A Mind is a Terrible Thing to Change: Confirmatory Bias in Financial

A Mind is a Terrible Thing to Change: Confirmatory
Bias in Financial Markets∗
Sebastien Pouget†, Julien Sauvagnat‡, and Stephane Villeneuve
§
April 22, 2014
∗
We are grateful to Nicholas Barberis, Bruno Biais, Sylvain Friederich, Ron Kaniel, Hyun-woo Lee, Thomas
Mariotti, Richard Payne, Matthew Rabin, Jean-Charles Rochet, Hersh Shefrin, Dimitri Vayanos, Wei Xiong, seminar
participants at Bristol University and Toulouse University, and participants to the 2010 American Finance Association
meeting and the fall 2011 NBER Behavioral Finance meeting for providing useful comments. A previous version of
this paper circulated under the title “Price formation with confirmation bias”. All errors are ours. Financial support
from the Agence Nationale de la Recherche (ANR-09-BLAN-0358-01) and from the Chaire SCOR at IDEI-R is
gratefully acknowledged.
†
Toulouse School of Economics (University of Toulouse-IAE-CRM-IDEI), 21 allee de Brienne, 31 000 Toulouse,
[email protected]
‡
Toulouse School of Economics and ENSAE-CREST, 3 avenue Pierre Larousse, 92245 Malakoff,
[email protected]
§
Toulouse School of Economics (University of Toulouse-CRM-IDEI), 21 allee de Brienne, 31000 Toulouse,
[email protected]
A Mind is a Terrible Thing to Change: Confirmatory Bias in Financial Markets
Abstract
This paper studies the impact of the confirmatory bias on financial markets. Building on
Rabin and Schrag (1999), we propose a model in which some traders misinterpret new evidence as confirming their prior beliefs regarding future asset cash flows. The confirmatory bias
provides a unified rationale for several stylized facts including excess volatility, excess volume
and momentum, and delivers novel predictions: differences of opinion and volume should be
larger when past subsequent returns have different signs. Using data on U.S. stocks, we find
strong empirical support for these predictions, suggesting that the confirmatory bias is at work
in financial markets.
Keywords: financial markets, psychological biases, confirmatory bias, momentum, bubbles,
trading strategies, volume, differences of opinion.
“A mind is a terrible thing to change... you believe stocks are going to outperform other assets,
and all you can hear are warnings of the bloodbath to come in the bond and commodity markets.
In short, your own mind acts like a compulsive yes-man who echoes whatever you want to believe.”
(Jason Zweig, in the Wall Street Journal, November 19, 2009)
The psychology literature defines the confirmatory bias as “the seeking or interpreting of evidence in ways that are partial to existing beliefs” (Nickerson, 1998).1 This bias is intimately related
to the dynamics of belief formation and thus appears particularly relevant in the context of trading
and investing activities. To the best of our knowledge, the present paper is the first to study how
the confirmatory bias affects asset pricing and trading volume in financial markets.
Building on Rabin and Schrag (1999), we propose a simple dynamic model of financial markets in which some traders are prone to the confirmatory bias: biased traders may misinterpret
information as confirming their prior views. In a framework with public information only, this bias
creates differences of opinion between rational speculators and confirmatory-biased traders over the
1
In his book surveying biases in human reasoning, Jonathan Evans, a leading scholar in psychology, refers to the
confirmatory bias as “the best known and most widely accepted notion of inferential error” (Evans, 1989). Starting
with the seminal contributions of Lord, Ross, and Lepper (1979) and Darley and Gross (1983), the confirmatory bias
has been extensively documented by psychologists. The prevalence of this bias has recently been confirmed by Hart
et al. (2009) in a meta-analysis based on 67 articles and more than 8,000 individuals.
1
interpretation of public information. These differences of opinion in turn give rise to trading. Speculators take opposite positions with respect to biased traders and thus have a corrective impact on
prices. Transaction costs however limit the effectiveness of corrective strategies causing the views
of both speculators and biased traders to be incorporated into asset prices.
To understand how the confirmatory bias may affect financial markets, consider that traders
initially hold positive views about a particular future asset cash flow. If subsequent information
is also positive, then all traders interpret it correctly. However, if this information is negative,
confirmatory-biased traders have a given probability to misinterpret the negative information for a
positive one. We indeed consider that information reaching the market is soft in the sense that it
reflects a lot of underlying pieces of news, as for example in a quarterly financial report, that may
be positive and negative. In this case, as documented by Bodenhausen (1988), the confirmatory
bias may induce some agents to neglect the news that are inconsistent with their priors, and thus
to misinterpret the overall meaning of the information.
Despite the arrival of negative news, some biased traders become even more positive than they
initially were. Other traders are instead more pessimistic because they have correctly interpreted
the news. This phenomenon, referred to as belief polarization, has for example been observed
during field experiments on political stock markets by Forsythe, Nelson, Neumann, and Wright
(1992). Depending on the weight of the various types of traders, it is thus possible for the asset
price to go up despite the arrival of negative public news.
Our model shows that the confirmatory bias provides a unified explanation for several stylized
facts, including excess volume (De Bondt and Thaler (1995)), excess volatility (Leroy and Porter
(1981), Shiller (1981)), momentum (Jegadeesh and Titman (1993)), and volume-based momentum
(Lee and Swaminathan (2000)). Some behavioral finance theories have been proposed to account
for these stylized facts (see for example Hirshleifer (2001) for a survey). Overconfidence may explain excess volume, excess volatility and even momentum when coupled with self-attribution bias
(see Benos (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and Odean (1998)). The representativeness heuristics may also rationalize momentum (see Barberis, Shleifer and Vishny (1998)
and Rabin and Vayanos (2010)). Finally, Hong and Stein (2007) argue that gradual information
flow and limited attention may explain excess volume, momentum and volume-based momentum.
To show the unique impact of the confirmatory bias, we derive novel theoretical predictions.
Our model predicts that, when some traders suffer from the confirmatory bias, differences of opinion
2
and volume should be larger when past subsequent returns have different signs. This is because conflicting information at the origin of the changes in return sign opens opportunities for biased traders
to misinterpret information. Misinterpretation then creates differences of opinion and translates
into volume.
We next test these novel theoretical predictions using data on U.S. stock markets over the
period 1982 to 2011. We follow the literature (see, e.g., Diether, Malloy, and Scherbina (2002))
and use dispersion in analysts’ earnings forecasts as a proxy for differences of opinion. We find
that forecasts dispersion and trading volume at the end of the fiscal year are significantly larger
when stock returns during the second and third quarters of the year are of opposite signs.2 These
empirical results suggest that the confirmatory bias is at work in financial markets.
Finally, we characterize what trading strategies are optimal for short-term traders who aim at
exploiting mispricings created by the confirmatory bias. We show that these strategies involve riding
bubble, thus complementing the analysis of Abreu and Brunnermeier (2002) on synchronization
risk and delayed arbitrage. This is in line with the evidence offered by Brunnermeier and Nagel
(2004) for the Dot Com bubble, and by Temin and Voth (2004) for the South Sea bubble. In
our model, such behavior of short-term traders is in sharp contrast with the behavior of long-term
speculators who have a corrective impact on prices.
Overall, our paper’s contribution is threefold. First, we propose a parsimonious and tractable
model in which departure from perfect rationality is well-grounded in psychology and driven by
only one parameter: when the severity of the confirmatory bias is null, we find back the perfectly
rational benchmark. Second, we show that the confirmatory bias alone offers a unified rationale
for several existing stylized facts, complementing previous explanations offered in the behavioral
finance literature. Finally, we deliver and test novel empirical predictions that follow from the
confirmatory bias.
1.
The model
We consider a pure exchange economy with one risky asset in fixed supply and a riskless asset in
perfectly elastic supply. The riskless asset rate of return is normalized to zero. There are 4 dates
of trading indexed by t ∈ {0, 1, 2, 3}. Consumption occurs only at date 4 when the risky asset
2
Our analyses control for numerous factors that are known to affect differences of opinion and volume, including
analyst coverage, market capitalization, standard deviation of daily raw returns, book-to-market ratio, and return on
asset. Finally, we obtain identical results when the sample is restricted to firms with fiscal year ending in December.
3
distributes a dividend v.3 The probability distribution of the random variable v depends on the
non observable state of the economy H or L. We assume that, conditional on the state of nature
X ∈ {H, L}, v has a Bernoulli distribution given by:
P(v = 1|X) = pX = 1 − P(v = 0|X) with pH > pL .
(1.1)
There is a continuum of traders with mass normalized to one indexed by j ∈ [0, 1]. A trader
pertains to one of two groups of agents. Speculators are represented by the subset [λ, 1], and biased
traders are represented by the subset [0, λ]. Therefore, the mass of speculators is 1 − λ while the
mass of biased traders is λ. Each trader is endowed with one unit of the risky asset and no cash.
Finally, information about the dividend payment is generated by the observation of public signals
st whose probability distribution depends on the state of nature. More precisely, we assume
1
P(st = 1|X = H) = P(st = −1|X = L) = θ > .
2
(1.2)
This public signal constitutes soft information in the sense that it opens up a scope for misinterpretation. For example, the public signal could represent the announcement of a change in
corporate strategy or a report regarding the prospects of a firm’s future products. We consider
below that traders who suffer from the confirmatory bias might ignore some pieces of information
that are not in line with their prior beliefs and thus end up misperceiving the overall meaning of
the public signal. On the contrary, we consider that the amount of dividend distributed at date 4
is hard information in the sense that it cannot be misperceived by biased traders.
In order to focus on the informational aspects of financial markets, we assume that traders are
risk neutral.4 Absent market frictions, risk neutrality implies that traders would stand ready to
exchange infinite amounts as long as prices do not equal their expectation of the asset value. This
would prevent the existence of an equilibrium since the market would not clear. In order to avoid
this phenomenon, we assume that traders incur an exogenous trading cost that is quadratic in the
quantity traded and parameterized by
c
2
> 0.5 The total cost of trading for trader j at date t with
3
One could extend the model to include an arbitrary number of periods, finite or infinite. In this case, dividend
distribution and consumption would take place at various dates across time. This would not affect our results.
4
This is in the spirit of Harris and Raviv (1993) and isolates our analysis from the influence of trading motives
based on risk sharing.
5
Alternatively, we could ensure existence of an equilibrium by assuming that traders can only trade up to a fixed
amount of shares as in Abreu and Brunnermeier (2002). This different modeling framework would not affect our
results.
4
a demand djt is thus equal to 2c (djt )2 , for all t. Trader j’s objective at each date is thus to maximize
2 Pt=3 j
j
c
j
+ v, conditional on information
expected wealth at date 4, W =
t=0 dt (v − Pt ) − 2 dt
available at this date.
The cost
c
2
can be viewed as an explicit transaction cost traders have to pay to submit orders or
as a proxy for the imperfect depth of financial markets. Imperfect market depth could be related
to inventory or adverse selection risks borne by liquidity providers (see Madhavan (2000) and
Biais, Glosten, and Spatt (2005) for surveys of the market microstructure literature dealing with
those issues). The transaction cost creates limits to arbitrage and opens the scope for potential
mispricings.6
In our model, differences of opinion emerge because all traders do not encode the public signal
in the same way. On the one hand, speculators are perfectly rational in the sense that they are
endowed with the actual probability model P. On the other hand, each individual biased trader
i ∈ [0, λ] is endowed with a different probability model Pi . When receiving a public signal st , biased
trader i actually sees a different signal σti , for every t. To incorporate the fact that individual biased
traders believe that they correctly perceive a signal st when they in fact observe σti , we consider
that, under Pi the probability distribution of σti is given by equation (1.2) and the probability
distribution of v is given by equation (1.1).7
To incorporate the confirmatory bias in our framework, we follow Rabin and Schrag (1999) and
assume that biased traders misinterpret public information when it is inconsistent with their prior
beliefs concerning the final dividend payment. To do so, we have to specify belief dynamics on the
state of the economy for each type of agents.
We assume that ex-ante, the two states of nature are equiprobable for each type of traders,
that is P(X = H) = Pi (X = H) =
1
2
for every i. We denote by µt = P (X = H|s1 , . . . , st ),
the beliefs of rational speculators (under the correct probability model P) given the information
they have received up to date t. To alleviate notation, we define the signal history up to time t
as ht = {s1 , . . . , st }. We denote by µit = Pi X = H|σ1i , . . . , σti , biased trader i’s beliefs (under
probability model Pi ) given the information he has received up to date t.
6
Alternative frameworks generating limits to arbitrage include noise trader risk as in De Long, Shleifer, Summers,
and Waldmann (1990), short horizons as in Shleifer and Vishny (1997), and synchronization risk as in Abreu and
Brunnermeier (2002). Barberis and Thaler (2001) survey this literature.
7
This implies that, before receiving the first public signal, biased traders have a correct understanding of the
statistical model underlying financial markets. As stated above, their bias derives only from their improper perception
of information. Except from this bias, biased traders maximize their expected utility, update their beliefs using Bayes’
rule, and have rational expectations.
5
We consider that a signal st is inconsistent with prior beliefs µit−1 if st has a different sign than
µit−1 − 21 , the difference between the conditional and the unconditional belief on the state of the
economy. When biased trader i is bullish (i.e., when he believes that the good state of the economy
is more likely), he might underreact to negative signals. Likewise, when biased trader i is bearish,
he might underreact to positive signals.
We can now formally define how the confirmatory bias affects biased traders’ perception of
information. The following definition precisely characterizes biased traders’ distortion of public
signals.
Definition 1.1. For i ∈ [0, λ], the information perceived by biased trader i is σ1i = s1 and, for
t ∈ {2, 3}, under the probability measure P:
σti = st 11(µi
1
t−1 − 2 )st >0
+ zti st 11(µi
1
t−1 − 2 )st <0
+ 11µi
1
t−1 = 2
st 11(µi
1
t−2 − 2 )st >0
+ zti st 11(µi
1
t−2 − 2 )st <0
, (1.3)
where zti is a random variable taking values in {−1, 1} with probability distribution P(zti = −1) = q,
and 11(.) is the indicator function that takes the value 1 if the condition is satisfied and 0 otherwise.
We assume that the random variables (zti )i∈[0,λ] are independent and that zti is independent from zsi
for s 6= t.
This definition deserves some comments. First, a biased trader changes the sign of a public
signal that is inconsistent with prior beliefs with probability q. The greater q, the more severe is the
confirmatory bias. Second, a proportion q of biased traders whose prior beliefs are inconsistent with
the signal misperceive the public signal st . This is because the random variable zti is independent
across biased traders and because traders form a continuum. Third, we assume, for simplicity, that
biased traders do not misperceive signals that are consistent with prior beliefs. Fourth, the third
element of the right-hand side of equation (1.3) indicates what distortion occurs when a biased
trader’s current belief has no valence (i.e., when µit−1 = µi0 =
1
2 ).
In this case, equation (1.3)
specifies that the bias depends on the valence of past beliefs µit−2 . Fourth, at date 1, biased traders
see the actual signal because they do not have formed an opinion yet. Their perception is thus not
biased at this date. We now turn to the analysis of the model.
6
2.
Equilibrium prices and rational benchmark
Our model features quadratic transaction costs. Individual asset demands are thus finite despite
risk neutrality and there exists an equilibrium. Standard arguments, developed in the Appendix,
show that prices in our financial market are weighted averages of speculators’ and biased traders’
beliefs given their respective information set. They are given in the following proposition where
we denote by Pt the conditional probability P(.|ht ) and by Pit the probability Pi (.|(z i h)t ), where
(z i h)t = {s1 , z2i s2 , . . . , zti st }.
Proposition 2.1. At each date t ∈ {0, 1, 2, 3}, the equilibrium price is given by:
λ
Z
Pit (v = 1) di + (1 − λ)Pt (v = 1)
Pt =
0
⇔
Z
Pt = pL + (pH − pL ) (1 − λ) µt +
λ
µit di .
0
The price is thus a linear function of traders’ average belief. Let us describe explicitly the
dynamics of equilibrium prices. Denote by Pt (ht ) the equilibrium price that prevails after a history
ht of actual public signals. At times t ∈ {0, 1}, all agents perceive correctly the public signal,
yielding:
1
P0 = pL + (pH − pL ) ,
2
P1 (1) = pL + (pH − pL )θ,
and
P1 (−1) = pL + (pH − pL )(1 − θ).
At time 2, the confirmatory bias may affect biased traders’ perception: they are indeed optimistic if s1 = 1 or pessimistic if s1 = −1. Each biased agent has a probability q to change the sign
of s2 . Due to the assumption of independence of the variables z i , the law of large numbers implies
that there is a proportion q of biased agents that have a bad perception of the public signal. As a
7
consequence, P2 (s1 , s2 ) can take three values:
P2 (1, 1) = pL + (pH − pL )θ(2, 2),
P2 (1, −1) = P2 (−1, 1) = pL + (pH − pL )[(1 − λq)θ(1, 2) + λqθ(2, 2)],
P2 (−1, −1) = pL + (pH − pL )θ(0, 2),
where θ(x, y) = P(v = 1|hy ∈ {X}), with X being the set of histories of y signals that include
exactly x positive signals. θ(x, y) is thus the probability that the future dividend is high given that
an agent has perceived x positive signals out of y. The explicit expression for θ(x, y) is provided in
the Appendix. Notice that the same function θ(x, y) applies both for rational and biased traders.
The difference in beliefs between these traders only comes from the fact that they may not perceive
the same number of positive and negative signals.
In our model, beliefs are symmetric around
1
2.
Without loss of generality, we thus focus on
paths ht such that s1 = 1. On these paths, P3 (s1 = 1, s2 , s3 ) can take three values:
P3 (1, 1, 1) = pL + (pH − pL )θ(3, 3),
P3 (1, 1, −1) = P3 (1, −1, 1) = pL + (pH − pL )[(1 − λq)θ(2, 3) + λqθ(3, 3)],
P3 (1, −1, −1) = pL + (pH − pL )[(1 − λq)θ(1, 3) + λq(1 − q)θ(2, 3) + λq 2 θ(3, 3)].
Before analyzing the impact of the confirmatory bias on market outcomes, it is useful to study
the benchmark case in which all traders are perfectly rational. Endogenous prices in this benchmark
are indicated by a star. This benchmark is nested in our model and corresponds to the case in
which λ = 0 or q = 0. In this case, we have Pt∗ = Pt (v = 1) = pL +(pH −pL )µt . Given the structure
of the uncertainty in our model, it is straightforward to show that the following proposition holds.
Proposition 2.2. When all traders are perfectly rational (that is, when λ = 0 or q = 0), market
outcomes are as follows:
• After the first public signal, expected prices are constant. In particular:
E (P3∗ |s1 = 1) = E (P2∗ |s1 = 1) = P1∗ (s1 = 1) .
8
• Expected returns are null. In particular:
E (P2∗ − P1∗ |P1∗ − P0∗ ) = E (P3∗ − P2∗ |P2∗ − P1∗ ) = 0.
• Volume is null at all date. In particular, for all t:
Vt∗ = 0.
• There is no correlation between returns and volume.
This proposition indicates that, when all traders are rational, there is no bubble. This is a
consequence of the fact that prices are a martingale. It also shows that there is no momentum. It
finally characterizes volume, volume at date t being defined as the sum of the quantities purchased
R1
at price Pt : Vt = 0 djt 11Ej (v)>Pt dj. In our model, there is no volume when all traders are rational.
t
This is because the only motive for trading, related to differences of opinion, is absent when all
traders are rational.
3.
Financial markets with the confirmatory bias
We now focus on the case in which some traders suffer from the confirmatory bias, i.e., on the case in
which λ > 0 and q > 0. The confirmatory bias then generates differences of opinion and systematic
mistakes. We first offer an explanation for various phenomena that have been documented in actual
financial markets: excess volatility, excess volume, momentum and volume-based momentum. We
then characterize optimal short-term trading strategies that involve bubble riding and sometimes
contrarian trading patterns. Finally, we highlight the distinctive impact of the confirmatory bias
by deriving new theoretical predictions on how past returns relate to future differences of opinion
and trading volume.
We start with an example of price and trade formation to explain how beliefs, prices and volume
evolve over time. For traders who suffer from the confirmatory bias, first impression matters: once
they have formed an opinion, biased traders tend to misperceive subsequent signals that contradict
their prior beliefs. To understand how the confirmatory bias influences financial markets, it is
thus particularly interesting to focus on mixed history paths, i.e., paths that include signals with
different signs.
9
3.1
An example of price and trade formation
Consider for example the mixed history path (s1 = 1, s2 = −1, s3 = −1). After observing s1 = 1,
traders become optimistic: their beliefs at time 1 are higher than at time 0, µ1 = µi1 = θ > µ0 = 12 .
As a result, the price increases, P1 − P0 > 0. Note that, at the first two dates, t = {0, 1}, there is
no volume because all traders have the same beliefs.
Differences of opinion arise after the occurrence of the negative signal s2 = −1. Rational
speculators perceive correctly this information and their beliefs come back to their initial level, i.e.,
µ2 = µ0 = 12 . Biased traders on the other hand are split in two groups. One group, representing
a proportion λ(1 − q) of the traders, perceives the negative signal correctly and ends up with the
correct belief µi2 = µ2 = 21 . The other group, in proportion λq, perceives a positive signal instead
of the actual negative one. These biased traders end up with an even more optimistic belief,
µi2 = θ(2, 2) =
θ2
θ2 +(1−θ)2
> µi1 . These differences of opinion generate trade. Note that it is not
clear whether the price actually decreases following the arrival of a negative public signal. Indeed,
P2 may be higher than P1 if the proportion of biased traders, λ, or the severity of the bias, q, are
large enough.
Finally, after the third public signal announcement (s3 = −1), rational speculators’ beliefs
become pessimistic, µ3 = 1 − θ < 21 . In this case, biased traders are split in three groups. Some
biased traders, who interpreted correctly all public signals, hold the same pessimistic beliefs as
rational speculators, µi3 = 1 − θ <
1
2;
these traders are in proportion λ(1 − q)2 . The second
group includes biased traders who have misinterpreted one of the last two signals. They hold
positive beliefs, µi3 = µ1 > 21 , and are in proportion 2λq(1 − q). The last group of biased traders
includes the ones who have misinterpreted the last two public signals. They hold extremely positive
beliefs, µi3 = θ(3, 3) =
θ3
θ3 +(1−θ)3
> µ1 , and are in proportion λq 2 . Again, differences of opinion
generate trading. The price P3 may increase despite the arrival of a negative public signal, when
the proportion of biased traders or the severity of the bias are large enough.
3.2
Explaining existing stylized facts on financial markets
We now show that the confirmatory bias provides a unified explanation for several empirically
documented phenomena including excess volume, excess volatility, momentum, and volume-based
momentum. First, De Bondt and Thaler (1995, p. 392) indicate that “the high trading volume on
organized exchanges is perhaps the single most embarrassing fact to the standard finance paradigm”.
10
Moreover, starting with Leroy and Porter (1981) and Shiller (1981), several contributions report
that asset prices are excessively volatile. These two phenomena constitute long-standing puzzles,
even if some behavioral explanations have already been proposed in the literature as discussed
below.
Jegadeesh and Titman (1993) document a momentum effect in the U.S. stock market: at
quarterly, semi-annual or yearly horizons, stocks that have performed well in the past have a better
performance than those that performed poorly. This result was confirmed in an international
context by Rouwenhorst (1998). Momentum has also been uncovered in other financial markets,
such as commodities and government bonds, for example by Asness, Moskowitz and Pedersen
(2013). Lee and Swaminathan (2000) further show that the momentum effect is stronger for stocks
with large past trading. Verardo (2009) indicates that this volume-based momentum is robust to the
inclusion of various control variables, including a stock’s media exposure and speed of information
diffusion.
Finally, several authors have argued that financial markets go through periods of bubbles and
crashes. For example, Shiller (2000) indicates that stock price run-ups followed by bursts tend to
occur after new technological discoveries that suggest a new “era” has begun. Xiong and Yu (2011)
offer clean evidence of a bubble on the Chinese warrant market.
The next proposition indicates that our model can accommodate these various stylized facts.
Proposition 3.3. When some traders are prone to the confirmatory bias (that is, when λ > 0 and
q > 0), market outcomes are as follows:
• There is excess volume at t ∈ {2, 3}:
Vt > Vt∗ = 0.
• There is excess volatility in asset prices at t = 2 and, for θ close to
t = 3; in this case:
Var(Pt ) > Var(Pt∗ ).
• There is momentum in asset prices at t = 1:
E(P2 − P1 |P1 − P0 > 0) > 0.
11
1
2
and q(1 + 4λ) > 2, at
• There is momentum in asset prices at t = 2 when λq > θ2 + (1 − θ)2 or when q > q¯ (the
analytical expression for q¯ is given in the Appendix):
E(P3 − P2 |P2 − P1 > 0) > 0,
• There is volume-based momentum in asset prices when λq > θ2 + (1 − θ)2 :
E(P3 − P2 |P2 − P1 > 0; V2 > 0) ≥ E(P3 − P2 |P2 − P1 > 0; V2 = 0).
• A bubble forms after a good initial public signal:
E (P3 |s1 = 1) > E (P2 |s1 = 1) > P1∗ (s1 = 1) .
The intuitions for these results are as follows. Volume arises directly because of differences of
opinion: biased and rational traders agree to disagree. The excess volatility result is more surprising
because biased traders underreact, rather than overreact, to some news. Volatility still arises in
our model because prices are drifting away from the fundamental value. Referring to the example
in the previous subsection, one can observe that, when a positive signal is followed by a negative
one, the price is higher than in the rational benchmark, P2 (1, −1) > P2∗ (1, −1). Our model being
symmetric, we also have P2 (−1, 1) < P2∗ (−1, 1). Prices at date 2 are thus more scattered than in
the rational benchmark, which explains the excessive volatility.
Momentum arises in our model because biased traders’ beliefs at a given date affect their future
misperception of public news. In particular, when prices increase, it is more likely that biased
traders are optimistic and therefore underreact to future bad news. As a result future prices tend
to be higher than they should, which creates the momentum effect.
Proposition 3.3 shows that volume-based momentum may prevail in our model. To study
volume-based momentum, one takes expectations of future returns conditional on past returns and
the level of past volume. Consider first history paths that include only positive news: returns are
positive but there is no volume because all traders have the same beliefs. The expected return is
high because biased traders are very optimistic and will likely misperceive future negative signals.
This effect goes against the presence of volume-based momentum because high past returns and
low volume are associated with high future expected returns.
12
On the other hand, on mixed history paths that begin with positive news, returns may be
positive and there is disagreement (and thus volume): some biased traders end up being overoptimistic pushing up expected returns. This is consistent with the volume-based momentum.
Overall, the volume-based momentum prevails if the intensity of the confirmatory bias is high
enough for mixed history paths to overcome the impact of history paths that include less conflicting
signals.8
In our model, a bubble forms after a first positive public signal is announced. This is because
biased traders become optimistic and are expected to misperceive future negative signals for positive
ones. This phenomenon amplifies with time which explains why prices are expected to grow. When
hard information reaches the market, in our model at date 4, prices revert to fundamentals.
Several behavioral models based on cognitive biases have been proposed to explain the above
results. Most of these models display differences of opinion. Overconfidence is one reason why
people may display differences of opinion and agree to disagree. It has been prominently cited to
explain excess volume (see for example, Odean (1998)) and excess volatility (see Daniel, Hirshleifer,
and Subrahmanyam (1998)). Daniel, Hirshleifer, and Subrahmanyam (1998) further show that,
coupled with the self-attribution bias, overconfidence may explain momentum. Finally, Scheinkman
and Xiong (2003) show that overconfidence can lead to speculative bubbles. The representativeness
heuristics has also been invoked to rationalize momentum (see Barberis, Shleifer and Vishny (1998)
and Rabin and Vayanos (2010)). Hong and Stein (2007) indicate that excess volume, momentum
and volume-based momentum may be explained by gradual information flow or limited attention.
Our model offers a complementary explanation to these phenomena and has the following
distinctive features: i) it provides a novel mechanism for differences of opinion, the confirmatory
bias, which is theoretically and empirically well-grounded in the psychology literature (see for
example, the survey on the confirmatory bias in Hart et al. (2009)), ii) it is parsimonious in the
sense that departures from perfect rationality are driven by only one parameter (when the severity
of bias, q, is null, we have the perfectly rational benchmark), and iii) it offers novel empirical
predictions that are derived and tested below.
8
This indicates that volume-based momemtum is not an artifact of differences-of-opinion models that display simple
momentum. Moreover, there exist alternative specifications of our model that would display simple momentum but
not volume-based momentum. For example, Rabin and Schrag (1999) consider that biased traders do not distort
information when µit−1 = µi0 . With this alternative specification, all our theoretical results remain valid except for
the presence of volume-based momentum.
13
3.3
Short-term trading strategies and the confirmatory bias
In the model so far, traders have a long-term horizon in the sense that they are only concerned
about the final dividend distributed at date 4. However, some traders in financial markets are
subject to liquidity constraints that prevent them from holding positions for an extended period of
time (see the analysis of Shleifer and Vishny (1997)). We thus investigate how traders with a short
horizon would behave when confronted with traders who suffer from the confirmatory bias.
To do so, we introduce in the model an additional risk neutral rational trader at each date.
These traders, referred to as hedge funds, are assumed to have a negligible mass and a one-period
horizon. The negligible mass implies that hedge funds’ trading behavior does not affect market
outcomes. As a consequence, all the pricing results derived above are still valid.
The short horizon implies that, at each date t, a hedge fund’s objective is to maximize, with
h
respect to dht (the demand of the hedge fund active at date t), next period expected wealth Wt+1
conditional on information at date t.9 The question that we aim to address here is whether or not
hedge funds have a corrective impact on the market and whether these funds use a contrarian or a
positive feedback strategy.
The following proposition characterizes the optimal hedge funds’ strategy.
Proposition 3.4.
• For t ∈ {1, 2}, the demand of a short-term trader is:
dht =
where µ
¯t = (1 − λ)µt +
pH − pL
(E(¯
µt+1 |ht ) − µ
¯t ) 211E(¯µt+1 |ht )−¯µt − 1 ,
c
Rλ
0
µit di represents traders’ average belief at time t.
• At t = 1, the short-term trader uses a positive feedback strategy.
• At t = 2, the short-term trader uses a positive feedback strategy most of the time but may also
use a contrarian strategy.
• At t = 2, the short-term trader rides bubbles.
Proposition 3.4 highlights the fact that a crucial dimension of a short-term strategy in our
model is to track the evolution of the average belief. Because rational beliefs are a martingale, this
is equivalent to tracking biased traders’ beliefs.
9
All the results hold as long as the horizon of hedge funds does not include the date at which the dividend is
distributed. If the horizon of hedge funds included this date, their behavior would be similar to the one of rational
speculators.
14
Proposition 3.4 shows that a short-term trader is sometimes using a contrarian strategy. This
occurs for example after s1 = 1 and s2 = −1. When the confirmatory bias is not very strong,
i.e., when λq < θ2 + (1 − θ)2 , the price falls at date 2: P2 (s1 = 1, s2 = −1) < P1 (s1 = 1). But
because the price P3 is expected to be higher than price P2 , the hedge fund buys. This result
is of interest because it shows that despite the fact that biased traders’ psychology inclines them
towards continuity in beliefs, a rational short-term trader might take a position that bets against
continuity in prices.
Proposition 3.4 shows that hedge funds ride bubbles instead of correcting them. After an initial
positive signal, when the price is too high at date t = 2, the hedge fund tends to buy. This is
in line with hedge fund behavior during the Dot Com bubble, as documented by Brunnermeier
and Nagel (2004), and with London-based bank Hoares trading behavior during the 1720s South
Sea bubble, as reported by Temin and Voth (2003). Such a trading behavior is in sharp contrast
with long-term speculators’ behavior in our model. These speculators indeed sell when the price is
above the rational fundamental value and buy when it is below, irrespective of future short-term
returns. This is because they hold their assets until the dividend is distributed and thus focus only
on fundamental value.
Another important difference with speculators’ strategy is that the short-term strategy requires
that hedge funds know parameters λ and q (in addition to the other parameters of the model).
This is not the case for long-term rational speculators who just need to evaluate whether prices are
too low or too high compared to their expectation of final dividends. Hedge funds could estimate
parameters λ and q using Bayesian techniques. Observe first that at date 1, the hedge fund cannot
learn anything so that it has to use its prior beliefs to form its demand. At date 2, the return
P2 − P1 reveals λq. Using the information on λq, the hedge fund can update its priors on λ and
on q and form its demand. An example of Bayesian learning for the case in which the hedge fund
initially believes that λ and q are uniformly distributed on [0, 1] is shown in the Appendix.10
3.4
Novel empirical predictions
In addition to providing a unified explanation for a variety of stylized facts, our model offers new
theoretical predictions that are tested in the next section. To complement our results on the
predictive power of past returns, we study how these returns are associated with future dispersion
10
The structural estimation of the model using Bayesian tools is left for future research.
15
of beliefs and future volume. We now state our most novel theoretical results.
Proposition 3.5. Consider that λq < θ2 + (1 − θ)2 so that subsequent returns can have different
signs. When some traders are prone to the confirmatory bias (that is, when λ > 0 and q > 0),
• Belief dispersion is higher when past returns have different signs:
E Var ν3 |(P1 −P0 )(P2 −P1 )<0 > E Var ν3 |(P1 −P0 )(P2 −P1 )>0 ,
where Var [ν3 ] is a random variable characterizing belief dispersion at date 3.
• Volume is higher when past returns have different signs:
E V3 |(P1 −P0 )(P2 −P1 )<0 > E V3 |(P1 −P0 )(P2 −P1 )>0 .
Belief dispersion is explicitly defined in the Appendix.11 . The intuition for Proposition 3.5 is
as follows. Observing changes in the sign of returns suggests that news with opposite information
content have accumulated. As a result, confirmatory-biased traders have plenty of occasions to
misperceive public signals that contradict their initial beliefs. This explains why, following returns
with different signs, differences of opinion are more acute in our model.
More precisely, consider that the sign of returns corresponds to the sign of the public signal.
As in the example provided above, consider that s1 = 1 and s2 = −1, so that P1 − P0 > 0 and
P2 − P1 < 0. After such a mixed history, the support of traders’ beliefs at date 3 spans a wide
range of values from a low θ(1, 3) = 1 − θ <
1
2
to a high θ(3, 3). This can be understood by looking
at the example in subsection 3.1 above.
Instead, consider that s1 = 1 and s2 = 1 such that P1 − P0 > 0 and P2 − P1 > 0. In this case,
the support of traders’ beliefs at date 3 lies between θ(2, 3) = θ >
1
2
and θ(3, 3) and is thus much
less spread out. In our model, differences of opinion are the main driver of volume; the second
result of Proposition 3.5 thus derives naturally from the first. These two results are tested in the
next section.
11
In our model, volume-based momentum appears only when λq > θ2 + (1 − θ)2 . Note however that it would be
possible to have our new predictions and volume-based momentum for the same parameter values if we were to add
an additional date of trading.
16
4.
Empirical analysis
This section provides an empirical test of our two novel theoretical predictions: when some traders
suffer from the confirmatory bias, differences of opinion and volume are larger when past returns
have different signs. We first describe the sample formation and then present the results.
4.1
Data
We use data on U.S. stocks from three sources. Data on analysts’ earnings forecasts come from the
Institutional Brokers Estimates System (I/B/E/S) database (and are available from 1982 onwards).
We follow the literature (e.g., Diether, Malloy, and Scherbina (2002)) and use the dispersion in
analysts’ earnings forecasts as a proxy for differences of opinion. From the Center for Research in
Security Prices (CRSP monthly file), we obtain monthly stock returns, closing stock prices, trading
volume and shares outstanding for stocks traded on NYSE, AMEX, and NASDAQ. We focus
on companies’ ordinary shares, that is, companies with CRSP share codes of 10 or 11. Finally,
COMPUSTAT provides us with firm-level accounting information. The sample period spans from
1982 to 2011.
Forecast dispersion is measured as the standard deviation of analysts’ forecasts scaled by the
prior year-end stock price to mitigate heteroskedasticity.12 Specifically, DISP−3,0 for stock i in
year t is the standard deviation multiplied by 100 of the most recent earnings forecast of each
analyst covering stock i in the last quarter of fiscal year t normalized by the stock price at the end
of fiscal year t − 1.13 Following earlier work, we exclude all observations with stock price lower than
5 dollars.
From CRSP, we compute RET−9,−3 , the cumulated six-month returns of a given stock over
the second and third quarters of the fiscal year,14 and T U RN−3,0 , the logarithm of the average
monthly share turnover in the last quarter of the fiscal year (defined as trading volume divided by
shares outstanding).
For every stock, we also construct COV ERAGE, the logarithm of the number of analysts who
covered the stock in the previous fiscal year; SIZE, the logarithm of the stock’s total market
12
As noted by Diether, Malloy and Scherbina (2002), there is a rounding error problem in the standard I/B/E/S
“Detail History” data set. We thus use data on analysts’ forecasts unadjusted for stock splits; we then scale analysts’
forecasts by the CRSP cumulative adjustment factor. We obtain similar, if anything stronger, results when using the
standard I/B/E/S data set.
13
The results are qualitatively similar when earnings forecasts dispersion is normalized by the absolute value of the
average or median earnings forecast.
14
Note that annual earnings of fiscal year t − 1 are almost always announced in the first quarter of fiscal year t.
17
capitalization (Compustat item CSHO × item PRCC F) computed at the end of the previous fiscal
year; SIGM A, the standard deviation of daily raw returns of the stock in the previous fiscal year;
LN BM , the logarithm of the book-to-market ratio defined as in Fama and French (2008) in year
t − 1; and ROA, firm return on assets, defined as operating income after depreciation (Compustat
item OIBDP - item DP) over total assets (item AT) computed at the end of the previous fiscal
year. Finally, to mitigate the effect of outliers, we trim all continuous variables at the first and
ninety-ninth percentiles of their respective empirical distribution.
Table 1 presents summary statistics on the variables of interest. We observe that, even after
trimming at the first and ninety-ninth percentiles, DISP−3,0 is right-skewed with mean (0.430)
noticeably larger than the median (0.173). The results are qualitatively unchanged when DISP−3,0
is trimmed at the fifth and ninety-fifth percentiles.
4.2
Results
Our theoretical model predicts that differences of opinion and volume should be higher when
investors received conflicting news in the previous months. To test this prediction, we examine
below whether analyst forecasts’ dispersion and turnover are larger when the sign of past returns
changed in the past six months. For this, we construct DIF SIGN−9,−3 which is a dummy that
equals one if the sign of the cumulated stock return in the second quarter of fiscal year t is not
the same as the sign of the cumulated stock return in the third quarter of fiscal year t, that is,
DIF SIGN−9,−3 = 11RET−9,−6 ∗RET−6,−3 <0 . Our theory predicts that DIF SIGN−9,−3 has a positive
effect on DISP−3,0 and T U RN−3,0 .
Belief dispersion. To examine the effect of DIF SIGN−9,−3 on belief dispersion, we estimate
the following equation:
DISP−3,0,it = α + δt + β DIF SIGN−9,−3,it + γ Controlsit + εit
(4.4)
where DISP−3,0,it measures belief dispersion for stock i in the last quarter of fiscal year t, δt
are year fixed effects, and εit is the error term.
We estimate different specifications of equation 4.4 which also potentially include industry or
stock fixed effects.15 By construction, DIF SIGN−9,−3 is correlated with the absolute value of
15
We construct the 48 Fama-French industry dummies using the firm’s 4 digit SIC industry code and the conversion
table in the Appendix of Fama and French (1997).
18
RET−9,−3 . To control for the direct effect of past returns on belief dispersion, equation 4.4 also
includes ten dummies indicating respectively ten deciles of RET−9,−3 . We use COV ERAGE,
SIZE, SIGM A, LN BM , and ROA as control variables. SIGM A controls for firm uncertainty,
which has been shown to be a key driver of belief dispersion.
The correlation between DIF SIGN−9,−3 and SIGM A is negative and low (the correlation
equals -0.005 with a p-value of 0.19), which mitigates the concern that the coefficient on DIF SIGN
captures part of the effect of firm uncertainty on belief dispersion. Because DISP−3,0 is strongly
persistent, it is likely that the εit are not independent from different observations of the same stock
i. We therefore cluster standard errors at the stock level to account for serial correlation of the
error term within the same stock. The sample period is from January 1982 to December 2011. The
coefficient of interest, β, measures the effect of past conflicting news on belief dispersion.
[Table 2 here]
Table 2 presents estimates of different specifications of equation 4.4. Columns [2] and [5] include
industry fixed effects whereas columns [3] and [6] include stock fixed effects. In columns [1] to [3],
we estimate equation 4.4 without control variables. The coefficient on DIF SIGN−9,−3 is always
positive and statistically significant (at least at the five percent confidence level), indicating that
past conflicting news are, as predicted, associated with higher dispersion in analysts’ forecasts.
In columns [4] to [6], we add the control variables. The coefficient on DIF SIGN−9,−3 becomes
smaller, but remains statistically significant in each specification.
The economic effect is also significant. For instance, in column [5], the coefficient on DIF SIGN−9,−3
equals 0.029. Given that the sample mean of DISP−3,0 equals 0.43, in this specification, a change
in the sign of past returns is associated with an increase in belief dispersion for the average firm of
around 7 percentage points.
Trading volume. To examine the effect of DIF SIGN−9,−3 on trading volume, we use the
same specification as in equation 4.4, except that the dependent variable is now T U RN−3,0 . We
include the bid-ask spread and a nasdaq dummy as additional control variables. The bid-ask spread
is used to control for stock liquidity. The nasdaq dummy controls for the fact that reported volume
for NASDAQ stocks includes dealer trades and are thus not directly comparable to reported volume
19
for NYSE and AMEX stocks.16
[Table 3 here]
Table 3 presents results that are consistent with those in Table 2. As predicted by our model, the
coefficient on DIF SIGN−9,−3 is positive and statistically significant at the one percent confidence
level in each specification. Focusing on column [5], in terms of economic significance, a change in the
sign of past returns is associated with an increase by 12 percent (=0.109/0.906) of one T U RN−3,0
standard deviation.
Overall, our empirical investigation provides new stylized facts on the link between past returns,
on one side, and differences of opinion and volume, on the other. These stylized facts are consistent
with the confirmatory bias affecting traders’ perception of information.
5.
Conclusion
This paper proposes a theory of price and volume formation based on the premise that some traders
are prone to the confirmatory bias. We model this cognitive bias by considering that biased traders
tend to misperceive public signals that are inconsistent with their prior views. We show that, in
the context of financial markets, this bias provides a rationale for various stylized facts including
excess volatility (documented by Shiller (1980) and by Leroy and Porter (1980)), momentum (documented by Jegadeesh and Titman (1993)), and volume-based momentum (documented by Lee
and Swaminathan (2000)).
The model also provides novel empirical predictions regarding the link between past returns, on
one hand, and differences of opinion and volume, on the other. In presence of confirmatory-biased
traders, differences of opinion and volume are expected to be larger when past returns exhibit
changes in sign. Such returns indeed indicate that both positive and negative news have been
released, giving ample room for biased traders to misinterpret information. As a result, the model
predicts more disagreement and thus more volume.
We test these novel predictions on U.S. stock market data over the period ranging from 1982
to 2011. For each stock and each year, we measure dispersion in analysts’ earnings forecasts (used
as a proxy for disagreement) and trading volume during the last quarter of the fiscal year. We
16
Results are very similar when NASDAQ stocks are excluded from the regressions.
20
regress these variables onto a dummy variable indicating whether returns during the second and
third quarters of the year were of opposite signs. We show that this dummy variable is positively
associated with forecasts dispersion as well as trading volume, even after controlling for a variety
of factors known to affect these variables. These results suggest that the confirmatory bias is at
work in financial markets.
In future research, it would be interesting to estimate the parameters of our model and, in
particular, the proportion of biased traders on financial markets and the magnitude of the confirmatory bias. This could be useful to evaluate the performance of a strategy designed to profit
from mistakes of confirmatory-biased traders. Finally, our model could be used to study optimal
corporate communication. Indeed, firms that are confronted to financial markets populated by investors prone to the confirmatory bias might have an interest in appropriately choosing the timing
of information releases.
21
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24
6.
Appendix
6.1
Belief dynamics
To set the stage for the analysis of the evolution of prices and volume, we start by studying belief
dynamics.
Lemma 6.1. For speculators, belief dynamics are given by:
P(X = H|s1 , . . . , st+1 ) = µt+1 = f (θ, µt )11{st+1 =1} + f (1 − θ, µt )11{st+1 =−1} ,
and for biased traders (i ∈ [0, λ]),
Pi (X = H|s1 , . . . , st+1 ) = µit+1 = f (θ, µit )11{σi
t+1 =1}
+ f (1 − θ, µit )11{σi
t+1 =−1}
,
where
f (θ, µ) =
θµ
.
θµ + (1 − θ)(1 − µ)
Proof. Because the law of the rational beliefs µt under P is the same as the one of biased beliefs µit
under the biased probability Pi , it is enough to focus on belief dynamics for rational agents. Using
Bayes’ formula, we have:
P(X = H|s1 , . . . , st , st+1 = 1) =
=
=
P(X = H; st+1 = 1|s1 , . . . , st )
P(st+1 = 1|s1 , . . . , st )
P(st+1 = 1|X = H)P(X = H|s1 , . . . , st )
P(st+1 = 1|s1 , . . . , st )
θµt
.
θµt + (1 − θ)(1 − µt )
Likewise, we have:
P(X = H|s1 , . . . , st , st+1 = −1) =
(1 − θ)µt
.
(1 − θ)µt + θ(1 − µt )
We conclude by noting that:
µt+1 = P(X = H|s1 , . . . , st , st+1 = 1)11{st+1 =1} + P(X = H|s1 , . . . , st , st+1 = −1)11{st+1 =−1} .
25
Note that for a belief history of length t, ht = (s1 , . . . , st ), the possible values for µt are:
θ(j, t) =
θj (1 − θ)t−j
.
θj (1 − θ)t−j + θt−j (1 − θ)j
(6.5)
where j is the number of signal with su = 1 for u ≤ t (this is similar to Rabin and Schrag, 1999).
We continue this subsection by giving some results concerning the probability distribution of the
public signal. Clearly,
P(s1 = 1) =
θ+1−θ
1
= .
2
2
Because of the symmetry of the model, the law of the belief process with path starting with s1 = −1
is obtained by changing θ in 1 − θ. Therefore, we only focus on belief paths with s1 = 1. We have:
P(s2 = 1|s1 = 1) =
=
P(s1 = 1; s2 = 1)
P(s1 = 1)
P(X = H)P(s1 = 1; s2 = 1|X = H) + P(X = L)P(s1 = 1; s2 = 1|X = L)
P(s1 = 1)
= θ2 + (1 − θ)2 ,
and thus,
P(s2 = −1|s1 = 1) = 2θ(1 − θ).
Likewise, it is straightforward to prove that:
P(s3 = 1|s1 = 1; s2 = 1) =
θ3 + (1 − θ)3
,
θ2 + (1 − θ)2
P(s3 = −1|s1 = 1; s2 = 1) =
θ2
θ(1 − θ)
,
+ (1 − θ)2
and
1
P(s3 = 1|s1 = 1; s2 = −1) = P(s3 = −1|s1 = 1; s2 = −1) = .
2
At time t, there exist 2t paths for the public signal history. Let us denote by ht = (s1 , s2 , . . . , st )
a path and (z i h)t = (s1 , z2i s2 , . . . , zti st ) biased trader i’s belief path. A speculator has a rational
estimation of the future dividend at time t given by P(v = 1|ht ) = pL + (pH − pL )µt . Biased trader
i has an estimation of the future dividend given by Pi (v = 1|(z i h)t ) = pL + (pH − pL )µit .
26
6.2
Equilibrium prices and rational benchmark
Proof of Proposition 2.1. At date 4, the final wealth of trader j, W j, is:
t=3 X
c j 2
j
W =
dt (v − Pt ) −
d
+ v,
2 t
j
t=0
with j = ∅ referring to an arbitrageur, and j = i ∈ [0, λ] referring to a biased trader. Since
traders only consume at the last date t = 4, their objective is to maximize their expected final
wealth conditional on their information. To solve for the optimal demands, we proceed backward.
Because traders are atomistic, they take prices as given. We start by solving the program of trader
j at period 3:
max Ej3
dj3
t=2 X
djt
t=0
!
c j 2
c j 2
j
d
+ d3 (v − P3 ) −
d
+v .
(v − Pt ) −
2 t
2 3
It is straightforward to check that the objective function is concave in dj3 . The first order
condition is thus necessary and sufficient to characterize the optimal demand at date 3:
dj3
Ej3 (v) − P3
=
.
c
Applying the same analysis backward, we obtain that:
djt =
Ejt (v) − Pt
, ∀t ∈ {0, 1, 2, 3} .
c
At date t ∈ {0, 1, 2, 3}, the market clearing condition is given by:
Z
0
λ
Eit (v) − Pt
di +
c
Z
1
(1−λ)
Et (v) − Pt
dj = 0
c
where 0 on the right-hand side corresponds to the fact that no new share is issued on the
market. The first pricing equation displayed in Proposition 2.1 derives from solving the market
clearing condition for Pt and noting that Ejt (v) = Pjt (v = 1).
The second pricing equation is obtained using the fact that Pjt (v = 1) = pL + (pH − pL )µjt , with
j = ∅ referring to an arbitrageur, and j = i ∈ [0, λ] referring to a biased trader.
We now analyze the benchmark case in which all traders are perfectly rational. Endogenous
27
prices in this benchmark are indicated by a star. This benchmark is nested in our model and
corresponds to the case in which λ = 0 or q = 0. In this case, we have Pt∗ = Pt (v = 1) =
pL + (pH − pL )µt . Given the structure of the uncertainty in our model, we have the following
proposition.
Proof of Proposition 2.2. The proof of the result on expected returns relies on the martingale
property of the belief process µt . The process µt is a martingale because:
E (µt+1 |s1 , . . . , st ) = E (P(X = H|s1 , . . . , st+1 )|s1 , . . . , st )
= E (E (11X=H |s1 , . . . , st+1 ) |s1 , . . . , st )
= E (11X=H |s1 , . . . , st )
= µt .
The second equality derives from the definition of a probability and the third one from the law of
iterated expectations.
To see that volume, defined by Vt =
is
Pt∗
= Pt (v = 1) =
Ejt
R1
0
djt 11Ej (v)>P ∗ dj, is null at each date, recall that the price
t
t
(v), for all j and t. The last result derives from the fact that volume is
constant.
We now study how asset prices, returns, and volume are influenced by the fact that some traders
are prone to the confirmatory bias. Statistical properties of equilibrium variables are evaluated
based on the true probability measure P because we take the viewpoint of an econometrician who
would observe independent repetitions of the model.
6.3
Stylized facts
The proof of Proposition 3.3 is divided in various paragraphs in order to consider separately the
different stylized facts.
6.3.1
Excess volume
Confirmatory bias induces excess volume. Consider for example the mixed history path (s1 =
1, s2 = −1). The equilibrium price along this path is:
P2 (1, −1) = pL + (pH − pL )(λqθ(2, 2) + (1 − λq)θ(1, 2)).
28
Therefore, there is a proportion λq of agents (corresponding to the biased traders who perceived
a positive signal at date 2 instead of the actual negative signal) who have a price estimation
pL + (pH − pL )θ(2, 2) which is higher than the equilibrium price. Hence, volume equals:
V2 (1, −1) =
6.3.2
(pH − pL )λq(1 − λq)
(θ(2, 2) − θ(1, 2)) > V2∗ (1, −1) = 0.
c
Excess volatility
Confirmatory bias may induce excess volatility. To show this result, it is useful to note that
E(Pt ) =
pH +pL
,
2
for t = 1, 2, 3. To see this, consider two opposite signal histories ht = {1, s2 , . . . , st }
and −ht = {−1, −s2 , . . . , −st }. The law of signals st being symmetric, P(ht ) = P(−ht ). Moreover,
Equation (6.5) implies that µjt (ht ) + µjt (−ht ) = 1. As a consequence, the pricing formula yields
Pt (ht ) + Pt (−ht ) = pH + pL . Finally,
E(Pt ) =
X
(P(ht )Pt (ht ) + P(−ht )Pt (−ht ))
ht
=
X
P(ht ) (Pt (ht ) + Pt (−ht ))
ht
= (pH + pL )
X
P(ht )
ht
= (pH + pL )P(s1 = 1)
=
pH + pL
.
2
We deduce from the price dynamics that:
P2 = P2∗ + ε2 and P3 = P3∗ + ε3 ,
where εi are the symmetric random variables given by:
ε2 (s1 , s2 ) = ε2 (1, −1)(11s1 =1;s2 =−1 − 11s1 =−1;s2 =1 )
= (pH − pL )λq(θ(2, 2) − θ(1, 2))(11s1 =1;s2 =−1 − 11s1 =−1;s2 =1 ),
29
indicating that ε2 (−1, 1) = −ε2 (1, −1) and ε2 (1, 1) = ε2 (−1, −1) = 0, and
ε3 (s1 , s2 , s3 ) = ε3 (1, 1, −1)(11s1 =1;s2 =1;s3 =−1 − 11s1 =−1;s2 =−1;s3 =1 )
+ ε3 (1, −1, 1)(11s1 =1;s2 =−1;s3 =1 − 11s1 =−1;s2 =1;s3 =−1 )
+ ε3 (1, −1, −1)(11s1 =1;s2 =−1;s3 =−1 − 11s1 =−1;s2 =1;s3 =1 )
= (pH − pL )λq(θ(3, 3) − θ(2, 3))(11s1 =1;s2 =1;s3 =−1 − 11s1 =−1;s2 =−1;s3 =1 )
+ (pH − pL )λq(θ(3, 3) − θ(2, 3))(11s1 =1;s2 =−1;s3 =1 − 11s1 =−1;s2 =1;s3 =−1 )
+ (pH − pL )λq(qθ(3, 3) + 2(1 − q)θ(2, 3) − (2 − q)θ(1, 3))(11s1 =1;s2 =−1;s3 =−1 − 11s1 =−1;s2 =1;s3 =1 ).
Thus, noting m =
pH +pL
,
2
we have:
Var(P2 ) = Var(P2∗ ) + Var(ε2 ) + 2E [(P2∗ − m)ε2 ] .
It is straightforward to see that E [(P2∗ − m)ε2 ] = 0. Indeed, because P ∗ (1, −1) = P ∗ (−1, 1) = m
and ε2 (1, 1) = ε2 (−1, −1) = 0, we have:
E [(P2∗ − m)ε2 ] = 0.
Therefore
Var(P2 ) > Var(P2∗ ).
There is thus always excess volatility in the price at date 2.
Regarding time t = 3, we first compute Var(ε3 ). Because E(ε3 ) = 0, we have Var(ε3 ) = E(ε23 ).
Moreover, we note that for every path h3 containing a change in the sign of the public signal
P(h3 ) =
θ(1−θ)
2
and
P3∗ (h3 ) − P3∗ (−h3 ) = (ph − pL )(2θ − 1).
Therefore
E(ε23 ) = θ(1 − θ) ε23 (1, 1, −1) + ε23 (1, −1, 1) + ε23 (1, −1, −1)
= θ(1 − θ)(pH − pL )2 λ2 q 2 2(θ(3, 3) − θ(2, 3))2 + [q(θ(3, 3) − θ(2, 3)) + (2 − q)(θ(2, 3) − θ(1, 3))]2
30
On the other hand,
E[(P3∗ − m)ε3 ] =
+
+
=
θ(1 − θ)
ε3 (1, 1, −1) (P3∗ (1, 1, −1) − P3∗ (−1, −1, 1))
2
θ(1 − θ)
ε3 (1, −1, 1) (P3∗ (1, −1, 1) − P3∗ (−1, 1, −1))
2
θ(1 − θ)
ε3 (1, −1, −1) (P3∗ (1, −1, −1) − P3∗ (−1, 1, 1))
2
θ(1 − θ)
(pH − pL )2 λq(2θ − 1)(2 − q)(θ(3, 3) + θ(1, 3) − 2θ(2, 3)).
2
Therefore,
Var(P3 ) − Var(P3∗ ) = θ(1 − θ)(pH − pL )2 λq λq 2(θ(3, 3) − θ(2, 3))2
Now, setting θ =
1
2
+
[q(θ(3, 3) − θ(2, 3)) + (2 − q)(θ(2, 3) − θ(1, 3))]2
+
(2θ − 1)(2 − q)(θ(3, 3) + θ(1, 3) − 2θ(2, 3))} .
+ η and making an expansion of Var(P3 ) − Var(P3∗ ) in a neighborhood of 21 , we
get
1
Var(P3 ) − Var(P3∗ ) = (pH − pL )2 λq(q(1 + 4λ) − 2)η + o(η).
4
This expression is positive if q(1 + 4λ) > 2. This shows that there is a parameter region for which
there is excess volatility in the price at date 3.
6.3.3
Momentum
Assume that t = 1. The random event {P1 − P0 > 0} coincides with the event {s1 = 1}. Therefore,
we have to prove that E(P2 − P1 |s1 = 1) > 0. But,
E(P2 − P1 |s1 = 1) = E(P2 (1, 1)11s2 =1 + P2 (1, −1)11s2 =−1 |s1 = 1) − ((pH − pL )θ + pL )
= (pH − pL ) [θ(2, 2)P(s2 = 1|s1 = 1) + ((1 − λq)θ(1, 2) + λqθ(2, 2))P(s2 = −1|s1 = 1)) − θ]
= (pH − pL )λq(θ(2, 2) − θ(1, 2))P(s2 = −1|s1 = 1) > 0
where the last equality follows from the martingale property of the belief process µt .
There is thus always momentum at time 1.
To prove momentum at t = 2, we need to know the histories that yield P2 − P1 > 0. Clearly,
31
on the event {s1 = 1; s2 = 1}, the return P2 − P1 is positive. On the event {s1 = 1; s2 = −1}, this
return equals:
P2 (1, −1) − P1 = (pH − pL )(λq(θ(2, 2) − θ(1, 2)) + θ(1, 2) − θ).
This is positive if and only if λq ≥
θ−θ(1,2)
θ(2,2)−θ(1,2)
= θ2 + (1 − θ)2 .
Therefore, two cases have to be considered. When λq ≥ θ2 + (1 − θ)2 , we have:
P(s1 = 1; s2 = 1)
P(P2 − P1 > 0)
P(s1 = 1; s2 = −1)
+ E(P3 − P2 |s1 = 1; s2 = −1)
P(P2 − P1 > 0)
E(P3 − P2 |P2 − P1 > 0) = E(P3 − P2 |s1 = 1; s2 = 1)
= E(P3 − P2 |s1 = 1; s2 = 1)(θ2 + (1 − θ)2 )
+ E(P3 − P2 |s1 = 1; s2 = −1)2θ(1 − θ).
Now,
E(P3 − P2 |s1 = 1; s2 = 1) = (P3 (1, 1, 1) − P2 (1, 1))P(s3 = 1|s1 = 1; s2 = 1)
+ (P3 (1, 1, −1) − P2 (1, 1))P(s3 = −1|s1 = 1; s2 = 1)
= λq(pH − pL )P(s3 = −1|s1 = 1; s2 = 1)(θ(3, 3) − θ(2, 3)),
where the last equality uses the martingale property of µt , that is:
θ(2, 2) = P(s3 = 1|s1 = 1; s2 = 1)θ(3, 3) + P(s3 = −1|s1 = 1; s2 = 1)θ(2, 3).
Similarly, we have:
E(P3 −P2 |s1 = 1; s2 = −1) =
λq(pH − pL )
((1 + q)θ(3, 3) + 2(1 − q)θ(2, 3) − (1 − q)θ(1, 3) − 2θ(2, 2)) .
2
32
Finally,
E(P3 − P2 |P2 − P1 > 0) = λq(pH − pL )θ(1 − θ)((2 + q)θ(3, 3)
+ (2(1 − q) − 1)θ(2, 3) − (1 − q)θ(1, 3) − 2θ(2, 2))
= λq(pH − pL )θ(1 − θ)[2(θ(3, 3) − θ(2, 2)) + q(θ(3, 3) − θ(2, 3))
+ (1 − q)(θ(2, 3) − θ(1, 3))] > 0.
There is thus momentum at time 2 when λq ≥ θ2 + (1 − θ)2 .
In the second case in which λq < θ2 + (1 − θ)2 , we have:
P(s1 = 1; s2 = 1)
P(P2 − P1 > 0)
P(s1 = 1; s2 = −1)
+ E(P3 − P2 |s1 = −1; s2 = 1)
P(P2 − P1 > 0)
E(P3 − P2 |P2 − P1 > 0) = E(P3 − P2 |s1 = 1; s2 = 1)
= E(P3 − P2 |s1 = 1; s2 = 1)(θ2 + (1 − θ)2 )
+ E(P3 − P2 |s1 = −1; s2 = 1)2θ(1 − θ).
By symmetry, we deduce from the first case:
E(P3 −P2 |s1 = −1; s2 = 1) =
λq(pH − pL )
((1 + q)θ(0, 3) + 2(1 − q)θ(1, 3) − (1 − q)θ(2, 3) − 2θ(0, 2)) .
2
Therefore,
E(P3 − P2 |P2 − P1 > 0) = λq(pH − pL )θ(1 − θ)((2 − q)θ(1, 3)
+ (1 − q)(θ(1, 3) − θ(2, 3)) + qθ(0, 3) − 2θ(0, 2))
= λq(pH − pL )θ(1 − θ)(2(θ(1, 3) − θ(0, 2)) + θ(1, 3) − θ(2, 3)
+ q (θ(0, 3) + θ(2, 3) − 2θ(1, 3))).
Observe that E(P3 −P2 |P2 −P1 > 0) is an increasing function of q because θ(0, 3)+θ(2, 3)−2θ(1, 3) ≥
0. Moreover, using Equation (6.5) and rearranging terms, we obtain:
2(θ(1, 3) − θ(0, 2)) + θ(1, 3) − θ(2, 3) = (2θ − 1)
33
1
−2
θ2 + (1 − θ)2
≤ 0.
As a consequence, when λq ≤ θ2 + (1 − θ)2 there is momentum only if:
q ≥ q¯ =
6.3.4
2(θ(1, 3) − θ(0, 2)) + θ(1, 3) − θ(2, 3)
.
θ(0, 3) + θ(2, 3) − 2θ(1, 3)
Volume-based momentum
To study volume-based momentum, we differentiate between histories with and without volume.
At date 2, V2 > 0 if and only if s1 s2 < 0. Using the computations made in the previous Paragraph
6.3.3, we have, when λq > θ2 + (1 − θ)2 :
E(P3 − P2 |P2 − P1 > 0; V2 > 0) = E(P3 − P2 |s1 = 1; s2 = −1)
=
λq(pH − pL )
((1 + q)θ(3, 3)
2
+ 2(1 − q)θ(2, 3) − (1 − q)θ(1, 3) − 2θ(2, 2))
and
E(P3 − P2 |P2 − P1 > 0; V2 = 0) = E(P3 − P2 |s1 = 1; s2 = 1)
= λq(pH − pL )P(s3 = −1|s1 = 1; s2 = 1)(θ(3, 3) − θ(2, 3)).
Defining p = P(s3 = −1|s1 = 1; s2 = 1) and using the martingale property θ(2, 2) = (1 − p)θ(3, 3) +
pθ(2, 3), we obtain:
E(P3 − P2 |P2 − P1 > 0; V2 > 0) − E(P3 − P2 |P2 − P1 > 0; V2 = 0) = λq(pH − pL )
1−q
(2θ(2, 3)
2
− (θ(3, 3) + θ(1, 3))) > 0.
The momentum effect is thus stronger when volume is higher. This effect is not here when λq <
θ2 + (1 − θ)2 . Indeed, when λq < θ2 + (1 − θ)2 , we have:
E(P3 − P2 |P2 − P1 > 0; V2 > 0) = E(P3 − P2 |s1 = −1; s2 = 1) < 0
while
E(P3 − P2 |P2 − P1 > 0; V2 = 0) = E(P3 − P2 |s1 = 1; s2 = 1) > 0.
34
6.3.5
Bubble
In Paragraph 6.3.3, we have proved that: E(P2 − P1 |s1 = 1) > 0 which is equivalent to E(P2 |s1 =
1) ≥ E(P1 |s1 = 1) = pL + (pH − pL )θ.
Moreover, we have:
E(P3 −P2 |s1 = 1) = E(P3 −P2 |s1 = 1, s2 = 1)P(s2 = 1|s1 = 1)+E(P3 −P2 |s1 = 1, s2 = −1)P(s2 = 1|s1 = 1)
We have already proved that:
E(P3 − P2 |s1 = 1, s2 = 1) > 0 and E(P3 − P2 |s1 = 1, s2 = −1) > 0.
Therefore,
E(P3 |s1 = 1) ≥ E(P2 |s1 = 1).
6.4
Short-term trading strategy
Proof of Proposition 3.4. The demand of the short-term trader, referred to as a hedge fund h,
at t ∈ {1, 2} is obtained by maximizing his objective function at date t with respect to his excess
demand dht :
c
c
dht (Et (Pt+1 ) − Pt ) − (dht )2 = dht (pH − pL )(Et (¯
µt+1 ) − µ
¯t ) − (dht )2 .
2
2
where µ
¯t = (1 − λ)µt +
Rλ
0
µit di represents traders’ average belief at time t.
We now characterize the short-term trading strategy of the hedge fund. Without loss of generality, we focus on the case in which s1 = 1, which corresponds to a positive return at time 1,
P1 (s1 = 1) − P0 > 0. In this case, the next expected return is positive because, as shown in
Paragraph 6.3.3, we have: E(P2 |s1 = 1) > P1 (s1 = 1). Therefore, the short-term trader after a
price increase from t = 0 to t = 1 buys at time 1. He thus always implements a positive feedback
strategy at time 1.
To analyze the hedge fund behavior at time 2, we first notice that E(P3 − P2 |s1 = 1; s2 = 1) =
λq(pH − pL )P(s3 = −1|s1 = 1; s2 = 1)(θ(3, 3) − θ(2, 3)) > 0. Thus, after a price increase from t = 1
to t = 2, a short-term trader buys and thus implements a positive feedback strategy.
35
We now analyze what happens after a price decrease from t = 1 to t = 2. Proceeding analogously
to Paragraph 6.3.4, we denote by p the probability P(s3 = −1|s1 = 1, s2 = 1). Along the path
h2 = (1, −1), the expected short-term return equals:
E(P3 |h2 ) − P2
1−q
1+q
= λq(pH − pL )
+ p − 1 θ(3, 3) + ((1 − q) − p)θ(2, 3) −
θ(1, 3)
2
2
1−q
1−q
(θ(3, 3) − θ(2, 3)) +
(θ(2, 3) − θ(1, 3))
= λq(pH − pL ) p −
2
2
1−q
= λq(pH − pL ) p(θ(3, 3) − θ(2, 3)) +
(2θ(2, 3) − (θ(3, 3) + θ(1, 3))) .
2
Using Equation 6.5, it is straightforward to show that the expression 2θ(2, 3) − (θ(3, 3) + θ(1, 3))
is always positive. This indicates that the hedge fund is always buying at time 2 after the history
h2 = (1, −1). When λq > θ2 + (1 − θ)2 , the price increases from time 1 to time 2 despite s2 = −1;
the hedge fund thus appears to use a positive feedback strategy at time 2. When λq < θ2 + (1 − θ)2 ,
the price decreases from time 1 to time 2 after s2 = −1; the hedge fund thus appears to implement
a contrarian strategy.
Overall, the hedge fund uses a contrarian strategy only when λq < θ2 + (1 − θ)2 in the event
s1 = 1, s2 = −1. This event has probability θ(1 − θ) < 14 . The hedge fund is thus more likely to
use a positive feedback strategy.
As shown in Proposition 3.3, the asset is overvalued at date t = 2 if s1 = 1. Moreover, we
have shown above that E(P3 |s1 = 1, s2 ) − P2 (s1 = 1, s2 ) > 0, which implies that the hedge fund is
buying at date 2 on any history path with s1 = 1. This proves that, at date 2, the hedge fund is
buying the asset despite knowing that it is overvalued.
To determine how much to buy or sell, hedge funds need to know the value of the parameters
Rλ i
B
B
λ and q. Indeed, their demand is proportional to Et (µB
t+1 ) − µt , with µt = 0 µt di representing
the average belief of biased traders. Equation 6.6 above shows that this expected change in belief
depends on q. At date t = 1, to estimate this change in belief, the hedge fund cannot do anything
else than using its a priori beliefs on λ and q. At date t = 2 on the contrary, the hedge fund can
form posterior beliefs about λ and q by conditioning his order on the level of the price P2 . We
offer below an example of such a Bayesian estimation for the case in which hedge funds initially
believe that λ and q are uniformly and independently distributed on [0, 1]. In that case, it is easy
36
to compute the joint distribution of (λq, q). Indeed, we have for any bounded function φ,
1Z 1
Z
E(φ(λq, q)) =
φ(xy, y) dxdy
0
0
1Z 1
Z
=
0
0
1
φ(u, v) 11u≤v dudv.
v
Hence, the density of the pair (λq, q) is given by f (x, y) = y1 11x≤y . Therefore, the marginal law of
λq is given by f (x) = − ln(x)110≤x≤1 and the conditional density of q knowing that λq = x0 is thus:
−
1 1
11x ≤y .
ln(x0 ) y 0
As a consequence, the best estimate for q observing λq is:
λq − 1
.
ln(λq)
6.5
6.5.1
Novel empirical predictions
Belief dispersion
Let us define by νt traders’ belief dispersion. For each history ht , νt (ht ) is a probability measure
on the finite space θ(x, t) for x ∈ {0, 1, . . . , t} representing traders’ belief distribution function. We
will define the average belief dispersion as the average variance of each probability measure νt (ht )
and the conditional average belief dispersion as the average of the conditional variance of each
probability measure.
Let us write precisely traders’ belief dispersion for t ∈ {0, 1, 2, 3}. Clearly,
ν0 = δ 1 ,
2
where δ stands for the Dirac measure. Because all traders have the same perception of time 1
signal, the distribution of biased traders’ beliefs at time 1 is:
1
ν1 = (δθ + δ1−θ ).
2
In particular, ν1 (1) = δθ and ν1 (−1) = δ1−θ and therefore Var ν1 (1) = Var ν1 (−1) = 0. The
average belief dispersion is thus E(Var ν1 ) = 0.
37
From time 2 onward, belief dispersion depends on the public signal’s path. More precisely, we
have:
ν2 = P(s1 = 1; s2 = 1)ν2 (1; 1) + P(s1 = 1; s2 = −1)ν2 (1; −1)
+ P(s1 = −1; s2 = 1)ν2 (−1; 1) + P(s1 = −1; s2 = −1)ν2 (−1; −1)
where
ν2 (1; 1) = δθ(2,2) , ν2 (1; −1) = qδθ(2,2) + (1 − λq)δθ(1,2) ,
ν2 (−1; −1) = δθ(0,2) , ν2 (−1; 1) = qδθ(0,2) + (1 − λq)δθ(1,2) .
At time 3, we have:
ν3 = P(s1 = 1; s2 = 1; s3 = 1)ν3 (1; 1; 1) + P(s1 = 1; s2 = 1; s3 = −1)ν3 (1; 1; −1)
+ P(s1 = 1; s2 = −1; s3 = 1)ν3 (1; −1; 1) + P(s1 = 1; s2 = −1; s3 = −1)ν2 (1; −1; −1)
+ P(s1 = −1; s2 = −1; s3 = −1)ν3 (−1; −1; −1) + P(s1 = −1; s2 = −1; s3 = 1)ν3 (−1; −1; 1)
+ P(s1 = −1; s2 = 1; s3 = −1)ν3 (−1; 1; −1) + P(s1 = −1; s2 = 1; s3 = 1)ν2 (−1; 1; 1)
where
ν3 (1; 1; 1) = δθ(3,3) , ν3 (1; 1; −1) = qδθ(3,3) + (1 − λq)δθ(2,3) ,
ν3 (1; −1; 1) = qδθ(3,3) +(1−λq)δθ(2,3) ; ν3 (1; −1; −1) = q 2 δθ(3,3) +2q(1−q)δθ(2,3) +((1−λ)+λ(1−q)2 )δθ(1,3) .
ν3 (−1; −1; −1) = δθ(0,3) , ν3 (−1; −1; 1) = qδθ(0,3) + (1 − λq)δθ(1,3) ,
ν3 (−1; 1; −1) = qδθ(0,3) +(1−λq)δθ(1,3) ; ν3 (−1; 1; 1) = q 2 δθ(0,3) +2q(1−q)δθ(1,3) +((1−λ)+λ(1−q)2 )δθ(2,3) .
We want to focus on cases in which subsequent returns can have different signs. As a result, we
consider that λq < θ2 + (1 − θ)2 , so that the set {(P2 − P1 )(P1 − P0 ) < 0} = {s1 = 1, s2 = −1}.
We want to prove that the conditional average belief dispersion is higher at time 2 along the path
{s1 = 1, s2 = −1} than along the path {s1 = 1, s2 = 1}.
But, on the set {s1 = 1, s2 = −1}, the belief distribution can only take two values ν3 (1; 1; 1) and
ν3 (1; 1; −1). Therefore, the conditional belief dispersion at time 2 when past returns have the same
38
sign is:
A = Var ν3 (1; 1; −1)P(s3 = −1|s1 = 1, s2 = 1) = Var ν3 (1; 1; −1)
θ2
θ(1 − θ)
,
+ (1 − θ)2
because Var ν3 (1; 1; 1) = 0.
On the other hand, the conditional belief dispersion at time 2 when past returns have different
signs equals:
B = Var ν3 (1; −1; 1)P(s3 = 1|s1 = 1, s2 = −1) + Var ν3 (1; −1; −1)P(s3 = −1|s1 = 1, s2 = −1)
=
1
(Var ν3 (1; −1; 1) + Var ν3 (1; −1; −1)).
2
Because Var ν3 (1; −1; 1) = Var ν3 (1; 1; −1), and because
6.5.2
θ(1−θ)
θ2 +(1−θ)2
< 21 , A < B.
Volume
A change in the sign of returns is only observable for λq < θ2 + (1 − θ)2 . In that case, we have
{(P1 − P0 )(P2 − P1 ) > 0} = {s1 s2 > 0}. By symmetry, it is enough to prove that:
E [V3 |s1 =1,s2 =−1 ] > E [V3 |s1 =1,s2 =1 ] .
But
E [V3 |s1 =1,s2 =1 ] = V3 (1, 1, −1)P(s3 = −1|s1 =1,s2 =1 ),
and
E [V3 |s1 =1,s2 =−1 ] =
V3 (1, −1, 1) + V3 (1 − 1, −1)
.
2
Because V3 (1, 1, −1) = V3 (1, −1, 1) and P(s3 = −1|s1 =1,s2 =1 ) < 21 , we conclude.
39
Table 1: SUMMARY STATISTICS
This table presents summary statistics. The sample consists of 62,576 stock-years (9,141 unique stocks) in the period
from January 1982 to December 2011 for which DISP−3,0 is non-missing. DISP−3,0 in year t is the standard deviation
multiplied by 100 of the most recent earnings forecast of each analyst covering the stock in the last quarter of fiscal
year t normalized by the stock price at the end of fiscal year t − 1. T U RN−3,0 is the logarithm of the average monthly
turnover, defined as trading volume divided by shares outstanding, in the last quarter of the fiscal year. RET−9,−3
is the six-month cumulated stock return over the second and third quarters of fiscal year t. DIF SIGN−9,−3 is a
dummy that equals one if the sign of the cumulated stock return in the second quarter of fiscal year t is not the same
as the sign of the cumulated stock return in the third quarter of fiscal year t. COV ERAGE is the logarithm of the
number of analysts who covered the stock in the previous fiscal year. SIZE is the logarithm of the stock total market
capitalization (Compustat item CSHO × item PRCC F computed at the end of the previous fiscal year). SIGM A is
the standard deviation of daily raw returns of the stock in the previous fiscal year. Return on assets, ROA, is defined
as operating income after depreciation (item OIBDP - item DP) over total assets (item AT) computed at the end of
the previous fiscal year. LN BM is book-to-market defined as in Fama and French (2008) in year t − 1. BIDASK
is the average bid-ask spread of the stock in the previous fiscal year. N ASDAQ is a dummy that equals one if the
stock is traded on the NASDAQ. All continuous variables are trimmed at the first and ninety-ninth percentiles. We
exclude all observations with stock price lower than $5.
Obs
Mean
SD
10th
50th
90th
Forecast Dispersion
DISP−3,0
61951
0.430
0.734
0.028
0.173
1.058
Trading Volume
T U RN−3,0
61316
-0.040
0.906
-1.233
-0.040
1.166
Stock Returns
RET−9,−3
DIF SIGN−9,−3
62565
62565
0.098
0.492
0.462
0.500
-0.293
0.000
0.046
0.000
0.483
1.000
Firm Characteristics
COV ERAGE
SIZE
SIGM A
LN BM
ROA
BIDASK
N ASDAQ
55429
60835
58908
54540
59805
49967
62576
2.076
6.483
0.028
-0.718
0.124
0.014
0.470
0.836
1.553
0.013
0.719
0.102
0.014
0.499
1.099
4.521
0.014
-1.691
0.019
0.001
0.000
2.079
6.372
0.025
-0.656
0.128
0.010
0.000
3.135
8.608
0.046
0.159
0.243
0.033
1.000
40
Table 2: DISPERSION IN ANALYST FORECASTS
DISP−3,0 in year t is the standard deviation multiplied by 100 of the most recent earnings forecast of each analyst
covering the stock in the last quarter of fiscal year t normalized by the stock price at the end of fiscal year t − 1.
DIF SIGN−9,−3 is a dummy that equals one if the sign of the cumulated stock return in the second quarter of fiscal
year t is not the same as the sign of the cumulated stock return in the third quarter of fiscal year t. COV ERAGE is
the logarithm of the number of analysts who covered the stock in the previous fiscal year. SIZE is the logarithm of
the stock total market capitalization (Compustat item CSHO × item PRCC F computed at the end of the previous
fiscal year). SIGM A is the standard deviation of daily raw returns of the stock in the previous fiscal year. Return on
assets, ROA, is defined as operating income after depreciation (item OIBDP - item DP) over total assets (item AT)
computed at the end of the previous fiscal year. LN BM is book-to-market defined as in Fama and French (2008) in
year t − 1. RET−3,−9 is the cumulated stock return over the second and third quarter of fiscal year t. All regressions
include ten dummies indicating ten deciles of RET−3,−9 and year fixed effects. Columns [2] and [5] also include
industry fixed effects. Columns [3] and [6] also include stock fixed effects. All continuous variables are trimmed at
the first and ninety-ninth percentiles. Standard errors, presented in parentheses, are clustered at the firm level. ∗ ,
and
∗∗∗
∗∗
denotes significance at the 10%, 5% and 1%, respectively. We exclude all observations with stock price lower
than $5. The sample period is from January 1982 to December 2011.
DEP. VARIABLE:
DIF SIGN−9,−3
[1]
0.050∗∗∗
(0.007)
[2]
0.053∗∗∗
(0.007)
Yes
Yes
No
No
Yes
Yes
Yes
No
61522
0.076
61522
0.101
COV ERAGE
SIZE
SIGM A
LN BM
ROA
Deciles of RET−9,−3
Year Fixed Effects
Industry Fixed Effects
Stock Fixed Effects
N
R2
DISP−3,0
[3]
[4]
0.016∗∗
0.031∗∗∗
(0.007)
(0.007)
0.070∗∗∗
(0.007)
-0.036∗∗∗
(0.004)
6.784∗∗∗
(0.420)
0.121∗∗∗
(0.007)
-1.396∗∗∗
(0.052)
Yes
Yes
Yes
Yes
No
No
Yes
No
61522
0.398
41
45675
0.164
[5]
0.029∗∗∗
(0.007)
0.067∗∗∗
(0.007)
-0.045∗∗∗
(0.004)
6.021∗∗∗
(0.452)
0.097∗∗∗
(0.007)
-1.444∗∗∗
(0.056)
Yes
Yes
Yes
No
[6]
0.015∗∗
(0.007)
0.100∗∗∗
(0.009)
-0.193∗∗∗
(0.010)
2.959∗∗∗
(0.601)
0.039∗∗∗
(0.009)
-0.945∗∗∗
(0.091)
Yes
Yes
No
Yes
45675
0.182
45675
0.449
Table 3: TRADING VOLUME
T U RN−3,0 in year t is the natural logarithm of the average trading volume in the last quarter of fiscal year t.
DIF SIGN−9,−3 is a dummy that equals one if the sign of the cumulated stock return in the second quarter of fiscal
year t is not the same as the sign of the cumulated stock return in the third quarter of fiscal year t. COV ERAGE is
the logarithm of the number of analysts who covered the stock in the previous fiscal year. SIZE is the logarithm of
the stock total market capitalization (Compustat item CSHO × item PRCC F computed at the end of the previous
fiscal year). SIGM A is the standard deviation of daily raw returns of the stock in the previous fiscal year. Return
on assets, ROA, is defined as operating income after depreciation (item OIBDP - item DP) over total assets (item
AT) computed at the end of the previous fiscal year. LN BM is book-to-market defined as in Fama and French
(2008) in year t − 1. BIDASK is the average bid-ask spread over the fiscal year t − 1. N ASDAQ is a dummy
that equals one if the stock is traded on the NASDAQ. RET−3,−9 is the cumulated stock return over the second and
third quarter of fiscal year t. All regressions include ten dummies indicating ten deciles of RET−3,−9 and year fixed
effects. Columns [2] and [5] also include industry fixed effects. Columns [3] and [6] also include stock fixed effects. All
continuous variables are trimmed at the first and ninety-ninth percentiles. Standard errors, presented in parentheses,
are clustered at the firm level.
∗
,
∗∗
and
∗∗∗
denotes significance at the 10%, 5% and 1%, respectively. We exclude
all observations with stock price lower than $5. The sample period is from January 1982 to December 2011.
DEP. VARIABLE:
DIF SIGN−9,−3
[1]
0.207∗∗∗
(0.008)
[2]
0.141∗∗∗
(0.007)
Yes
Yes
No
No
Yes
Yes
Yes
No
73783
0.253
73783
0.348
COV ERAGE
SIZE
SIGM A
LN BM
ROA
BIDASK
N ASDAQ
Deciles of RET−9,−3
Year Fixed Effects
Industry Fixed Effects
Stock Fixed Effects
N
R2
T U RN−3,0
[3]
[4]
0.064∗∗∗
0.126∗∗∗
(0.006)
(0.008)
0.225∗∗∗
(0.010)
0.065∗∗∗
(0.007)
33.876∗∗∗
(0.595)
-0.020∗∗
(0.009)
0.699∗∗∗
(0.062)
-12.649∗∗∗
(0.639)
0.099∗∗∗
(0.015)
Yes
Yes
Yes
Yes
No
No
Yes
No
73783
0.739
42
43595
0.453
[5]
0.109∗∗∗
(0.008)
0.187∗∗∗
(0.009)
0.078∗∗∗
(0.007)
27.298∗∗∗
(0.567)
0.002
(0.009)
0.311∗∗∗
(0.063)
-12.381∗∗∗
(0.624)
0.158∗∗∗
(0.015)
Yes
Yes
Yes
No
[6]
0.071∗∗∗
(0.007)
-0.009
(0.010)
0.125∗∗∗
(0.011)
15.172∗∗∗
(0.586)
-0.021∗∗
(0.010)
0.594∗∗∗
(0.075)
-4.606∗∗∗
(0.678)
0.322∗∗∗
(0.028)
Yes
Yes
No
Yes
43595
0.492
43595
0.768