ prior
belief heterogeneity is defined as the cross-sectional dispersion in investors’ beliefs about a firm’s
prospects prior to the arrival of a new piece of information. 1 The consensus effect is defined as a
measure of the extent to which investors’ beliefs diverge or converge as a result of an information
release. Consensus decreases when beliefs diverge and increases when they converge. Differential
interpretation is defined as the differential belief revision across individual investors as a result of
observing a new common public signal. An excellent anecdotal example of the differential
interpretation construct noted in Kandel and Pearson (1995, hereafter KP) appears in a New York
Times article by L.M. Fisher (1993, p. D4), who observed that:
“after Apple Computer Inc. announced a decline in earnings for its second fiscal
quarter, analysts rushed to revise their estimates for the year. Some revised them
downward, as one might expect, but some raised their estimates and others even
issued new buy recommendations.” It appears that the analysts disagreed because
they used different models of the computer industry to interpret the public
announcement, for the Times goes on to state that “the Apple bulls contend that
the pricing pressure on the company will abate in the second half of the year as
new products become more available, and sales will continue to grow. The bears
say that Apple has been promising earnings growth for some time now, and that
maintaining margins will get harder, not easier.”
Our empirical analysis is based on a sample of 26,169 quarterly earnings announcements by
1,995 firms between 1984 and 2008, inclusive. We employ the dispersion in individual financial
analysts’ forecasts of annual earnings per share (EPS) immediately before each sampled quarterly
earnings announcement as our proxy for investors’ prior belief heterogeneity. Our proxy for the extent
of differential interpretation of each quarterly earnings signal is the standard deviation of the revisions
in the same individual analysts’ forecasts of annual EPS immediately after each announcement,
deflated by the absolute value of the mean of the analysts’ annual EPS forecasts immediately before
each quarterly announcement. 2 Finally, our proxy for the consensus effect is the posterior dispersion in
the same analysts’ annual EPS forecasts immediately after each quarterly earnings announcement,
relative to their belief dispersion indicated in their prior forecasts. Given the novelty of the differential
1
2
See also Sarkar and Schwartz (2009) and Hong and Stein (2007).
We also employ the variance of the difference in the rankings of analysts forecasts before and after
each quarterly earnings announcement, normalized by the average rank before the announcement, as an
alternative proxy.
3
interpretation and the consensus effect metrics in the literature, we provide numerical examples (see
appendix) to show that our empirical proxies for differential interpretation and the consensus effect are
closely linked to their theoretical constructs, are intuitive and also well behaved.
The results indicate that trading volume is significantly positively related to the proxies for
prior belief heterogeneity, differential interpretation and the consensus effect of quarterly earnings
signals. These results hold even after controlling for the (positively associated) volume effects of the
magnitude of the price effects of quarterly earnings signals, the magnitude of quarterly earnings
surprise, as well as the volume effects of firm size. This empirical evidence corroborates the
theoretical results of Karpoff (1986), Varian (1989), DR, and HV that these three variables -- prior
belief heterogeneity, differential interpretation and the consensus effect of information events -- are
fundamental determinants of trading volume. As determinants of trading volume, they are also
determinants of trading volume reactions to financial and accounting information.
This paper makes a number of contributions. First, it serves as a reminder that the effects of
heterogeneous prior beliefs, differential interpretation, and the consensus effect on trade are distinct
and that trading volume reactions to news events are more complicated than is normally reflected in
most empirical investigations of the topic. Second, it presents reasonable measures of investors’
differential interpretation and the consensus effect of news. Finally, it demonstrates that trading
volume reactions to news reflect the effects of these three factors, thereby corroborating the
theoretical models of financial economists.
The rest of the paper proceeds as follows. Section 2 discusses the theoretical underpinnings
and prior studies motivating our expectations. Research methodology issues relating to sample
design, data collection, operational definition of variables, and model specification are discussed in
section 3, while section 4 presents empirical tests and results. A summary and conclusions are
provided in section 5.
2. Theory, Prior Studies and Hypotheses
2.1. Theoretical Motivation
Theoretical models of trading volume suggest that information-based trading is a
consequence of three fundamental determinants. First, investors’ prior belief heterogeneity causes
4
them to hold positions that they must modify in the light of new information such as a quarterly
earnings announcement. Second, when investors interpret new information differentially, they will
revise their prior beliefs differentially and motivate a new round of trading to reequilibrate individual
asset portfolios. These first two distinct motives for trade -- heterogeneous prior belief and
differential interpretation -- were modeled by Karpoff (1986), Varian (1989) and DR. 3 HV advance
the consensus effect of information releases as a third distinct reason for trade. That is, ceteris
paribus, trading volume increases (decreases) as consensus decreases (increases) because of
increased (decreased) diversity in investors’ beliefs about the value of the asset.
Karpoff (1986), Varian (1989) and DR all base their theoretical models on heterogeneous
investors who periodically and idiosyncratically revise their beliefs. As noted above, they establish
two distinct ways in which the arrival of new information affects trading volume -- heterogeneous
prior beliefs and differential interpretation. Heterogeneous prior beliefs refer the situation where
investors hold diverse beliefs about firm’s fundamental value prior to the arrival of new public
information about the firm. They show that when investors’ prior beliefs are different, they hold
different asset positions accordingly. As a result, the arrival of new public information, such as a
quarterly earnings announcement causes them to modify their prior positions and that induces
trading. Consistent with the above theoretical definition, we measure heterogeneous prior beliefs as
the standard deviation across the individual analysts’ forecasts of annual EPS immediately before
each quarterly earnings announcement, divided by either (1) the absolute value of the mean
individual analysts’ prior annual EPS forecasts (APrMAF), or (2) prior stock price, denoted as PrDf
and PrDp, respectively. Intuitively, PrDf and PrDp may be characterized as the variance in the
individual analysts’ prior beliefs, normalized by their average prior beliefs and stock price,
respectively.
Karpoff (1986), Varian (1989) and DR argue that differential interpretation occurs when
investors differ in how much they revise their beliefs in reaction to a public announcement. The
differential belief revision is reflected in how much individual investors’ revisions deviate from the
average. Thus, given investors’ prior beliefs, the theoretical measure of differential interpretation is an
increasing function of the variance of investors’ belief revisions following the observation of a public
3
See also Dontoh and Ronen (1993) and Kim and Verrecchia (1997).
5
signal. When all investors revise their beliefs by the same magnitude in the same direction such that
there is no variation in their belief revision, they have identical interpretations and the measure of
differential interpretation is, by definition, zero. Consistent with this theoretical definition, we measure
differential interpretation as the standard deviation of the revision in the same individual analysts’
annual EPS forecasts from immediately before to immediately after each quarterly earnings
announcement (SDrev), divided by either (1) the absolute value of the mean individual analysts’ prior
annual EPS forecasts (APrMAF), or (2) prior stock price, denoted as DIf, and DIp, respectively.
Intuitively, DIf and DIp may be characterized as the variance in the individual analysts’ belief revision,
normalized by their average prior beliefs and stock price, respectively. When analysts have identical
interpretations of information, there is no incremental trading beyond that caused by prior belief
heterogeneity. On the other hand, when they interpret new information (e.g., quarterly earnings
reports) differentially, they will also revise their prior beliefs (e.g., forecasts of annual EPS)
differentially provided the precision of their prior information was identical. Some revise their prior
beliefs optimistically, while others are more pessimistic. Analysts’ differential revisions resulting from
their various interpretations induce additional trading volume beyond that resulting from the
heterogeneity of prior beliefs.
HV present a partially revealing rational expectations model of competitive trading and
demonstrate the influence of the consensus effect of information releases on volume of trade. They
define the consensus effect as a measure of the extent to which investors’ beliefs diverge (decrease in
consensus) or converge (increase in consensus) as a result of an information release. HV show
analytically that, ceteris paribus, a decrease (an increase) in consensus among investors is associated
with an increase (a decrease) in trading volume. Consistent with HV’s theoretical definition, we use
the posterior belief dispersion (PsDf) in analysts’ annual EPS forecasts immediately following each
quarterly earnings announcement relative to the prior belief dispersion (PrDf) in their annual
forecasts as our proxy for the consensus effect (CEf). Consequently, CEf = PsDf / PrDf = 1.0 would
imply no change in consensus among the analysts, or no convergence or divergence in the analysts’
beliefs. CEf < 1.0 would imply an increase in consensus among the analysts, or a convergence in
6
beliefs; and CEf > 1.0 would imply a decrease in consensus among the analysts, or a divergence in
beliefs. 4
The theoretical results of Karpoff (1986), Varian (1989), DR, KP and HV form the basis of
our hypotheses that the magnitude of trading volume reaction to the information in quarterly earnings
announcements is an increasing function of (1) the heterogeneity of investors’ prior beliefs, (2)
differential interpretation and (3) the consensus effect of quarterly earnings signals. Other reasons for
trade include endowment differences, trading for liquidity, risk tolerance, tax, and portfolio
rebalancing considerations. However, because we examine specific earnings announcements, it is
unlikely that trading for these other reasons (other than volume effects of how the information in
quarterly earnings announcements affects the above three fundamental determinants) will
systematically affect our results.
2.2. Prior Studies
Ajinkya, Atiase, and Gift (1991, hereafter AAG) provide empirical evidence on the general relation
between trading volume and belief heterogeneity for an almost continuous flow of information that
analysts implicitly use in their periodic revisions of annual EPS forecasts. The results in AAG
indicate a significant positive association between the dispersion in analysts’ forecasts of annual EPS
and the volume of trading. However, AAG do not examine the volume effects of differential
interpretation or the consensus effect. KP suggest differential interpretation of public signals as a
plausible explanation for the empirical observation that there is significant abnormal volume around
anticipated public announcements even when prices do not change in response to the announcement.
They also provide extensive evidence to show that analysts interpret public signals differentially.
However, their study does not address the relation between trading volume and differential
interpretation or heterogeneous prior beliefs, or the consensus effect. Bamber et al. (1997) report that
trading volume around earnings announcements is positively related to the complement of the
correlation between relative positions of individual analysts’ pre and post forecasts, as well as the
dispersion in prior beliefs and the change in dispersion. However, as noted by them, their
4
We also employ an alternative proxy for the consensus effect (CEp) based on price-deflated variables defined in
Section II. However, for simplicity, we limit the discussion in this section to only mean forecast deflated variables
PrDf, PsDf, DIf, and CEf.
7
complement of analysts’ forecast correlation measure is quite distinct from the differential
interpretation construct addressed in this paper. Thus, Bamber et al. (1997) do not address or control
for the volume effects of differential interpretation. Bamber, Barron, and Stober (1999) examine the
relation between differential interpretation and trading volume but do not control for, or address, the
relation between trading volume and prior belief dispersion or the consensus effect. To date, no prior
study has examined the relation between trading volume and all three fundamental determinants
advanced by financial economists—heterogeneous prior beliefs, differential interpretation, and the
consensus effect in a multivariate setting to determine whether only a subset of the three variables or
all three variables are empirically important. As noted above, if the evidence shows that all three
variables are empirically important, then empirical trading volume models that exclude or fail to
control for any of these determinants are most likely misspecified with biased estimated coefficients.
In examining the relation between trading volume, prior belief heterogeneity, differential
interpretation and the consensus effect, it is necessary to control for the magnitude of quarterly
earnings signals. The motivation for controlling for the magnitude of quarterly earnings signals
follows from prior evidence (e.g., Karpoff (1987), Kim and Verrecchia (1991), and Atiase and
Bamber (1994)) that trading volume is positively related to unexpected information revealed by an
earnings announcement.
2.3. Hypotheses
From the preceding discussion, we posit that trading volume is:
1. a positive function of prior belief heterogeneity,
2. a positive function of the extent of differential interpretation of each quarterly earnings signal,
3. a positive function of the consensus effect, and
4. a positive function of the magnitude of quarterly earnings signals.
3. Research methodology
3.1. Sample Design and Data Collection
Sample-firm observations meet the following ten selection criteria. The firm must (1) be a
member of the New York Stock Exchange or the American Stock Exchange; (2) have a December 31
fiscal year end; (3) be listed on the COMPUSTAT Merged Fundamental Annual File; (4) be listed on
8
the COMPUSTAT Merged Fundamental Quarterly File; (5) be listed on the CRSP Daily Stock Securities Database; (6) be listed on the Institutional Brokers Estimate System (I/B/E/S) Detail
History Database; (7) be listed on the I/B/E/S Summary History Database; (8) have actual first,
second or third quarter earnings information available on both the COMPUSTAT Merged
Fundamental Quarterly File and I/B/E/S Summary History Databases and the corresponding
quarterly earnings announcement date available on the COMPUSTAT Merged Fundamental
Quarterly File 5 ; (9) have at least three analysts contributing to the mean quarterly EPS forecasts
during the month immediately before each sampled quarterly earnings announcement; and (10) have
at least three most current individual analysts’ new forecasts of annual EPS within 45 days before
each sampled quarterly earnings announcement, and the same individual analysts that issued a
forecast in the prior period must have a matching posterior forecast in the I/B/E/S database within 30
days following each quarterly earnings announcement whether or not their posterior forecast differed
from their prior forecast.
With respect to criteria (9) and (10), replication of the analysis with a minimum of five
analysts resulted in a smaller sample but substantially similar inferences. Criterion (1) is imposed to
avoid a possible difference in information environment due to exchange listing or “exchange effect.”
Criterion (2) facilitates matching COMPUSTAT quarterly earnings announcement dates with the
most current I/B/E/S forecasts associated with the sampled quarterly earnings announcements. 6 The
COMPUSTAT Merged Fundamental Annual File is used to screen for criterion (2). Criteria (3)
through (8) provide assurance of access to data required for the study. The purpose of criterion (9) is
to enhance the statistical stability of each mean quarterly EPS forecast. Criterion (10), perhaps the
most restrictive criterion, is imposed to minimize contamination by other events or information
5
Fourth quarter earnings announcements are excluded to minimize ambiguity about the fiscal year with which
analysts' annual EPS forecasts are associated. Analysts start to forecast annual EPS for the following fiscal year
after the fourth quarter.
6
This sample selection criterion is often employed in studies that use analysts’ forecasts (e.g., O'Brien (1988),
AAG, and Atiase and Bamber (1994)). Restriction of the sample to firms with December 31 fiscal year-ends likely
biases the sample in favor of large firms (Smith and Pourciau (1988)). Thus, to check on the robustness of our
results with respect to the “firm size-related differential information hypothesis” or “size effect” (Atiase (1985),
Freeman (1987)), we control for firm size directly in our empirical Model IIb specified below. The results of
estimating Model IIb reported under Sensitivity Analysis show that controlling for firm size does not affect our
inferences. We also follow Atiase and Bamber (1994) and repeated the analysis after partitioning the sample into
two groups based on firm size, and for both subsamples, the results are similar to those reported here. Finding
similar results in the “relatively smaller firms” subsample suggests that any bias in favor of larger firms induced by
the sample selection criteria is unlikely to affect significantly our study's inferences.
9
affecting individual analysts’ prior and posterior annual EPS forecasts as well as minimize the
differential lag between the individual analysts’ prior and posterior annual EPS forecasts and enhance
the precision of the measures of the prior dispersion in annual EPS forecasts, differential
interpretation, and the consensus effect of quarterly earnings signals. In particular, our measures of
differential interpretation and the consensus effect require prior and posterior forecasts of annual EPS
by the same individual analysts. This criterion was originally employed by Ajinkya, Atiase, and Gift
(1995) and subsequently followed by Bamber et al. (1997) and others. The data set that meets all the
above selection criteria consists of 26,169 observations associated with quarterly earnings
announcements by 1,995 firms between 1984 and 2008, inclusive.
For each sampled firm, data on daily returns, trading volume and total shares outstanding
(adjusted for stock splits and stock dividends) are obtained from the CRSP Daily Stock - Securities
Database. The dates of quarterly earnings announcements and actual quarterly earnings information
are retrieved from the COMPUSTAT Merged Fundamental Quarterly File. The actual quarterly
earnings figures per the COMPUSTAT Merged Fundamental Quarterly File are cross-checked with
the actual quarterly earnings figures from the I/B/E/S Summary Database for consistency. Data for
the computation of the prior dispersion in analysts’ annual earnings forecasts, the differential
interpretation of quarterly earnings signals, and the magnitude of quarterly earnings signals are
retrieved from the I/B/E/S Detail and Summary Database, the COMPUSTAT Merged Fundamental
Quarterly File, and the CRSP Daily Stock - Securities Database.
3.2. Variable Definitions
3.2.1. Trading Volume Reaction (TV)
Two trading volume measures are used. Following AAG and Atiase and Bamber (1994), the
first measure of trading volume is based on the percentage of firm i’s shares traded on day t (Vit)
surrounding each quarterly earnings announcement. The daily percentages of shares traded are
cumulated over four different windows -- two short windows and two long windows -- and divided
by the number of trading days in each window to give the average daily percentage of shares traded
in each of the four windows. As noted above, to be included in the sample, a firm must have at least
three most current individual analysts’ new forecasts of annual EPS within 45 days before each
10
sampled quarterly earnings announcement, and the same individual analysts must have a matching
posterior forecast in the I/B/E/S database within 30 days following each quarterly earnings.
The variable tb is the date that the first of the analysts forecasts annual EPS (within 45 days)
before each quarterly earnings announcement, and ta is the date the last of the analysts revises his
forecast (within 30 days) after each quarterly earnings announcement. Then, the four windows are
defined as follows:
1) first short window: days −1 and +1, relative to a quarterly earnings announcement,
2) second short window: from day −1 to day +5, relative to a quarterly earnings announcement,
3) first long window: from day −1 to day ta +1, relative to a quarterly earnings announcement,
4) second long window: from day tb −1 to day ta +1.
The motivation for the short windows stems from prior research (Morse (1981), among
others) which suggests that although the bulk of the trading volume reaction occurs on days −1 to +1,
relative to the earnings announcement date, abnormally high trading persists up to five days after the
announcement. However, the long windows may better capture the volume effects of prior belief
heterogeneity and differential interpretation as well as the consensus effect, since more of the
analysts’ prior and posterior annual EPS forecasts are likely to occur during these spans of time. By
using average daily percentage of shares traded we control for the differences in the length of the
four windows. These (unadjusted) trading volume metrics are denoted APVOL3, APVOL7,
APVOLLW1, and APVOLLW2 for the four windows, respectively.
The second and the principal measure of trading volume, Vmadjit, adjusts for the overall
market level of trading by dividing each firm i’s daily percentage of shares traded (Vit) by the daily
percentage of shares traded on the overall market on day t (Vmt). This yields each firm i’s daily
percentage of shares traded relative to the daily percentage of shares traded on the overall market on
day t (i.e., Vmadjit = Vit/Vmt) 7 . Again, the resulting daily market-adjusted percentage of firm i’s
shares traded on day t, Vmadjit, are cumulated over the same four event windows defined above and
divided by the number of days in each window. This measure gives the average daily market7
Daily percentage of shares traded on the overall market is proxied by the daily total shares traded by all firms
(approximately 2,002 firms on average) on the CRSP database that were actively traded throughout the study period
(1984-2008), divided by the total shares outstanding for the same firms.
11
adjusted percentage of firm i’s shares traded in each of the four windows. These market-adjusted
trading volume metrics are denoted MAVOL3, MAVOL7, MAVOLLW1, and MAVOLLW2 for the
four windows, respectively.
In all our empirical analysis described below, the results based on the market-adjusted trading
volume metrics (MAVOL) are essentially identical to the results based on the unadjusted trading
volume metrics (APVOL). Thus, for brevity, we only report results based on the market-adjusted
trading volume metrics (MAVOL).
3.2.2. Prior and Posterior Belief Dispersion in Analysts’ Annual EPS Forecasts (PrD and PsD) 8
We employ two alternative measures of PrD. The numerator is the standard deviation across
the individual analysts’ forecasts of annual EPS immediately before each quarterly earnings
announcement (σprior). The alternative denominators are (i) the absolute value of the mean of the
individual analysts’ forecasts of annual EPS immediately before each quarterly announcement, i.e.,
∑ f ib
/ n , 9 or (ii) the most current I/B/E/S stock price before each quarterly earnings announcement,
pb. That is:
(1)
) / n − 1]
∑(
[annual
EPS forecast of analyst i before each quarterly earnings announcement
2
− fb
1
2
σprior
=
f ib
=
f ib
= mean annual EPS forecast over n analysts before each quarterly earnings
fib
announcement (i.e.,
∑ fib / n , where i = 1, ..., n.)
The resulting metrics are denoted
(i)
PrDf = σprior / (
(ii)
PrDp = σprior / pb.
∑ f ib / n ), and
Similarly, we employ two alternative measures of posterior belief dispersion, PsD. The
numerator is the standard deviation across the individual analysts' forecasts of annual EPS made
within 30 days after each quarterly earnings announcement. The alternative deflators are (i) the
absolute value of the mean of the individual analysts’ forecasts of annual EPS immediately before
8
In this study, we use the terms prior belief heterogeneity and prior belief dispersion interchangeably.
Observations with mean EPS forecasts between $–.05 and $.05 are omitted due to the metric's sensitivity to
small denominators. Pincus (1983) and O'Brien (1988) have used similar cutoff rules. Inclusion of these “small
denominator” observations does not affect the overall tenor of the results.
9
(2)
(3)
12
each quarterly announcement (APrMAF), or (ii) the most current I/B/E/S stock price before each
quarterly earnings announcement. The resulting posterior belief dispersion metrics are denoted PsDf
and PsDp, respectively. Similar metrics for PrD based on I/B/E/S summary data are employed in
prior studies (e.g., AAG and Elliott and Philbrick (1990)). In this study however, we construct our
measures of PrD from the most current individual analysts forecasts based on I/B/E/S detailed data
that is purged of potentially outdated forecasts. Thus, we address the concern expressed about prior
studies using I/B/E/S summary data (as opposed to the I/B/E/S detailed data) that the association
between PrD and trading volume may be partially affected by outdated analyst forecasts.
3.2.3. Differential Interpretation of Quarterly Earnings Signals (DI)
As noted above, we measure differential interpretation (DI) as the standard deviation of the
revision in individual analysts’ forecasts of annual EPS, SDrev, divided by either (i) the absolute
value of the mean analysts’ annual EPS forecast immediately before each quarterly earnings
b
announcement, or (ii) pb (defined above). Specifically, let f i be the annual EPS forecast for analyst
a
i, and f i be the same analyst’s forecast after the quarterly earnings announcement. The revision in
analyst i’s annual forecast is Δfi = f ia − f ib . The deviation in analyst i’s revision from the average
analysts’ annual EPS forecast revision equals Δfi – Δ f , where Δ f = ΣΔfi/n. Thus,
SDrev =
[∑ (
Δf i − Δf
)
2
]
/ n −1
1
2
, where i = 1, ..., n.
(4)
The resulting differential interpretation (DI) metrics, DIf and DIp, are given by:
∑ f ib / n ), and
(i)
DIf = SDrev / (
(ii)
DIp = SDrev / pb.
(5)
(6)
In addition to the above two parametric measures of differential interpretation (DIf and DIp), we also
use a nonparametric measure of differential interpretation denoted DI2. DI2 is the variance of the
differences in the rankings of analyst forecasts before and after each quarterly earnings
announcement, normalized by the average rank before the announcement. A potential drawback of
the nonparametric measure of differential interpretation, DI2, is that it would miss some true
differences in analysts’ forecast revisions. For example, so long as the rankings in analysts’ forecasts
of annual EPS before and after each quarterly earnings announcement do not change, DI2 will always
be zero, implying identical interpretation even when the variance in their revisions is greater than
13
zero, which would imply differential interpretation (See e.g., Scenario B in Figure 1 in the appendix).
On the other hand, an advantage of the nonparametric measure of differential interpretation over the
parametric measure is that while the parametric measure implies a particular functional form of the
differential belief revisions, the nonparametric measure (e.g., DI2) does not.
Like all proxies, ours may contain other potential limitations. In particular, it should be noted
that DIf, DIp, and DI2 reflect differential belief revisions across analysts. 10 In any case, prior research
suggests that analysts’ earnings forecasts and forecast revisions are reasonable surrogates for
investors’ beliefs and belief revisions. 11 Thus, the above-noted potential limitation notwithstanding,
DIf, DIp, and DI2 are reasonably good empirical proxies for the unobservable differential
interpretation construct.
3.2.4. Consensus Effect (CE)
As discussed in Section 2, HV define the consensus effect as a measure of the extent to which
investors’ beliefs diverge (decrease in consensus) or converge (increase in consensus) as a result of an
information release. Consistent with HV’s theoretical definition, we use the posterior belief dispersion
(PsDf and PsDp) in analysts’ annual EPS forecasts immediately following each quarterly earnings
announcement relative to the prior belief dispersion (PrDf and PrDp) in their annual forecasts as our
proxies for the consensus effect (CE). These metrics are denoted CEf and CEp, respectively. Since
PsDf and PrDf are both standardized by the same deflator, APrMAF, and PsDp and PrDp are both
deflated by the same prior stock price, CEf and CEp are equal by construction and may be denoted as
simply CE.
10
Variation in analysts' belief revisions may underestimate variation in the belief revisions of a broader, more
heterogeneously informed set of investors. Any such error in the proxy is likely to work against finding the
hypothesized positive incremental relation between DIf (or DIp) and trading volume reaction.
11
KP (p. 833), for example, note several other good reasons for using analysts' forecasts (and forecast revisions)
as surrogates for investors' beliefs (and belief revisions) in the following terms:
... many investors and money managers, and the equity research analysts whose forecasts we study, are
intelligent, well-trained, sophisticated individuals with a great deal at stake. There are substantial rewards
for success and few barriers to entry, and agents have tremendous incentives to take account of others'
information: “If she wants to trade with me, why should I trade with her?” Also, “If her forecast is
different from mine, shouldn't I update mine to reflect the information in hers?” If we find that agents fail
to take full account of others' information in a well-developed financial market, we can reasonably expect
that other agents in other settings will also fail to do so.
14
3.2.5. Quarterly Earnings Signal (QES)
We use two different proxies for the magnitude of quarterly earnings signals (QES). The first
measure is the magnitude of the price reaction (i.e., the magnitude of the CRSP excess returns)
associated with each quarterly earnings announcement as a proxy for the magnitude of the price
effects of quarterly earnings signals (QES). We use two alternative excess returns metrics denoted
ARES2 and ARES7. These are the absolute value of the CRSP excess returns, cumulated over the
following 2- and 7-day windows relative to the quarterly earnings announcement date:
ARES2:
two-day window from day −1 to day 0,
ARES7:
seven-day window from day −1 to day +5.
The second measure is the magnitude of unexpected quarterly earnings (UQE). Unexpected
quarterly earnings metrics are defined as the absolute value of the difference between actual quarterly
EPS and the mean analysts’ forecasts of the quarterly EPS immediately before each quarterly
earnings announcement, deflated either by the absolute value of the mean analysts’ forecasts of the
quarterly EPS or the most current I/B/E/S stock price before each quarterly earnings announcement
denoted UQEf and UQEp, respectively.
3.2.6. Firm Size (FSIZE)
Firm Size is defined as the market value of common equity just before a Quarterly Earnings
Announcement.
3.3. Model Specification
The relation investigated in this study is of the form:
TV = f(PrD, DI, CE, QES)
The primary method of analysis is regression analysis. We use logarithmic transformations of both
the dependent and independent variables to reduce departures of the regression errors from normality
as well as to reduce skewness in the data. 12 The skewness in the trading volume data is consistent
12
Comparing the degree of departure from normality of the regression errors when the raw variables are used in
model estimations to that when log-transformed variables are used, the following representative results in favor of
log transformations are observed. For the estimation of Model I (specified below) with the three-day volume (i.e.,
MAVOL3 versus LMAVOL3) and PrDf versus LPrDf, DI2 versus LDI2, CE versus LCE, the skewness and kurtosis
15
with prior empirical evidence in Ajinkya and Jain (1989). They report that a natural log
transformation mitigates any nonnormality of percentage of shares traded data (which may translate
to regression errors). The log-transformed variables are labeled LTV (LMAVOL3, LMAVOL7,
LMAVOLLW1, and LMAVOLLW2), LPD (LPDf and LPDp), LDI (LDIf, LDIp, LDIs, and LDI2), LCE
(LCEf and LCEp) and LQES (LARES2, LARES7, LUQEf, and LUQEp).
The models estimated are of the form:
Model I:
LTV = α 0 + α1 LPrD + α 2 LDI + α 3 LCE + u
(7)
Model II:
LTV = β 0 + β1 LPrD + β 2 LDI + β 3LCE + β 4 LQES + u'
(8)
Based on the discussion in section 2, in all the models, all the coefficients of the independent
variables are expected to be positive. All the models are estimated twice, once with mean forecastdeflated variables and once with price-deflated variables.
4. Empirical Tests and Results
4.1. Descriptive Statistics
Table 1 summarizes various percentile values of the distributions of the raw dependent and
independent variables. For the dependent variable, trading volume, the mean market-adjusted daily
percentage of shares traded are 2.057% and 1.760% for two short windows MAVOL3, MAVOL7 and
1.684%, and 1.492% for the two long windows MAVOLLW1, and MAVOLLW1. The trading volume
reactions with respect to the 3-day and 7-day short windows (MAVOL3 and MAVOL7, respectively)
are consistent with prior evidence (Morse (1981) and Bamber (1987)) which suggests that the bulk of
trading volume reaction occurs on days −1 to +1, relative to the earnings announcement date, although
abnormally high trading volume persists up to five days after the announcement. The mean consensus
effect metric, CE, is 1.340, suggesting a decrease in consensus, on average, in reaction to the
announcements. However, the median CE is 0.924. Indeed, 56.7% of CEs are less than 1.000,
suggesting approximately 57% (43%) of the quarterly earnings signals are associated with an increase
(a decrease) in consensus. The range for each dependent and independent variable is quite large, and
of the error terms are 5.285 and 52.299, respectively, for the case of the raw variables, diminishing to 0.168 and
0.379, respectively, for the log-transformed variables.
16
each variable exhibits some degree of positive skewness (means exceed medians). The logarithmic
transformations significantly reduce the skewness. 13
Insert Table 1 about here
4.2. Univariate Analysis
Table 2 presents the pairwise correlations among the empirical proxies for the theoretical
constructs. The Pearson product-moment (Spearman rank-order) correlation coefficients between the
alternative measures of the dependent variable (LTV) and each of the alternative measures of the four
independent variables (LPrD, LDI, LCE, and LQES) and between the measures of the independent
variables themselves are shown below (above) the diagonal. As expected, the proxies for prior belief
dispersion (LPrDf and LPrDp) are all significantly positively correlated with the measures of trading
volume (α < 0.01). Also, the proxies for differential interpretation (LDIf, LDIp, and DI2) are
significantly positively correlated with the measures of trading volume (α < 0.01) in all twelve
estimations. Table 2 also reveals that the proxies for the magnitude of quarterly earnings signals
LQES, namely LARES2, LARES7, LUQEf, and LUQEp are all significantly positively associated
with trading volume reactions to quarterly earnings announcements (α < 0.01). On the other hand,
three of the four correlations between the proxy for the consensus effect (LCE) and trading volume
reactions are not significantly different from zero. Indeed, the correlation between LCE and LPrDp is
significantly negative. The latter results suggest that either there is no unconditional relation between
the proxy for the consensus effect and volume of trade or the relation is significantly negative. Notice
however that the proxy for the consensus effect (LCE) is significantly negatively correlated with the
proxies for prior belief dispersion (LPrDf and LPrDp, α ≤ 0.01), and the proxies for prior belief
dispersion (LPrD) and differential interpretation (LDI) are significantly positively correlated with
each other. This suggests that any inference on the relation between consensus effect and volume of
trade from the simple correlation between the proxies for the consensus effect and trading volume
(without controlling for prior belief dispersion and differential interpretation) is misleading. Also, all
13
Since the maximum values of some of the raw variables are large relative to the 99th percentile values (not
reported), we winsorized the data at 99%. We also conducted an analysis of potential outliers. Specifically, we
replicated our analyses based on the original data. The results of the original sample are qualitatively identical to
those of the winsorized sample. Thus the results are not unduly influenced by extreme observations. It should be
noted that the log transformations moderate the effect of large raw values anyway.
17
the three main independent variables, LPrD, LDI, and LCE are significantly positively correlated with
LQES in 19 out of 20 estimations. Taken together, the inter-correlations among the variables raise the
following questions that we address in our empirical analysis:
i.
Do all three variables advanced by theory as stimulating trade following an information event
namely, prior heterogeneous beliefs, differential interpretation and the consensus effect, provide
significant explanation for trading volume reaction to new information, or does only a subset of
the variables explain the reaction and the other variables are empirically unimportant?
ii.
Does the relation between trading volume and the variables prior heterogeneous beliefs,
differential interpretation and the consensus effect persist after controlling for the
magnitude of quarterly earnings signals? 14
The next section reports results of multivariate analyses with Models I and II that address these
questions.
Insert Table 2 about here
4.3. Multivariate Analyses
Models I and II are each estimated twice, once with mean forecast-deflated variables and once
with price-deflated variables. The models are also estimated with the nonparametric differential
14
There are several reasons that can be conjectured as to why a subset of the three variables may be empirically
unimportant. The three determinants of trading volume: prior belief dispersion, the magnitude of differential
interpretation, and the consensus effect are (various transformations or scalings of) the dispersion of prior beliefs,
the dispersion of belief revisions, and the dispersion of posterior beliefs. An analyst’s posterior forecast is the prior
forecast plus the revision, i.e., if x is the prior forecast, y is the forecast revision, and z is the posterior forecast, then
z = x + y and var(z) = var(x) + var(y) + 2cov(x, y). Thus, for example, it can be argued that if cov(x, y) = 0, then the
three variables var(x), var(y), and var(z) actually only contain two pieces of independent information, and var(z), the
variance of the posterior forecast, which reflects the consensus effect, does not contain additional information
beyond the variances of x and y. This is because if cov(x, y) =0 (i.e., x and y are uncorrelated), then knowledge of
var(z) will be redundant since it will be uniquely determined by var(x) and var(y). In that case, given prior belief
dispersion and the magnitude of differential interpretation, the volume effects of the consensus effect wil be
empirically unimportant. In general, it can be shown that given var(x) and var(y), belief divergence or convergence
following an earnings signal depends on whether the change in forecast variance defined by var(y) + 2cov(x, y) is
greater than zero, less than zero, or equal to zero. More specifically, beliefs converge (i.e., CEf < 1) resulting in an
increase in consensus whenever var(y) + 2cov(x, y) < 0. Beliefs diverge (i.e., CEf > 1) resulting in a decrease in
consensus whenever var(y) + 2cov(x, y) > 0, and there is no divergence or convergence in beliefs (i.e., CEf = 1)
whenever var(y) + 2cov(x, y) = 0. Thus, given prior heterogeneous beliefs and a given magnitude of differential
interpretation,whether the consensus effect is empirically unimportant depends crucially on the var(y) + 2cov(x, y)
term and this is an empirical issue. Similarly, as noted above, the proxies for prior belief dispersion (LPrD) and
differential interpretation (LDI) are significantly positively correlated with each other. If the positive and significant
correlation between the proxies for prior belief dispersion (LPrD) and differential interpretation (LDI) are high
enough, it is possible that one of the two variables would subsume the other and make it empirically unimportant.
Again, this is an empirical issue.
18
interpretation proxy, DI2, instead of the parametric LDIf, and LDIp. Table 3 presents the results for
Model I, the regression of trading volume metrics (LTV) on proxies for prior belief dispersion (LPrD),
differential interpretation (LDI), and the consensus effect (LCE) without any controls for the
magnitude of quarterly earnings signals (LQES). 15 The results of estimating the model with mean
forecast-deflated variables and the differential interpretation proxies LDI2 and LDIf appear in panels A
and B, respectively. These results strongly support our expectations by documenting positive
incremental relations between trading volume and the proxies for prior belief heterogeneity,
differential interpretation, and the consensus effect. First, the coefficients of the proxy for prior belief
heterogeneity, LPDf, is positive and significant at α levels of < 0.01 in all (8 out of 8) estimations.
This result extends the empirical result in AAG that trading volume is positively related to prior belief
heterogeneity to a specific information event -- quarterly earnings signals. Second, the conditional
results on the estimated coefficients of LDI2 and LDIf are positive and statistically significant at α
levels of < 0.01 in all (8 out of 8) estimations. Third, the conditional results on the estimated
coefficients of LCE (the proxy for the consensus effect of quarterly earnings signals) are all positive
and statistically significant at α levels of < 0.01 in all (8 out of 8) estimations. In sum, the results of
Model I reported in Table 3 show that absent any controls for the magnitude of quarterly earnings
signals, all three main independent variables, prior belief dispersion, differential interpretation, and
the consensus effect, provide significant explanation for trade.
The results for Model II, the regression of trading volume metrics (LTV) on proxies for prior
belief dispersion (LPrD), differential interpretation (LDI), and the consensus effect (LCE) with
controls for the magnitude of quarterly earnings signals (LQES) are reported in Table 4. The results
of estimating the model with mean forecast-deflated variables, the differential interpretation proxy,
LDI2, and the proxies for the magnitude of quarterly earnings signals, LQES: LARES2, LARES7,
and LUQEf appear in panels A, B, and C, respectively, while the results of estimating the model with
mean forecast-deflated variables, the differential interpretation proxy, LDIf, and the proxies for the
magnitude of quarterly earnings signals LQES, namely LARES2, LARES7, and LUQEf appear in
panels D, E, and F, respectively. Again, these results strongly support our expectations. In all cases,
15
The regression results for the models specified with the parametric measure of differential interpretation, DIp
are qualitatively similar to those for the models specified with DIf. Thus for brevity, only the results for the
differential interpretation metrics, DI2 and DIf are reported and discussed further.
19
the coefficients of LARES2, LARES7, and LUQEf are positive and significant (α < 0.01), indicating
that the proxies for the magnitude of quarterly earnings signals are positively related to trading
volume reactions to quarterly earnings announcements. 16 This result extends the prior evidence on
annual earnings in Atiase and Bamber (1994) to quarterly earnings announcements. Taken together,
the above results show that the magnitude of quarterly earnings signals is an important control
variable.
More importantly, the results reported in Table 4 document positive incremental relations
between trading volume and the proxies for prior belief heterogeneity, differential interpretation, and
the consensus effect even after controlling for the price effects of quarterly earnings signals. First,
the coefficients of the proxies for prior belief heterogeneity, LPDf, are still positive and statistically
significant at α levels of < 0.01 in 21 out of 24 estimations. Second, the conditional results on the
estimated coefficients of LDI2 and LDIf, the proxies for differential interpretation of quarterly
earnings signals, are still positive and statistically significant at α levels of < 0.01 in all (24 out of 24)
estimations. Third, the conditional results on the estimated coefficients of LCE, the proxy for the
consensus effect of quarterly earnings signals, are also still positive and statistically significant at α
levels of < 0.05 in 16 out of the 24 estimations (12 are significant at α < 0.01). 17, 18 The positive and
significant coefficients of the proxies for prior belief dispersion and differential interpretation, along
with the significantly positive coefficient of the proxy for the consensus effect, show that even after
controlling for the positive volume effects of the proxies for the magnitude of the price effects of
quarterly earnings signals, trading volume is significantly positively related to all three determinants:
analysts’ prior belief heterogeneity, differential interpretation, and the consensus effect of quarterly
earnings signals.
Insert Tables 3 and 4 about here.
16
This result is consistent with the finding in the finance literature that trading volume is, in general, positively
associated with the magnitude of returns (e.g., Harris (1986), and Karpoff (1987)).
17
We also estimated each of the above models specified with the unadjusted trading volume metric (LAPVOL)
as the dependent variable and included the average daily percentage of shares traded on the overall market
(LMAVOL) as an independent (control) variable (i.e., models of the general form: LAPVOL = f(LPrD, LDI, LCE,
LQES, LMAVOL)). Again, the results of these estimations are essentially identical to those reported in tables 3 and
4 discussed above and are therefore not reported.
18
The t-statistics of the estimated coefficients of LPD, LDI, and LCE are generally larger for the models
specified with the long trading volume windows relative to the short windows (tables 3 and 4), suggesting that the
long windows are better at capturing the volume effects of LPD, LDI, and LCE.
20
4.4. Sensitivity Analysis
We now turn to a sensitivity analysis. Our results thus far show that trading volume reaction
to quarterly earnings signals is significantly positively related to the dispersion in prior beliefs,
differential interpretation, as well as the consensus effect of news. However, as noted earlier, Table
2 indicates the proxies for prior belief dispersion (LPrDf and LPrDp) are highly positively correlated
with the proxies for differential interpretation (LDIf, LDIp, and LDI2). Intuitively, it can be argued
that a wider spread of prior belief will lead to greater differential belief revisions (i.e., differential
interpretation). This raises another interesting question as to whether trading volume is still related
to differential interpretation after normalizing it for prior belief dispersion. We address this question
by constructing a measure of differential interpretation per unit of prior belief dispersion DI/PrD
denoted DIPerPrD by estimating the following models:
Model Ia:
LTV = α 0A + α 1A LPrD + α 2A LDIPerPrd + α 3A LCE + u"
Model IIa:
LTV = β 0A + β1A LPrD + β A2 LDIPerPrD + β 3A LCE + β A4 LQES + u' ' '
(9)
(10)
The results of estimating Model Ia, the regression of trading volume metrics LTV on proxies
for prior belief dispersion LPrD, differential interpretation per unit of prior dispersion LDIPerPrD,
the consensus effect LCE, without controls for the magnitude of quarterly earnings signals (LQES)
are presented in panel A of Table 5. These results show that the estimated coefficients of prior belief
dispersion (LPrD), α 1 , are significantly positive at α levels of < 0.01 in all four estimations, and the
A
estimated coefficients of LCE, α 3A , are positive and statistically significant at α levels of < 0.01 in all
four estimations. More importantly, the estimated coefficients of differential interpretation per unit of
A
prior dispersion (LDIPerPrD), α 2 , are also significantly positively related to trading volume at α
levels of < 0.01 in all four estimations.
The results of estimating Model IIa, the regression of trading volume metrics (LTV) on
proxies for prior belief dispersion (LPrD), differential interpretation per unit of prior dispersion
(LDIPerPrD), the consensus effect (LCE), with control for the proxies for the magnitude of quarterly
earnings signals, LQES: LARES2, LARES7, and LUQEf appear in panels B, C, and D of Table 5,
A
respectively. These results show that the estimated coefficients of prior belief dispersion (LPrD), β 1
, are significantly positive at α levels of < 0.01 in all 12 estimations. The estimated coefficients of
21
LCE, β 3A , are positive and statistically significant at α levels of < 0.10 in 10 out of 12 estimations (7
of 12 are significant at α < 0.05). Also, the estimated coefficients of the LQES control variables are
all positive and statistically significant at α levels of < 0.01 in all 12 estimations. More importantly,
A
the estimated coefficients of differential interpretation per unit of prior dispersion (LDIPerPrD), β 2 ,
are also significantly positively related to trading volume at α levels < 0.01 in all 12 estimations,
even after controlling for LPrD, LCE, and LQES.
Next we examine the sensitivity of our results with respect to control for firm size measured
as the market value of common stock traded just before quarterly earnings announcements. Our
results thus far show that trading volume reaction to quarterly earnings signals is significantly
positively related to the dispersion in prior beliefs, differential interpretation, and the consensus
effect of news even after controlling for the positive volume effects of the proxies for the magnitude
of quarterly earnings signals, LQES: LARES2, LARES7 and LUQEf. An alternative way of
controlling for the magnitude of quarterly earnings signals is to control for firm size. The “firm sizerelated differential information hypothesis” or “size effect” advanced and corroborated in Atiase
(1985) and further corroborated by Freeman (1987) suggests that the amount of firm specific
earnings-related predisclosure information production and dissemination is an increasing function of
firm size. Thus the amount of unexpected information conveyed by an earnings report should be
inversely related to firm size, ceteris paribus. We address this question as to whether trading volume
is still related to proxies for prior heterogeneous beliefs, differential interpretation and the consensus
effect persist after controlling for volume effects of firm size by estimating the following model:
Model IIb:
LTV = β 0B + β1B LPrD + β B2 LDI + β 3B LCE + β B4 LFSIZE + u' ' ' '
(11)
The results of estimating Model IIb, the regression of trading volume metrics (LTV) on
proxies for prior belief dispersion (LPrD), differential interpretation (LDI), the consensus effect
(LCE), and firm size control (LFSIZE) are reported in Table 6. For brevity, we report only the results
for LTV, LPrDf, LDI2, LCE, and LFSIZE. Consistent with the “firm size-related differential
information hypothesis”, the results show that the estimated coefficients of the firm size (LFSIZE)
B
control variable, β4 , are significantly negative at α levels of < 0.01 in all four estimations. More
importantly, the results reported in Table 6 document positive incremental relations between trading
22
volume and the proxies for prior belief heterogeneity, differential interpretation, and the consensus
effect even after controlling for firm size. First, the coefficients of the proxy for prior belief
heterogeneity, LPDf, are still positive and statistically significant at α levels of < 0.01 in all four
estimations. Second, the conditional results on the estimated coefficients of LDI2, the proxy for
differential interpretation of quarterly earnings signals, are still positive and statistically significant at
α levels of < 0.01 in all four estimations. Third, the conditional results on the estimated coefficients
of LCE, the proxy for the consensus effect of quarterly earnings signals, are also still positive and
statistically significant at α levels of < 0.10 in all four estimations (2 of 4 are significant at α < 0.05).
The positive and significant coefficients of the proxies for prior belief dispersion and differential
interpretation, along with the significantly positive coefficient of the proxy for the consensus effect,
show that here again even after controlling for the volume effects of the firm size, trading volume is
significantly positively related to all three determinants: analysts’ prior belief heterogeneity,
differential interpretation, and the consensus effect of quarterly earnings signals. Thus, our results are
robust with respect to the firm size control as well.
Insert Tables 5 and 6 about here
5. Summary and Conclusions
Models of financial economists (Karpoff (1986), Varian (1989), Holthausen and Verrecchia
(1990), and Dontoh and Ronen (1993)) have demonstrated that an information event can stimulate
trade for three fundamental reasons. The first reason is that differences in investors’ prior beliefs
cause them to take positions that must be unwound in light of new information. A second reason is
that investors can interpret new information differently, thus revising their prior beliefs differentially
and motivating a re-shuffling of assets to new owners. Third, given heterogeneous prior beliefs, a
given magnitude of differential interpretation can result in a decrease or an increase in consensus,
thereby inducing more or less trade, i.e., the consensus effect. Although these effects are wellunderstood theoretically, empirical studies that investigate trading volume reaction to the arrival of
new information have tended to combine the three fundamental motives for trade following the
23
release of new information thereby masking their distinct incremental effects on trade. Thus,
empirically, it is not clear whether trading in response to news generally reflects any one of these
explanations or some combination of them. Indeed, if a subset of these variables dominates and the
others are empirically unimportant, then that would suggest that trade occurs primarily due to the
dominant variables. On the other hand, if evidence shows that all three reasons provide significant
explanation for trading, then that will suggest that the three reasons for trade that appear in the
theoretical literature (heterogeneous prior beliefs, differential interpretations, and the consensus
effect) are all empirically important, and therefore empirical trading volume models that exclude or
fail to control for any of these determinants are misspecified with biased estimated coefficients.
This study is motivated by the desire to sort out the three motives for trade empirically and
shed light on these issues by conducting an empirical investigation as to whether whether trading in
response to information disclosures generally reflects a subset or all three fundamental motives
predicted by theory. 19
We find that trading volume is significantly positively related to all three fundamental
determinants of trading volume: prior belief heterogeneity, differential interpretation, and the
consensus effect of quarterly earnings signals. The evidence corroborates the theoretical results of
Karpoff (1986), Varian (1986), Dontoh and Ronen (1993), as well as Holthausen and Verrecchia
(1990). The empirical findings serve as a reminder that the effects of heterogeneous prior beliefs,
differential interpretation, and the consensus effect on trade are distinct and incremental to each other
and that trading volume reactions to news events are more complicated than is normally reflected in
most empirical investigations of the topic. Second, it presents reasonable measures of investors’
differential interpretation and the consensus effect resulting from information disclosures. Third, it
demonstrates that trading volume reactions to news reflect the effects of heterogeneous prior beliefs,
19
We employ a number of methodological improvements over prior studies. In particular, we construct our
measures of prior belief dispersion from the most current individual analysts’ forecasts based on I/B/E/S detailed
data that are purged of potentially outdated forecasts. Thus, we address the concern expressed about prior studies
using I/B/E/S summary data (as opposed to the I/B/E/S detailed data) that the association between prior belief
dispersion and trading volume may be partially confounded by outdated analysts’ forecasts. We also control for the
magnitude of the associated price reactions. In so doing, we address the concern that studies using the dispersion in
analysts’ forecasts as a proxy for investors’ prior belief heterogeneity need to include the magnitude of the
associated price change to control for the magnitude of the price effects of the quarterly earnings signals
(Abarbanell, Lanen, and Verrecchia (1995)).
24
differential interpretation, and the consensus effect of the news, thereby corroborating the theoretical
models of financial economists.
Our findings suggest that empirical trading volume models that exclude or fail to control for
any of these determinants may suffer from a correlated omitted variables problem and hence would be
misspecified and estimated coefficients biased resulting in wrong inferences. A possible extension to
our study therefore would be to re-investigate previous empirical trading volume models that do not
control for the determinants suggested by theory. Additional research that empirically investigates
and provides evidence on the issue of potential model misspecification and coefficient bias would
contribute to our understanding of how the market processes information releases such as earnings
announcements.
25
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Morse, D. C. 1981. “Price and Trading Volume Reaction surrounding Earnings Announcements: A
Closer Examination.” Journal of Accounting Research 19: 374-83.
O’Brien, P. C. 1988. “Analysts’ Forecasts as Earnings Expectations.” Journal of Accounting and
Economics 10: 53-83.
Pfleiderer, P. C. 1984. “The Volume of Trade and the Variability of Prices: A Framework for
Analysis in Noisy Rational Expectations Equilibrium” Working paper. Stanford University.
Pincus, M. 1983. “Information Characteristics of Earnings Announcements and Stock Market
Behavior.” Journal of Accounting Research 21: 155-83.
Sarkar, A. and R. A. Schwartz. 2009. “Market sidedness: Insights into motives for trade initiation.”
Journal of Finance 64: 375-423.
Smith, D. B. and S. Pourciau. 1988. “A Comparison of the Financial Characteristics of December
and non-December Year-end Companies.” Journal of Accounting and Economics 10: 335-44.
Varian, H. R. 1989. “Differences of Opinion in Financial Markets.” Financial Risk: Theory,
Evidence, and Implications: Proceedings of the 11th Annual Economic Policy Conference of the
Federal Reserve Bank of St. Louis: 3-37.
27
FIGURE 1
Annual EPS forecasts of analysts A1, A2, and A3 before a quarterly EPS announcement (Tb) and
their revised forecasts after the quarterly EPS announcement (Ta) a
Scenario A:
Divergence with flip (prior dispersion < posterior dispersion, decrease in consensus).
APrMAF = 10
Ai = Forecast of Annual EPS by Analyst i
SDPF=5
PrDf = 5/10=0.5
PsDf = 8/10=0.8
SDrev = 13
Ai
DIf = 13/10=1.3
CEf = 0.8/0.5 = 1.6
18
A1
15
A2
1
10
5
A3
2
Tb
Ta
Time
Scenario B:
Divergence without flip (prior dispersion < posterior dispersion, decrease in consensus).
APrMAF = 10
Ai = Forecast of Annual EPS by Analyst i
SDPF=5
PrDf = 5/10=0.5
PsDf = 8/10=0.8
SDrev = 3
Ai
DIf = 3/10=0.3
CEf = 0.8/0.5 = 1.6
18
A1
15
A2
10
10
5
A3
2
Tb
Ta
Time
Scenario C:
Convergence without flip (prior dispersion > posterior dispersion, increase in consensus).
APrMAF = 10
SDPF=5
Ai = Forecast of Annual EPS by Analyst i
PrDf = 5/10=0.5
PsDf = 2/10=0.2
SDrev = 3
DIf = 3/10=0.3
Ai
CEf = 0.2/0.5 = 0.4
A1
15
A2
10
12
10
8
A3
5
Tb
Ta
Time
a
APrMAF is the absolute value of the mean individual analysts' prior annual EPS forecast; SDPF is the standard
deviation of mean individual analysts' prior annual EPS forecast; PrDf is the standard deviation of the individual
analysts' forecasts of annual EPS, divided by the absolute value of the mean individual analysts' prior annual EPS
forecast (APrMAF); DIf is the standard deviation of the revisions in the same individual analysts' forecasts of annual
EPS after each quarterly earnings announcement, divided by APrMAF; PsDf is the standard deviation of the
individual analysts' posterior annual EPS forecasts, divided by APrMAF; and CEf is the posterior belief dispersion
(PsDf) relative to the prior belief dispersion (PrDf).
Appendix:
Empirical Proxies for Differential Interpretation and the Consensus Effect
This appendix shows that our empirical proxies for the magnitude of differential interpretation, DIf,
DIp, and DI2 are closely linked to the theoretical differential interpretation construct of Karpoff (1986), and
Varian (1989). The appendix also demonstrates that the empirical proxies for the consensus effect, CEf and
CEp, are closely linked to the theoretical construct of Holthausen and Verrecchia (1990; hereafter HV). We
provide numerical examples to show that DIf, DIp, and DIs as well as CEf and CEp are intuitive constructs.
Finally, we show that the information in (and the volume effects of) the consensus effect is unique and over
and above the information in (and the volume effects of) prior belief dispersion and differential
interpretation. For simplicity, the numerical analysis focuses on the forecast deflated variables DIf and CEf.
As argued by Karpoff (1986), and Varian (1989), differential interpretation occurs when investors
differ in how much they revise their beliefs in reaction to a public announcement. The differential belief
revision is reflected in the deviation of individual investors’ belief revisions from investors’ average belief
revisions. Consistent with the above definition, we measure DIf as the standard deviation of the revision in
the same individual analysts’ forecasts of annual EPS from immediately before to immediately after each
quarterly earnings announcement, SDrev, divided by the absolute value of the mean individual analysts’
prior annual EPS forecast (APrMAF). Intuitively, DIf may be characterized as the variance in the individual
analysts’ belief revision, normalized by their average prior beliefs.
HV define the consensus effect as a measure of the extent to which investors' beliefs diverge or
converge as a result of an information release. HV show analytically that a decrease (an increase) in
consensus among investors is associated with an increase (a decrease) in trading volume, ceteris paribus.
Consistent with HV’s theoretical definition of the consensus effect, we use the posterior belief dispersion
(denoted PsDf ) in analysts’ annual EPS forecasts immediately after each quarterly earnings announcement
relative to the prior belief dispersion (denoted PrDf) in the annual forecasts of the same individual analysts
as our proxy for the consensus effect -- CEf . Thus, for example, CEf = PsDf / PrDf = 1.0 would imply no
change in consensus among the analysts, i.e., no convergence or divergence in the analysts’ beliefs. By the
same token, CEf < 1.0 would imply an increase in consensus among the analysts (convergence in beliefs)
and CEf > 1.0 would imply a decrease in consensus among the analysts (divergence in beliefs). Thus, our
empirical proxy CEf is closely linked to HV’s consensus effect construct.
We assume that analysts’ expectations (annual EPS forecasts) are representative of investors'
expectations. We also limit our analysis to the annual EPS forecasts of three individual analysts i, Ai,
where i = 1, 2, 3. Each individual analyst i issues an annual forecast, Ai, at time Tb immediately before a
quarterly earnings announcement, and each individual analyst revises his forecast at time Ta, immediately
after each quarterly EPS announcement. Again, for simplicity but without any loss in generality, the
absolute value of the mean individual analysts' prior annual EPS forecasts (APrMAF), and the prior
dispersion in the individual analysts' forecasts (PrDf) at time Tb, are all held constant in all the scenarios
analyzed below. These assumptions enable us to better focus on the behavior of DIf and CEf. The analysis
is done in two phases. We first hold prior dispersion, PrDf, and the consensus effect, CEf, constant and
analyze the behavior of DIf and its volume effects. Then, we hold PrDf and DIf constant and examine the
behavior of CEf and its volume effects. Figure 1 depicts three scenarios, A, B, and C, that illustrate the
distinct volume effects of our empirical proxies for differential interpretation (DIf) and the consensus effect
(CEf).
Insert Figure 1 about here
In scenarios A, B, and C, each individual analyst i issues an annual EPS forecast of 15, 10, and 5,
respectively, at time Tb immediately before a quarterly earnings announcement, and then revises his forecast
at time Ta, immediately after each quarterly EPS announcement. Thus, in all the scenarios, the prior
dispersion in the individual analysts’ EPS forecasts (PrDf) and the absolute value of their mean prior
forecasts (APrMAF) are held constant at time Tb. Also, for the purposes of the illustration, analyst A2’s
revised forecast following the quarterly earnings announcement is held constant at 10 in all three scenarios.
Scenarios A and B provide an illustration of the behavior of the differential interpretation proxy, DIf, and
the distinct volume effect of DIf beyond trading induced by prior belief dispersion, PrDf, and the consensus
effect, CEf. In both of these scenarios, PrDf and CEf are held constant. In scenario A, after the quarterly
earnings announcement, the optimistic analyst A1 interprets the announcement as very bad news, and
revises his annual EPS forecast downward by 13 to 2, and the pessimistic analyst A3 interprets the
announcement as very good news and changes his annual EPS forecast upward by 13 to 18. In scenario B,
after the quarterly earnings announcement, the optimistic analyst A1 interprets the announcement as good
news and revises his annual EPS forecast up by 3 to 18, while the pessimistic analyst A3 interprets the
announcement as bad news and changes his annual EPS forecast down by 3 to 2. The differential forecast
revisions due to differential interpretation will induce further trading between the analysts beyond that due
to the dispersion in the three analysts’ prior annual EPS forecasts, PrDf, and the consensus effect, CEf.
However, since the variance of belief revision (annual EPS forecast revision) is greater in scenario A than
in scenario B, the magnitude of the differential interpretation proxy, DIf, and its distinct volume effects
beyond those of PrDf and CEf, will be greater in scenario A than in scenario B.
A comparison of scenarios B and C illustrates the behavior of the consensus effect proxy, CEf, and
its distinct volume effects beyond that attributed to PrDf and DIf. These two scenarios depict the same
magnitude of differential interpretation, DIf. As noted above in scenario B, after the quarterly earnings
report, the optimistic analyst A1 interprets the report as good news and revises his annual EPS up by 3 from
15 to 18, and the pessimistic analyst A3 interprets the report as bad news and revises his annual EPS
forecast down by 3 from 5 to 2. In scenario C, the optimistic analyst A1 interprets the quarterly earnings
report as bad news and revises his annual forecast down by 3 from 15 to 12, while the pessimistic analyst
A3 interprets the report as good news and revises his annual forecast up by 3 from 5 to 8. These differential
forecast revisions induce additional trading between the analysts beyond the volume effects of the
dispersion in their prior annual forecasts. In any case, since the variance of the annual forecast revisions in
scenarios B and C are equal, the magnitude of differential interpretation and its expected trading volume
effect are the same. Thus, in scenarios B and C, both the proxies for prior belief dispersion, PrDf, and the
magnitude of differential interpretations, DIf, are held constant. Notice however that in scenario B, after the
quarterly earnings announcement, the three analysts’ beliefs diverge (CEf greater than 1.0) implying a
decrease in consensus and an increase in trading volume as a result. In scenario C the three analysts’ beliefs
converge, (CEf less than 1.0) implying an increase in consensus, thereby implying a lower trading volume.
Thus, controlling for prior belief dispersion and differential interpretation, a decrease in consensus as in
scenario B will be associated with a higher trading volume, whereas an increase in consensus as in scenario
C will be associated with a lower trading volume.a Thus, our empirical proxies for differential
interpretation and the consensus effect are closely linked to their theoretical constructs.
_____________________________
a
It should be noted that the amount and nature of trade are only suggested by the diagrams. Trading will
depend on how news in the quarterly earnings announcement changes demand. Since demand changes reflect not
only changes in expected earnings (which the diagrams in Figure 1 reflect), but also information precision, risk
tolerance, and wealth, scenarios A, B, and C do not necessarily imply that trades will occur only between analysts
A1 and A3. Even analyst A2, whose beliefs do not change, could be a net buyer or seller after the announcement,
depending on whether the changes in analyst A1’s and A3’s demands create a net increase or decrease in demand.
Table 1
Descriptive statistics
a
Distribution of raw dependent and independent variables; N=26,169; 1984-2008
MAVOL3 MAVOL7 MAVOLLW1 MAVOLLW2
PrDf
PrDp
DIf
DIp
Mean
2.057
1.760
1.684
1.492
0.219
0.007
0.224
0.007
Std. Dev.
2.401
1.787
1.640
1.340
4.031
0.041
4.197
0.110
Maximum
11.550
8.800
8.067
6.579
1.970
0.065
1.885
0.060
90%
4.199
3.448
3.281
2.783
0.229
0.021
0.207
0.012
75%
2.411
2.101
2.011
1.784
0.093
0.013
0.081
0.005
50%
1.353
1.241
1.208
1.123
0.040
0.006
0.035
0.002
25%
0.798
0.777
0.775
0.745
0.019
0.003
0.016
0.001
10%
0.508
0.522
0.530
0.525
0.010
0.001
0.008
0.000
Minimum
0.029
0.044
0.039
0.011
0.000
0.000
0.000
0.000
FSIZE (x 1,000)
DI2
DIPerPrD
CE
ARES2
ARES7
UQEP
UQEF
Mean
0.807
1.324
1.340
0.039
0.049
0.006
0.582
8,925,895.4
Std. Dev.
0.912
4.874
4.870
0.044
0.051
0.161
5.790
23,231,539.2
Maximum
4.129
7.888
7.981
0.201
0.236
0.053
7.333
108,069,000.0
90%
1.939
1.967
1.962
0.089
0.110
0.008
0.613
19,173,600.0
75%
1.143
1.257
1.257
0.052
0.067
0.003
0.233
7,193,160.0
50%
0.533
0.898
0.924
0.026
0.035
0.001
0.091
2,621,990.0
25%
0.167
0.616
0.631
0.010
0.014
0.000
0.033
1,059,310.0
10%
0.000
0.364
0.410
0.003
0.007
0.000
0.008
472,288.0
Minimum
0.000
0.000
0.010
0.000
0.000
0.000
0.000
17,708.9
Table 1 (continued)
a
MAVOL3, MAVOL7, MAVOLLW1, and MAVOLLW2 are the average of a sampled firm's daily
percentage of shares traded relative to the daily percentage of shares traded on the overall market:
MAVOL3
-two-day window, from day –1 to day +1,
MAVOL7
-seven-day window, from day –1 to day +5,
MAVOLLW1
-first long window, from day –1 to day TA +1,
MAVOLLW2
-second long window, from day TB –1 to day TA +1
where day 0 is the quarterly earnings announcement date, TB is the date the least most current of the
individual analysts forecasts of annual EPS is made (within 45 days) before each sampled quarterly
earnings announcement, and TA is the date the last of the same individual analysts revises his/her
forecast of annual EPS (within 30 days) after each quarterly earnings announcement.
PrDf and PrDp are the standard deviation of the individual analysts' forecasts of annual EPS, divided by
the absolute value of the mean individual analysts' prior annual EPS forecast and stock price,
respectively.
DIf and DIp are the standard deviation of the revisions in the same individual analysts' forecasts of
annual EPS after each quarterly earnings announcement, divided by the absolute value of mean
individual analysts' prior annual EPS forecast and stock price, respectively.
DI2 is the variance of the difference in the rankings of analyst forecast before and after each quarterly
earnings announcement normalized by the average rank before the announcement.
DIPerPrD is differential interpretation per unit of prior belief dispersion (i.e., DIf/PrDf or DIp/PrDp)
where PrDf, PrDp, DIf, DIp are defined.
CE is the posterior belief dispersion, PsDf and PsDp, relative to the prior belief dispersion, PrDf and
PrDp, respectively, where PsDf and PsDp are defined as the standard deviation of the individual analysts'
posterior annual EPS forecasts, divided by the absolute value of the mean individual analysts' prior
annual EPS forecast and stock price, respectively, and PrDf and PrDp are defined above.
ARES2 and ARES7 are the absolute value of the market model residual returns, cumulated over the
following 2-day and 7-day windows relative to the quarterly earnings announcement date:
ARES2
ARES7
---
two-day window, from day –1 to day 0,
seven-day window, from day –1 to day +5.
UQE is Unexpected Quarterly Earnings.
FSIZE is Firm Size, defined as the market value of common equity just before Quarterly Earnings
Announcement.
Table 2
Pearson product-moment and Spearman rank-order correlation coefficientsa between pairs of (log-transformed) regression variablesb;
N=26,169; 1984-2008
LMAVOL3 LMAVOL7 LMAVOLLW1 LMAVOLLW2
0.944
LMAVOL3
LPrDf
LPrDp
LDIf
LDIp
LDI2 LDIPerPrD LCE LARES2 LARES7 LUQEP
LUQEF
LFSIZE
0.897
0.847
0.092
0.071 0.126 0.103 0.096
0.045
-0.005 0.312
0.250
0.111
0.138
-0.046
0.946
0.893
0.128
0.110 0.154 0.134 0.094
0.038
-0.002 0.283
0.272
0.129
0.152
-0.037
0.920
0.125
0.106 0.149 0.128 0.085
0.033
-0.011 0.259
0.248
0.122
0.146
-0.034
0.186
0.173 0.190 0.176 0.096
0.009
-0.017 0.176
0.173
0.130
0.150
-0.019
0.915 0.789 0.721 0.138
-0.237
-0.196 0.015
0.038
0.412
0.445
-0.322
0.717 0.802 0.139
-0.044
-0.181 -0.006
0.021
0.495
0.404
-0.339
0.918 0.422
0.313
-0.009 0.044
0.060
0.447
0.478
-0.310
0.421
0.303
-0.002 0.024
0.043
0.528
0.437
-0.337
0.504
-0.020 0.028
0.016
0.052
0.137
0.142
0.349 0.048
0.045
0.102
0.099
-0.045
0.025
0.027
0.125
0.123
-0.044
0.626
0.059
0.073
-0.018
0.069
0.081
-0.046
0.899
-0.304
LMAVOL7
0.948
LMAVOLLW1
0.904
0.949
LMAVOLLW2
0.856
0.898
0.923
LPrDf
0.086
0.117
0.113
0.169
LPrDp
0.073
0.109
0.103
0.168
0.873
LDIf
0.118
0.137
0.130
0.160
0.690
0.580
LDIp
0.110
0.137
0.128
0.173
0.660
0.771 0.871
LDI2
0.071
0.069
0.063
0.067
0.109
0.118 0.446 0.419
LDIPerPrD
0.062
0.051
0.047
0.029
-0.127 -0.132 0.605 0.433 0.473
LCE
0.007
0.007
-0.002
-0.014
-0.256 -0.263 -0.003 0.004 -0.026
0.210
LARES2
0.263
0.234
0.215
0.135
0.019
0.007 0.037 0.031 0.020
0.021
0.029
LARES7
0.214
0.232
0.210
0.140
0.034
0.025 0.044 0.045 0.009
0.016
0.029 0.514
LUQEP
0.113
0.130
0.122
0.131
0.408
0.507 0.384 0.532 0.107
0.052
0.113 0.057
0.066
LUQEF
0.094
0.105
0.098
0.097
0.341
0.317 0.318 0.340 0.088
0.041
0.093 0.041
0.048
0.838
LFSIZE
-0.141
-0.180
-0.184
-0.212
-0.312 -0.357 -0.251 -0.198 0.095
0.022
-0.040 -0.007 -0.040
-0.316
-0.270
-0.198
Table 2 (continued)
a
The table shows Pearson product-moment (and Spearman rank-order) correlation coefficients between
pairs of log-transformed variables below (above) the diagonal. Correlation coefficients equal to or
greater than 0.014 (0.012) are significantly different from zero at α levels of ≤ 0.01 ( ≤ 0.05).
b
The variables are the logarithmic transformations of the raw variables MAVOL3, MAVOL7,
MAVOLLW1, MAVOLLW2, PrDf, PrDp, DIf, DIp, DI2, DIPerPrD, CEf, CEp, ARES2, and ARES7.
MAVOL3, MAVOL7, MAVOLLW1, and MAVOLLW2 are the average of a sampled firm's daily
percentage of shares traded relative to the daily percentage of shares traded on the overall market:
MAVOL3
-two-day window, from day –1 to day +1,
MAVOL7
-seven-day window, from day –1 to day +5,
MAVOLLW1
-first long window, from day –1 to day TA +1,
MAVOLLW2
-second long window, from day TB –1 to day TA +1
where day 0 is the quarterly earnings announcement date, TB is the date the least most current of the
individual analysts forecasts of annual EPS is made (within 45 days) before each sampled quarterly
earnings announcement, and TA is the date the last of the same individual analysts revises his/her
forecast of annual EPS (within 30 days) after each quarterly earnings announcement.
PrDf and PrDp are the standard deviation of the individual analysts' forecasts of annual EPS, divided by
the absolute value of the mean individual analysts' prior annual EPS forecast and stock price,
respectively.
DIf and DIp are the standard deviation of the revisions in the same individual analysts' forecasts of
annual EPS after each quarterly earnings announcement, divided by the absolute value of mean
individual analysts' prior annual EPS forecast and stock price, respectively.
DI2 is the variance of the difference in the rankings of analysts forecasts before and after each quarterly
earnings announcement normalized by the average rank before the announcement.
DIPerPrD is differential interpretation per unit of prior belief dispersion (i.e., DIf/PrDf or DIp/PrDp).
CE is the posterior belief dispersion, PsDf and PsDp, relative to the prior belief dispersion, PrDf and
PrDp, respectively, where PsDf and PsDp are defined as the standard deviation of the individual analysts'
posterior annual EPS forecasts, divided by the absolute value of the mean individual analysts' posterior
annual EPS forecast and stock price, respectively.
ARES2 and ARES7 are the absolute value of the market model residual returns, cumulated over the
following 2-day and 7-day windows relative to the quarterly earnings announcement date:
ARES2
ARES7
---
two-day window, from day –1 to day 0,
seven-day window, from day –1 to day +5.
UQE is Unexpected Quarterly Earnings.
FSIZE is Firm Size, defined as the market value of common equity just before Quarterly Earnings
Announcement.
Table 3
Results of regressions of trading volume metrics (LTV) on proxies for prior belief dispersion (LPrD),
differential interpretation (LDI) and consensus effect (LCE) a,b; N=26,169; 1984-2008.
Model I: LTV = α 0 + α1 LPrD+ α 2 LDI + α 3 LCE + u
Panel A
Independent Variables
α̂ 3 , LCE
Adjusted R2
αˆ 1 , LPrDf
α̂ 2 , LDI2
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.05519
(13.55)***
0.01670
(9.96)***
0.03659
(4.86)***
0.0119
LMAVOL7
0.06920
(19.00)***
0.01383
(9.22)***
0.04211
(6.26)***
0.0183
LMAVOLLW1
0.06307
(17.97)***
0.01193
(8.26)***
0.02967
(4.58)***
0.0160
LMAVOLLW2
0.08727
(27.09)***
0.01049
(7.91)***
0.02880
(4.84)***
0.0317
α̂ 3 , LCE
Adjusted R2
Dependent Variable, LTV
Independent Variables
Panel B
αˆ 1 , LPrDf
α̂ 2 , LDIf
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.00768
(1.32)*
0.05923
(12.52)***
0.01273
(1.64)*
0.0140
LMAVOL7
0.03088
(5.96)***
0.04789
(11.31)***
0.02282
(3.29)***
0.0199
LMAVOLLW1
0.02760
(5.53)***
0.04406
(10.81)***
0.01192
(1.78)**
0.0178
LMAVOLLW2
0.06093
(13.29)***
0.03321
(8.87)***
0.01542
(2.51)***
0.0323
Dependent Variable, LTV
Table 3 (continued)
a
The table shows estimated coefficients and t statistics (in parentheses) for the respective independent
variables in the model. ***, **, and * indicate one-tailed significance at the 1%, 5%, and 10% levels
respectively
b
The variables are the logarithmic transformations of the raw variables MAVOL3, MAVOL7,
MAVOLLW1, MAVOLLW2, PrDf, DIf, DIs, DI2, CE.
MAVOL3, MAVOL7, MAVOLLW1, and MAVOLLW2 are the average of a sampled firm's daily
percentage of shares traded relative to the daily percentage of shares traded on the overall market:
MAVOL3
-three-day window, from day –1 to day +1,
MAVOL7
-seven-day window, from day –1 to day +5,
MAVOLLW1
-first long window, from day –1 to day TA +1,
MAVOLLW2
-second long window, from day TB –1 to day TA +1
where day 0 is the quarterly earnings announcement date, TB is the date the least most current of the
individual analysts forecasts of annual EPS is made (within 45 days) before each sampled quarterly
earnings announcement, and TA is the date the last of the same individual analysts revises his/her
forecast of annual EPS (within 30 days) after each quarterly earnings announcement.
PrDf is the standard deviation of the individual analysts' forecasts of annual EPS before each quarterly
earnings announcement, divided by the absolute value of the mean individual analysts' prior annual EPS
forecast.
DIf is the standard deviation of the revisions in the same individual analysts' forecasts of annual EPS
after each quarterly earnings announcement, divided by the absolute value of mean individual analysts'
prior annual EPS forecast.
DI2 is the variance of the differences in the rankings of analyst forecasts before and after each quarterly
earnings announcement, normalized by the average rank before each quarterly earnings announcement.
CE is the posterior belief dispersion, PsDf and PsDp, relative to the prior belief dispersion, PrDf and
PrDp, respectively, where PsDf and PsDp are defined as the standard deviation of the individual analysts'
posterior annual EPS forecasts, divided by the absolute value of the mean individual analysts' prior
annual EPS forecast and stock price, respectively.
Table 4
Results of regressions of trading volume metrics (LTV) on proxies for prior belief dispersion
(LPrD), differential interpretation (LDI) and consensus effect (LCE) and the magnitude of
Quarterly Earnings Signal (LQES)a,b; N=26,169; 1984-2008.
Model II: LTV = β0 + β1 LPrD + β2LDI + β3LCE + β4LQES+ u'
Panel A
Independent Variables
Dependent
Variable, LTV
Adjusted R2
β̂1 , LPrDf
β̂2 , LDI2
β̂3 , LCE
β̂4 , LARES2
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.05081
(12.92)***
0.01543
(9.54)***
0.02535
(3.49)***
0.17160
(44.07)***
0.0801
LMAVOL7
0.06572
(18.55)***
0.01281
(8.79)***
0.03315
(5.06)***
0.13674
(38.96)***
0.0721
LMAVOLLW1
0.06000
(17.50)***
0.01104
(7.83)***
0.02177
(3.44)***
0.12054
(35.49)***
0.0612
LMAVOLLW2
0.08550
(26.78)***
0.00997
(7.59)***
0.02424
(4.11)***
0.06953
(21.98)***
0.0492
Adjusted R2
Panel B
Independent Variables
Dependent
Variable, LTV
β̂1 , LPrDf
β̂2 , LDI2
β̂3 , LCE
β̂4 , LARES7
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.04937
(12.39)***
0.01638
(10.00)***
0.02657
(3.61)***
0.14652
(35.12)***
0.0563
LMAVOL7
0.06358
(17.91)***
0.01352
(9.26)***
0.03243
(4.95)***
0.14157
(38.10)***
0.0699
LMAVOLLW1
0.05817
(16.93)***
0.01166
(8.26)***
0.02123
(3.34)***
0.12345
(34.32)***
0.0584
LMAVOLLW2
0.08433
(26.40)***
0.01033
(7.86)***
0.02373
(4.02)***
0.07414
(22.17)***
0.0495
Table 4 (continued)
Panel C
Independent Variables
Dependent
Variable, LTV
Adjusted R2
β̂1 , LPrDf
β̂2 , LDI2
β̂3 , LCE
β̂4 , LUQEf
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.03915
(8.95)***
0.01580
(9.43)***
0.02163
(2.82)***
0.02292
(9.95)***
0.0156
LMAVOL7
0.05503
(14.07)***
0.01303
(8.70)***
0.02889
(4.22)***
0.02025
(9.83)***
0.0219
LMAVOLLW1
0.05016
(13.31)***
0.01121
(7.76)***
0.01762
(2.67)***
0.01845
(9.29)***
0.0192
LMAVOLLW2
0.07995
(23.09)***
0.01008
(7.60)***
0.02196
(3.62)***
0.01047
(5.74)***
0.0329
Adjusted R2
Panel D
Independent Variables
Dependent
Variable, LTV
β̂1 , LPrDf
β̂2 , LDIf
β̂3 , LCE
β̂4 , LARES2
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.00731
(1.31)*
0.05429
(11.89)***
0.00351
(0.47)
0.17113
(43.98)***
0.0819
LMAVOL7
0.03058
(6.07)***
0.04395
(10.68)***
0.01547
(2.29)**
0.13637
(38.88)***
0.0734
LMAVOLLW1
0.02734
(5.60)***
0.04059
(10.19)***
0.00545
(0.83)
0.12017
(35.40)***
0.0627
LMAVOLLW2
0.06078
(13.37)***
0.03121
(8.41)***
0.01169
(1.92)**
0.06931
(21.92)***
0.0497
Table 4 (continued)
Panel E
Independent Variables
Dependent
Variable, LTV
Adjusted R2
β̂1 , LPrDf
β̂2 , LDIf
β̂3 , LCE
β̂4 , LARES7
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.00459
(0.81)
0.05606
(12.12)***
0.00404
(0.53)
0.14576
(34.97)***
0.0580
LMAVOL7
0.02789
(5.52)***
0.04481
(10.87)***
0.01442
(2.13)**
0.14097
(37.95)***
0.0710
LMAVOLLW1
0.02500
(5.12)***
0.04138
(10.37)***
0.00460
(0.70)
0.12289
(34.18)***
0.0598
LMAVOLLW2
0.05937
(13.06)***
0.03160
(8.51)***
0.01103
(1.81)**
0.07373
(22.04)
0.0499
Adjusted R2
Panel F
Independent Variables
Dependent
Variable, LTV
β̂1 , LPrDf
β̂2 , LDIf
β̂3 , LCE
β̂4 , LUQEf
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
−0.00499
(−0.84)
0.05582
(11.78)***
−0.00029
(−0.04)
0.02205
(9.57)***
0.0174
LMAVOL7
0.01964
(3.70)***
0.04485
(10.58)***
0.01126
(1.60)*
0.01957
(9.49)***
0.0233
LMAVOLLW1
0.01740
(3.40)***
0.04130
(10.12)***
0.00143
(0.21)
0.01777
(8.95)***
0.0208
LMAVOLLW2
0.05516
(11.73)***
0.03165
(8.43)***
0.00948
(1.52)*
0.01005
(5.51)***
0.0334
Table 4 (continued)
a
The table shows estimated coefficients and t statistics (in parentheses) for the respective independent
variables in the model. ***, **, and * indicate one-tailed significance at the 1%, 5%, and 10% levels
respectively.
b
The variables are the logarithmic transformations of the raw variables MAVOL3, MAVOL7,
MAVOLLW1, MAVOLLW2, PrDf, DIf, DIs, DI2, CE, ARES2, and ARES7.
MAVOL3, MAVOL7, MAVOLLW1, and MAVOLLW2 are the average of a sampled firm's daily
percentage of shares traded relative to the daily percentage of shares traded on the overall market:
MAVOL3
-three-day window, from day –1 to day +1,
MAVOL7
-seven-day window, from day –1 to day +5,
MAVOLLW1
-first long window, from day –1 to day TA +1,
MAVOLLW2
-second long window, from day TB –1 to day TA +1
where day 0 is the quarterly earnings announcement date, TB is the date the least most current of the
individual analysts forecasts of annual EPS is made (within 45 days) before each sampled quarterly
earnings announcement, and TA is the date the last of the same individual analysts revises his/her
forecast of annual EPS (within 30 days) after each quarterly earnings announcement.
PrDf is the standard deviation of the individual analysts' forecasts of annual EPS before each quarterly
earnings announcement, divided by the absolute value of the mean individual analysts' prior annual EPS
forecast.
DIf is the standard deviation of the revisions in the same individual analysts' forecasts of annual EPS
after each quarterly earnings announcement, divided by the absolute value of mean individual analysts'
prior annual EPS forecast.
DI2 is the variance of the differences in the rankings of analyst forecasts before and after each quarterly
earnings announcement, normalized by the average rank before each quarterly earnings announcement.
CE is the posterior belief dispersion, PsDf and PsDp, relative to the prior belief dispersion, PrDf and
PrDp, respectively, where PsDf and PsDp are defined as the standard deviation of the individual analysts'
posterior annual EPS forecasts, divided by the absolute value of the mean individual analysts' prior
annual EPS forecast and stock price, respectively.
ARES2 and ARES7 are the absolute value of the market model residual returns, cumulated over the
following 2-day and 7-day windows relative to the quarterly earnings announcement date:
ARES2
-two-day window, from day –1 to day 0,
ARES7
-seven-day window, from day –1 to day +5.
UQE is Unexpected Quarterly Earnings.
Table 5
Results of regressions of trading volume metrics (LTV) on proxies for prior belief dispersion (LPrD),
differential interpretation (LDIPerPrD), consensus effect (LCE) and the magnitude of the quarterly
earnings signal (LQES)a,b; N=26,169; 1984-2008.
A
A
A
A
Model Ia: LTV = α 0 + α1 LPrD + α 2 LDIPerPrD+ α 3 LCE + u"
A
A
A
A
A
Model IIa: LTV = β 0 + β1 LPrD + β 2 LDIPerPrD+ β 3 LCE + β 4 LPEQES+ u' ' '
Panel A
Independent Variables
Adjusted R2
α̂1 , LPrDf
α̂1 , LDIPerPrD
α̂1 , LCE
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.06311
(15.55)***
0.03889
(11.23)***
0.02084
(2.72)***
0.0129
LMAVOL7
0.07562
(20.83)***
0.03061
(9.88)***
0.02971
(4.34)***
0.0188
LMAVOLLW1
0.06876
(19.66)***
0.02813
(9.43)***
0.01828
(2.77)***
0.0168
LMAVOLLW2
0.09191
(28.62)***
0.02070
(7.56)***
0.02041
(3.37)***
0.0315
Dependent
Variable, LTV
Independent Variables
Panel B
Dependent
Variable, LTV
Adjusted R2
β̂1 , LPrDf
β̂2 , LDIPerPrD
β̂3 , LCE
β̂4 , LARES2
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.05817
(14.85)***
0.03636
(10.88)***
0.01063
(1.44)*
0.17154
(44.07)***
0.0811
LMAVOL7
0.07168
(20.30)***
0.02859
(9.49)***
0.02157
(3.24)***
0.13672
(38.97)***
0.0726
LMAVOLLW1
0.06529
(19.11)***
0.02635
(9.04)***
0.1111
(1.72)**
0.12049
(35.49)***
0.0619
LMAVOLLW2
0.08991
(28.24)***
0.01968
(7.25)***
0.01627
(2.71)***
0.06957
(21.99)***
0.0490
Table 5 (continued)
Panel C
Independent Variables
Dependent
Variable, LTV
Adjusted R2
β̂1 , LPrDf
β̂2 , LDIPerPrD
β̂3 , LCE
β̂4 , LARES7
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.05708
(14.38)***
0.03733
(11.03)***
0.01148
(1.53)*
0.14615
(35.05)***
0.0571
LMAVOL7
0.06979
(19.73)***
0.02910
(9.65)***
0.02066
(3.10)***
0.14129
(38.03)***
0.0702
LMAVOLLW1
0.06368
(18.60)***
0.02681
(9.19)***
0.01039
(1.61)*
0.12319
(34.25)***
0.0589
LMAVOLLW2
0.08886
(27.90)***
0.01992
(7.34)***
0.01568
(2.61)***
0.07396
(22.11)***
0.0492
Adjusted R2
Panel C
Independent Variables
Dependent
Variable, LTV
β̂1 , LPrDf
β̂2 , LDIPerPrD
β̂3 , LCE
β̂4 , LUQE
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.04678
(10.69)***
0.03703
(10.70)***
0.00674
(0.87)
0.02276
(9.88)***
0.0165
LMAVOL7
0.06115
(15.62)***
0.02896
(9.35)***
0.01721
(2.48)***
0.02017
(9.79)***
0.0223
LMAVOLLW1
0.05562
(14.75)***
0.02663
(8.93)***
0.00692
(1.03)
0.01832
(9.23)***
0.0199
LMAVOLLW2
0.08438
(24.34)***
0.01985
(7.24)***
0.01391
(2.26)**
0.01050
(5.75)***
0.0327
Table 5 (continued)
a
The table shows estimated coefficients and t statistics (in parentheses) for the respective independent
variables in the model. ***, **, and * indicate one-tailed significance at the 1%, 5%, and 10% levels
respectively.
b
The variables are the logarithmic transformations of the raw variables MAVOL3, MAVOL7,
MAVOLLW1, MAVOLLW2, PrDf, DIPerPrD, CE, ARES2, and ARES7.
MAVOL3, MAVOL7, MAVOLLW1, and MAVOLLW2 are the average of a sampled firm's daily
percentage of shares traded relative to the daily percentage of shares traded on the overall market:
MAVOL3
-three-day window, from day –1 to day +1,
MAVOL7
-seven-day window, from day –1 to day +5,
MAVOLLW1
-first long window, from day –1 to day TA +1,
MAVOLLW2
-second long window, from day TB –1 to day TA +1
where day 0 is the quarterly earnings announcement date, TB is the date the least most current of the
individual analysts forecasts of annual EPS is made (within 45 days) before each sampled quarterly
earnings announcement, and TA is the date the last of the same individual analysts revises his/her
forecast of annual EPS (within 30 days) after each quarterly earnings announcement.
PrDf is the standard deviation of the individual analysts' forecasts of annual EPS before each quarterly
earnings announcement, divided by the absolute value of the mean individual analysts' prior annual EPS
forecast.
DIPerPrD is defined as DI divided by PrD.
DI is the standard deviation of the revision in the same individual analysts' annual EPS forecasts from
immediately before to immediately after each quarterly earnings announcement divided by the absolute
value of the mean individual analysts' annual EPS forecast immediately before each quarterly earnings
announcement.
PrD is the standard deviation of the individual analysts' forecasts of annual EPS immediately before
each quarterly earnings announcement, divided by the absolute value of the mean individual analysts'
annual EPS forecast immediately before each quarterly earnings announcement.
CE is the posterior belief dispersion, PsDf and PsDp, relative to the prior belief dispersion, PrDf and
PrDp, respectively, where PsDf and PsDp are defined as the standard deviation of the individual analysts'
posterior annual EPS forecasts, divided by the absolute value of the mean individual analysts' prior
annual EPS forecast and stock price, respectively.
ARES2 and ARES7 are the absolute value of the market model residual returns, cumulated over the
following 2-day and 7-day windows relative to the quarterly earnings announcement date:
ARES2 -ARES7
two-day window, from day –1 to day 0,
-seven-day window, from day –1 to day +5.
UQE is Unexpected Quarterly Earnings.
Table 6
Results of regressions of trading volume metrics (LTV) on proxies for prior belief dispersion
(LPrD), differential interpretation (LDI) and consensus effect (LCE) and Firm Size (LFSIZE)
N=26,169; 1984-2008.
B
B
B
B
B
Model II: LTV = β0 + β1 LPrD + β2 LDI + β3 LCE + β4 LFSIZE+ u' ' ' '
Independent Variables
Dependent
Variable, LTV
Adjusted R2
β̂1 , LPrDf
β̂2 , LDI2
β̂3 , LCE
β̂4 , LFSIZE
coefficient
estimate
coefficient
estimate
coefficient
estimate
coefficient
estimate
LMAVOL3
0.02421
(5.62)***
0.02153
(12.82)***
0.01625
(2.16)**
−0.07897
(−20.73)***
0.0278
LMAVOL7
0.03478
(9.07)***
0.01919
(12.84)***
0.01951
(2.91)***
−0.08773
(−25.87)***
0.0428
LMAVOLLW1
0.02877
(7.79)***
0.01728
(12.01)***
0.00715
(1.31)*
−0.08743
(−26.79)***
0.0423
LMAVOLLW2
0.05329
(15.76)***
0.01578
(11.98)***
0.00648
(1.30)*
−0.08661
(−28.98)***
0.0618
Table 6 (continued)
a
The table shows estimated coefficients and t statistics (in parentheses) for the respective independent
variables in the model. ***, **, and * indicate one-tailed significance at the 1%, 5%, and 10% levels
respectively.
b
The variables are the logarithmic transformations of the raw variables MAVOL3, MAVOL7,
MAVOLLW1, MAVOLLW2, PrDf, DIf, DIs, DI2, CE, ARES2, and ARES7.
MAVOL3, MAVOL7, MAVOLLW1, and MAVOLLW2 are the average of a sampled firm's daily
percentage of shares traded relative to the daily percentage of shares traded on the overall market:
MAVOL3
-three-day window, from day –1 to day +1,
MAVOL7
-seven-day window, from day –1 to day +5,
MAVOLLW1
-first long window, from day –1 to day TA +1,
MAVOLLW2
-second long window, from day TB –1 to day TA +1
where day 0 is the quarterly earnings announcement date, TB is the date the least most current of the
individual analysts forecasts of annual EPS is made (within 45 days) before each sampled quarterly
earnings announcement, and TA is the date the last of the same individual analysts revises his/her
forecast of annual EPS (within 30 days) after each quarterly earnings announcement.
PrDf is the standard deviation of the individual analysts' forecasts of annual EPS before each quarterly
earnings announcement, divided by the absolute value of the mean individual analysts' prior annual EPS
forecast.
DIf is the standard deviation of the revisions in the same individual analysts' forecasts of annual EPS
after each quarterly earnings announcement, divided by the absolute value of mean individual analysts'
prior annual EPS forecast.
DI2 is the variance of the differences in the rankings of analyst forecasts before and after each quarterly
earnings announcement, normalized by the average rank before each quarterly earnings announcement.
CE is the posterior belief dispersion, PsDf and PsDp, relative to the prior belief dispersion, PrDf and
PrDp, respectively, where PsDf and PsDp are defined as the standard deviation of the individual analysts'
posterior annual EPS forecasts, divided by the absolute value of the mean individual analysts' prior
annual EPS forecast and stock price, respectively.
FSIZE is Firm Size, defined as the market value of common equity just before Quarterly Earnings
Announcement.