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. 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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

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