Investor disagreement: the modern approach

INVESTOR DISAGREEMENT: THE MODERN

APPROACH

Fernando Ferreira da Luz Barbosa

INVESTOR DISAGREEMENT: THE MODERN

APPROACH

Fernando Ferreira da Luz Barbosa

Dissertação submetida como requisito parcial para

conclusão do Mestrado em Economia

Ficha catalográfica elaborada pela Biblioteca Mario Henrique Simonsen/FGV

Barbosa, Fernando Ferreira da Luz

Investor disagreement: the modern approach / Fernando Ferreira da Luz Barbosa. – 2015.

35 f.

Dissertação (mestrado) - Fundação Getulio Vargas, Escola de Pós-Graduação em Economia.

Orientador: Caio Ibsen Rodrigues de Almeida. Inclui bibliografia.

1. Teoria da informação em finanças. 2. Mercado financeiro. 3. Investimentos. 4. Previsão econômica. 5. Modelos econômicos. I. Almeida, Caio Ibsen Rodrigues de. II. Fundação Getulio Vargas. Escola de Pós-Graduação em Economia. III. Título.

Agradecimentos

Agradeço ao Professor Caio Almeida por toda orientação e apoio, neste e nou-tros projetos. Ao Professor Marco Bonomo, pelas discussões que ajudaram a dar um rumo a este trabalho. Ao Professor Felipe Iachan, pelas sugestões e comentários valiosos.

Abstract

Disagreement between economists is a well know fact. However, it took a long time for this concept to be incorporated in economic models. In this survey, we review the consequences and insights provided by recent models. Since disagreement between market agents can be generated through differ-ent hypotheses, the main differences between them are highlighted. Finally, this work concludes with a short review of nowcasting using google trends, emphasizing advances connecting both literatures.

Contents

1 Introduction 8

2 Disagreement Models 9

2.1 Gradual information flow . . . 10

2.2 Limited attention . . . 18

2.3 Heterogeneous priors . . . 20

2.4 Overconfidence . . . 24

2.5 Short-sale constraints . . . 26

3 Nowcasting 26

Investor Disagreement: The modern Approach

Fernando Luz Barbosa

Abstract

Disagreement between economists is a well know fact. However, it took a long time for this concept to be incorporated in economic models. In this survey, we review the consequences and insights provided by recent models. Since disagree-ment between market agents can be generated through different hypotheses, the main differences between them are highlighted. Finally, this work concludes with a short review of nowcasting using google trends, emphasizing advances connect-ing both literatures.

Keywords: Disagreement, differences of opinion, nowcasting, google trends, Dif-ferential interpretations, limited attention

1

Introduction

One of the key assumptions of CAPM is what is usually calledhomothetic expecta-tions. Under this hypothesis, rational investors, if exposed to the same information set, will have the exact same estimate of the asset’s future payoff distribution. In other words, all investors evaluate a company in the exact same way.

In the real world, however, economists are known to disagree about everything, from future inflation to companies valuation. In an early attempt to understand the consequence of disagreement, Miller [1977] showed how, together with short sale constraints, disagreement could generate bubbles in financial markets. The intuition is that in the presence of short sale constraints, the valuations of optimists will be reflected in a stock’s price, but the valuations of pessimists will not. Thus, even if the valuation of the average investor is unbiased, the stock price will be biased upward.

In this survey, we will go over modern developments in the literature that fea-tures disagreement among agents. We will argue that the possibility of agents “agreeing to disagree” has provided important insights. Additionally, we will dis-cuss mechanisms proposed by the literature to explain how disagreement among agents arises; namely: gradual information flow, limited attention and heterogeneous priors.

This survey is organized as follows. In the next section, we discuss disagree-ment models and review the literature on theoretical mechanisms that generate disagreement, as well as related empirical results. In section 3, we present nowcast-ing tools and argue that they can be invaluable for understandnowcast-ing the disagreement phenomenon and its causes.

2

Disagreement Models

A traditional benchmark in finance is that the return of a stock can only be effec-tively forecast with measures of its riskiness. Since this will not be the focus of our project, it suffices to say that this hypothesis has been shown to be inconsistent with the data; i.e. a large catalog of variables with no apparent connection to risk has been show to forecast stock returns.1

Also, the notion derived from standard models that rational agents should trade only for portfolio re-balancing needs is not consistent with the huge trad-ing volume observed in stock markets, especially durtrad-ing bubbles.

Two different branches of the theoretical literature that try to explain these anomalies must be mentioned. The first of them tries to understand what pre-vents rational investors to eliminate this predictability in returns, making a profit in the process. Short sales, for example, might be costly or simply not available for traders. Alternatively, DeLong et al. [1987] introduce “noise traders” whose irra-tional beliefs can drive prices away from the fundamental. When betting against noise traders, rational investors must take in consideration that the misspricing can increase even more. In other words, the fluctuations in noise traders beliefs gen-erate risk which in turn make (risk averse) rational investors less aggressive than they would otherwise be.

Another possible argument as to why arbitrage might not be ubiquitous in the stock market comes from Shleifer and Vishny [1997]. They argue that arbitrage would likely come from professionals that manage other people’s money. That would give rise to agency problems which might prevent them from acting ag-gressively as arbitrageurs.

The other branch of the theoretical literature seeks to explain the observed anomalies. To tackle this type of problem, it will be seen that the need to relax rationality hypotheses is often inevitable. 2 We are particularly interested in

mod-1_{One example is the phenomenon known as mid-term momentum, that is: the tendency for stocks}

that have had unusually high past returns or good earnings news to continue to deliver relatively strong returns over the subsequent six to twelve months (and vice-versa for stocks with low past returns or bad earnings news).Other examples are post earnings drift returns and longer run fundamental reversion. See Fama and French [1988] for a classical article on the subject, and Cochrane [2001] for a more detailed account.

2_{Assumptions of rationality provide scientific discipline, and abandoning the rationality paradigm}

els that use different forms of heterogeneity to explain such anomalies.

In particular, we will focus on models that rely on a particular type of hetero-geneity, that Hong and Stein [2007] calldisagreement models. They are based on the following mechanisms, which will be explained in detail later:

1. gradual information flow,

2. limited attention,

3. heterogeneous priors

In the classical literature, agents infer information from other agents’ behavior. The mere act of trying to sell a stock signals to other agents the belief that the stock is overvalued. The other agents subsequently use that information to update their beliefs. This leads to a lower stock price, which prevents the sale from taking place. This type of behavior would lead to low levels of trade in equilibrium. It has already been mentioned that this does not conform to the data. The psychological bias know as overconfidence can help getting around this problem. We will go over this in the next sections.

By introducing disagreement with either one of the aforementioned mecha-nisms, high volumes of trade can be generated. These mechanisms also provide in-sights about bubbles. In fact, by introducing short-sale constraints in this “hetero-geneous beliefs” environment, bubbles might be inflated because agents who think assets are overvalued cannot sell these assets short. Therefore, as “pessimists” are being kept out of the market, prices tend to reflect “optimists” beliefs. This causes prices to go up.

We explain these mechanisms in greater detail in the three following subsec-tions.

2.1

Gradual information flow

According to this hypothesis, information diffuses slowly across agents. The basic idea is that much like the real world, agents do not have access to the information at the exact same time. So, for example, perhaps a scientist will have access to information about a new procedure to cure cancer before the average investor. That can obviously lead to profits, if the scientist can anticipate before the crowd that this new discovery is going to increase earnings of a given company.

The concept becomes clear once the case of EntreMed, documented by Huber-man and Regev [2001], is introduced. In 1998, the New York Times front-page had a story of a breakthrough in cancer research, to which EntreMed had licensing rights. EntreMed’s stock price immediately soared, going from 12 to 85 dollars in less than a week. Afterwards, the stock price stabilized above the 30 dollar level.

Figure 1: EntreMed Historical price and volume. Chart from Huberman and Regev [2001].

event stock prices did respond to the news, but not nearly as dramatically as when the story hit the front-page.

Figure 1 shows EntreMed historical price (and volume) chart. The first small spike, in November 1997, corresponds to the first time the research breakthrough hit the news. It was reported in the journal Nature and in various popular news-papers, including the Times itself. In May 1998 we can see a huge spike in prices and trading volume following the New York Times front-page.

Thus, enthusiastic public attention induced a permanent rise in share prices, even though no genuinely new information had been presented. What happened with EntreMed greatly exemplifies what gradual information is trying to capture: if information had flown perfectly that first time, then showing old information on the first page of the Times should not have had any effects on prices.

We hope the EntreMed case was enough to convince you that gradual informa-tion flow is a reasonable assumpinforma-tion. But more important than that, including this hypothesis can generate some interesting results in already well know models. In Mankiw and Reis [2001] the authors propose to replace the so called “New Keyne-sian Phillips Curve” , which is build upon sticky prices , with a sticky information model. In other words, replacing Calvo prices with gradual information flow.

“this model is hard to square with the facts. Laurence Ball (1994a) shows that the model yields the surprising result that announced, credible dis-inflations cause booms rather than recessions. Jeffrey Fuhrer and George Moore (1995) argue that it cannot explain why inflation is so persistent. Gregory Mankiw (2001) notes that it has trouble explaining why shocks to monetary policy have a delayed and gradual effect on inflation. All of these problems arise from the same source: Although the price level is sticky in this model, the inflation rate can change quickly. By contrast, empirical analyses of the inflation process (e.g., Robert Gordon, 1996) typically give a large role to ’inflation inertia.’ ”

In a few paragraphs we will see how small changes at the original model, i.e. substituting sticky prices for sticky information, can generate results that better fit the data and stylized facts about inflation and monetary police.

Remember that in sticky prices, firms don’t get to choose when they are going to adjust prices. Every period, a company has a (λ) probability of being able to change her charged price. This probability is the same for every firm, independent of how long ago was her last price adjustment.

This model is populated by identical monopolistically competitive firms. Be-cause of that, the desired price p∗_t is a function of the overall price level p and outputy. Of course, the desired price maximizes profit for that moment in time.

p∗_t =pt+αyt

But remember that firms can rarely charge the desired price, since price adjust-ment happens at random times. Consequently, when a firm has the opportunity to change its price, it is set equal to the average desired price until the next adjust-ment.

xt=λ

∞

X

j=0

(1−λ)jEtp∗_t+j

In other words, the adjusted price (xt) is a weighted average of all future de-sired prices. Of course, the distant future carries a smaller weight because there is a big chance the company will be able to reset their prices before that date.

As expected, the economy price level is an average of all firms prices. Some were chosen in the distant past, and some were chosen at the present.

pt=λ

∞

X

j=0

(1−λ)jxt−j

Wrapping all up gives us the so called “New Keynesian Phillips Curve”.

πt= αλ 2

Now, let’s insert Gradual Information flow to the model. The firms’ desired price doesn’t change.

p∗_t =pt+αyt

The model is still populated by identical monopolistically competitive firms. Analogous to Calvo prices, in sticky information just aλfraction of the firms adjust theirexpectationsevery period. That means all the other companies are mak-ing their decisions based on outdated expectations. Similar to our last hypothesis, here each firm has the same probability (λ) of updating expectations the next pe-riod, regardless of how long ago was the last update.

Since we drop the sticky price hypothesis, all companies can adjust their prices on every period they intend to.

xj_t =Et−jp∗t

The subscriptjindicates how outdated are the expectations.

Again, the price level (pt) is going to be an average of all prices in the economy.

pt=λ

∞

X

j=0

(1−λ)j_xj t

=λ

∞

X

j=0

(1−λ)jEt−j(pt+αyt)

Finally, this equation for the price level yields the following equation for the inflation rate.

πt= αλ

1−λyt+λ

∞

X

j=0

(1−λ)jEt−1−j[πt+α(yt−yt−1)]

Notice the importance of expectations formed in the past for this sticky infor-mation Phillips curve.

These small changes to the model end up making all the difference. In the fig-ure below, Mankiw and Reis [2001] compare how sticky information, sticky prices and adaptive expectations respond to a sudden disinflation. This experiment rep-resents a sudden and permanent shift in the money supply growth rate.

As we can see, at the sticky prices model, the change in monetary police is in-stantly incorporated to the price adjustments. That is, when given the opportunity to adjust prices, the agents change their rate of price increase according to the new economic conjuncture. In this model, prices are sticky, but there is no inflation inertia. However, at the sticky information model, a fraction of the firms keeps in-creasing their prices at the same rate they used to. That lasts until the information slowly spreads throughout the economy and all the agents adjust their expecta-tions. This behavior creates inflation inertia. Notice the lag between change in monetary police and maximum response in inflation, a well know stylized fact.

Now let’s focus on what happens with the output.

prices as if nothing has happened. With a constant money supply and rising prices, the economy experiences a recession before output recovers.

What about announced disinflations? Suppose the disinflation of the last ex-ample is announced and credible two years in advance.

Note how Calvo prices create the weird result where a economic boom follows disinflation.

In conclusion, substituting sticky information for sticky prices creates a lag be-tween change in monetary police and inflation response, a well know stylized fact. At the same time, it creates inflation inertia, a phenomenon not possible with Calvo prices unless we impose an auto-regressive structure onm.

In another paper, Mankiw, Reis and Wolfers (2003) make a case study of the Volker disinflation period. After Volcker was nominated chairman of the Board of Governors of the Federal Reserve Board, the inflation started to drop from 11% per year to 4% accompanied by a contractionary monetary police. This sudden change in policy and its effects on agents expectations is show at figure 2. As we can see, expectations slowly change, first increasing dispersion in agents expectations and finally regrouping at the new (smaller) inflation.

Figure 3 plots the distribution of inflation expectations predicted by Mankiw et al. [2003] using a VAR application of the sticky information model. Note how the model predictions are similar to the actual data. At the sticky information model, information flows slowly through the economy. At the beginning of the disinfla-tion process a few agents update their informadisinfla-tion sets, lowering their infladisinfla-tion expectations. This flattens out the distribution and the dispersion increases as seen in the actual data. The gap between pre-Volker expectations and more recently up-dated beliefs creates a bi-modal distribution. As more agents update their beliefs, the mass of this distribution shifts from the right peak to the left, lowering the av-erage inflation expectation. Ultimately, the distribution resumes it’s normal single peak shape, now around a lower inflation level. Althought predictions look ”too sharp”, the model succesfully accounts for the broad features of the real process seen in figure 2.

Figure 2: Inflation Expectations Through the Volcker Disinflation. Chart from Mankiw et al. [2003].

2.2

Limited attention

Economists often argue that errors are independent across individuals, and there-fore cancel out in equilibrium. However, people share similar heuristics, those that worked well in our evolutionary past. So on the whole we should be subject to some similar biases. Systematic biases (common to most people, and predictable based upon the nature of the decision problem) have been documented in a vast literature in experimental psychology. Kahneman and Tversky [2000] provide sev-eral examples in their book.

Remember that in Gradual Information Flow, agents were irrational in the sense that they could not extract information from price movements and trade pattern. In limited attention, we go a little bit further. Here, not only can agentsnotextract information from prices but they also cannot absorb all the information available about the companies.

In other words, this hypothesis says that cognitively-overloaded investors pay attention to only a subset of publicly available information. Living in a world connected by internet, we all know it is pretty hard to keep up with the gigantic amount of information we pass through every day. That being said, limited atten-tion sounds quite intuitive.

It is worth mentioning that in order to generate interesting results about volume and prices, it is usually assumed that traders do not take into account that they are basing their valuation upon only a subset of all information.

On the empirical side, DellaVigna and Pollet [2009] showed interesting ev-idence of the limited attention hypothesis. They observed that when firms an-nounced their earnings on a Friday, the subsequent level of trade associated with it was smaller than when they announced it on other days of the week. According to the authors, this happens because investors get distracted during the weekend.

The careful reader might have noticed that there are conceptual similarities be-tween limited attention and gradual information flow. Furthermore, both frame-works rely on traders “agreeing to disagree”: their valuations are made using their information set only; they do not make inferences from market prices. This type of behavior occurs, for example, when agents are overconfident. (For a detailed description of overconfidence, see section 2.4.)

In these models, disagreement usually arises because not every agent pays at-tention to a new information. As with gradual information flow hypothesis, here, disagreement arises because investors are looking at different information sets.

asset values when the country in question is also featured in a front-page story in the New York Times. This conclusion is also corroborated by the EntreMed case, discussed in the last section, where a front-page with no new information seems to have had a tremendous impact on stock prices.

One way of formalizing the idea that agents have a limited capacity when it comes to processing information comes from the work of Sims [2003] on what is calledRational Inattention.3 Rational Inattention is the ideia that agents are rational but limited in their capacity to process information. Sims [2003] makes use of cod-ing theory developed by engineercod-ing science. The act of processcod-ing information by agents is modeled as if information is passing through a “channel” with finite capacity; because of that, economic agents may not be able to respond arbitarily quickly and precisily to market signals.

To be more specific, suposeX is a macroeconomic variable that agents would like to keep track. S = X+εis a noisy signal. If agents don’t observe neither of the variables, they have only the information about the distribution of X. We could measure the uncertainty about X using the concept of entropy.

H(X) =E[−log2(f(x))]

wheref(x)is the pdf of X.

However, if an agent has acess to signalS, that information reduces uncertainty about X. In other words, even if it’s not possible to perfectly identifyX, observ-ing the signal allows for a better prediction of the non-observable macro variable. The amount of information acquired throughS is the change in entropy, i.e. the reduction in uncertainty aboutX.

I(X;S) =H(X)−H(X|S)

Now we could model limited attention as a bound on information flow. That is:

I(X;S)≤κ (1)

IfXis assumed to be normally distributed, and the noiseεis a consequence of information processing limited capacity, normally distributed with mean zero and independent ofX, then:

H(X) = 1

2log2(2πeσ 2 X)

H(X|S) = 1

2log2(2πeσ 2 X|S)

And consequently we can rewrite the capacity restriction, i.e. equation 1, as

σ2 X σ_X2_|S ≤2

2κ

That means there is a lower bound to variance reduction.

σ_X2_|S ≥ σ 2 X 22κ

Note how a bigger capacityκ allows a smaller conditional variance. Also, if there is more uncertainty aboutX,Swill be less precise. In other words, a bigger σ2_X implies a tigther constraint onσ_X2_|S (ceteris paribus).

In a rational inattention enviroment, agents optimize their actions bounded by capacity restrictions. Supose a firm is solving a tracking problem. She chooses her charged price, P, to track the optimal price, P∗. The firm loss for charging the wrong price is E[(Pt−Pt∗)2]. Here,P is chosen on the basis of an optimally coded transmission of data aboutP∗through a finite-capacity information channel. Formally the firm solves the following problem.

min

b,c E[(Pt−P

∗

t) 2

]

subject to

P_t∗= n X

s=0

asεt−s, Pt= n X

s=0

bsεt−s+ n X

s=0 csνt−s

I[Pt;Pt∗]≤κ

Whereεandν are both i.i.d. N(0,1) stochastic processes. Clearly, if there is no capacity restriction, the firm would choose to perfectly trackP∗. That is, the firm would choosebs=as, cs = 0∀s. In figure 4, extracted from Sims [2003] we see the solution for the case whereais a simple linearly declining sequence of weights. In this caseκis exogenously set to 0,641 andn= 31. Note how the capacity restriction makesPtrespond more slowly to recent shocks. That is a direct consequence of the largest difference betweenbsandasfor smaller values ofs. In other words, when t=τ a spike inετhas a bigger impact inPτ∗ than inPτ. Also note thatcis sharply

peaked near zero, implying that high frequency variation in Pt is dominated by noise.

2.3

Heterogeneous priors

Figure 4: Optimal price and charged price response to shocks.Chart from Sims [2003].

To give a more precise meaning to the terms used in the above paragraph, con-sider an agent who receives a random public signal s ∈ S that he uses to learn about a parameterθ∈Θ. The (Bayesian) posterior density (or pmf) onθsatisfies

f(θ|s) = f(s|θ)f(θ) f(s) =

f(s|θ)f(θ)

R ˜

θ∈Θf(s|θ˜)f(˜θ)dθ˜

Intuitively, the conditional density functionf(s|θ) gives the likelihood of ob-serving signalswhen the underlying state isθ. This likelihood function is called the agent’sinterpretationof the public signal, or hiseconomic model. Also,f(θ|s)is called the agent’sposterior belief. One way to measure disagreement among agents is therefore to evaluate a “distance” between the posterior beliefs of these agents.

In this section, i.e. heterogeneous priors, we include all models where agents agree to disagree about a specific parameter. That could mean investors have dif-ferent prior beliefs aboutf(θ)(as usual), about the interpretation off(θ|s)or even about other parameters.

To see why a setting in which agents have different interpretations can lead to more disagreement – i.e., to a “divergence” of their posterior beliefs – upon the arrival of public information, consider the following example.4

In a hypothetical country, two Bayesian learners (F andG) don’t know if the

president is corrupt or not. Then the relevant set of parameters isΘ ={corrupt,not corrupt}. The prior beliefs of agents that the president is corrupt are π_cF andπG_c .

Further-more, a news releasesis made available which has two possible contents: either corruption isfound(s=C) or corruption isnot found(s=N C).

The point about differential interpretations is that agents may react very dif-ferently after observing the signal. It may sound reasonable that when agents

serve a signal s = C – that is, a news release which points to corruption – they update their beliefs placing “more probability” on the stateθ=corrupt. However, a supporter of the ruling party might argue that the signalsrefers to corruption in the economy as a whole. In his view this is no direct proof of the president in-volvement. This agent may actually sees=Cas evidence that the ruling party is fighting corruption. In other words, more corruption is being brought up but not necessarily more corruption is happening.

Moreover, even if both agents believe that a signal s = C indicates a higher probability ofθ =corrupt (relative toθ = not corrupt), they might have different subjective views of the credibility of the news.

In short, their attitude with respect to the public signal would be summarized, in this example, by the subjective conditional probability mass functions

pK(s=σ|θ) K∈ {I, J}

that is, the probability of observing signal σ ∈ {C, N C} when the true state isθ. For example, if

pF_{(s=C|θ=corrupt)}

pF_{(s=C|θ=not corrupt)} <1<

pG_{(s=C|θ=corrupt)}

pG_{(s=C|θ=not corrupt)} (2)

thenF believes that observings=C is evidence that the president is not corrupt, whileGthinks it is evidence that it is corrupt. Now, suppose that both agents share the same priors5, and that a signals=Carrives. In this case, if they interpret the signal as in (2), they necessarily start disagreeing after observing the signal.

One important feature of the models based on differential interpretations is that, even though the different interpretations of public signals are common knowl-edge, every agent thinks his interpretation is the correct one and that the others’ interpretations are wrong. In other words, agents “agree to disagree” about the in-terpretation of public signals. Moreover, agents are assumed to be limited in their ability to extract information from prices using their knowledge of the interpreta-tion of others. In some sense, therefore, rainterpreta-tionality is not common knowledge in this context.

The hypothesis that agents agree to disagree may sound stringent to some read-ers, but it did not come out of nowhere. Empirical puzzles concerning the time-series properties of trading volume, and its relationship to price, had been previ-ously addressed by rational expectations models without much success. According to Harris and Raviv [1993], “rational expectations generate disagreement through private information.”6 (p. 475) When agents are exposed to public signals in these models, on the other hand, trade is not generated. This is difficult to reconcile with the empirical evidence which shows that trading volume typically spikes around

5_{by same priors we mean: same prior distribution about the set of relevant statesθ}

an earnings announcement.

Other documented empirical properties of the volume of trade, as summarized by Harris and Raviv [1993] include “the positive correlation between volume and absolute price changes or price variance over time” (p. 478), and a “positive rela-tion between volume and the absolute change in the mean forecast of analysts.” (p. 478) The results generated by the differential interpretation model presented by Harris and Raviv [1993] are consistent with these empirical regularities in the volume of trade. In their model, however, trading volume is only generated when there is a price change.

It is important to mention that in more recent papers other mechanisms than differential interpretations have been able to generate a spike in trading volume during an earnings announcement. In Kondor [2012], a scenario where a group of agents have higher-order beliefs is emulated in a rational expectations equilibrium. When there is a public announcement – modeled in the paper as an exogenous increase in the precision of the public signal – there is greater disagreement about “second order beliefs” , which leads to a greater volume of trade.

Finally, Harrison and Kreps [1978] show that a heterogeneous priors mecha-nism can be used to generate a bubble due to a resale option. In their model, agents have different beliefs about the Markov process of a dividend. One of the agents believes economic cycles are shorter and tend to revert faster than the other agent believes. Under short sale constraints this scenario makes equilibrium prices ex-ceed the agents valuation of buying the asset and holding it forever. This happens because once we are in a bad state the short cycle believer is willing to pay more just for having the opportunity to resell the asset once the economy recovers. Since they are willing to pay a premium over their valuations today in hope of reselling the asset, overpricing will occur.

As the reader might still be skeptical about heterogeneous priors hypotheses, we include a very persuasive defense of these models by Xiong [2013]:

“If heterogeneous beliefs are derived from heterogeneous priors, one may argue that as individuals obtain sufficient information over time, learning will eventually cause their beliefs to converge. While appeal-ing, this argument does not always hold true. Endogenous learning ex-plains one reason. In the multi-armed bandit problem studied by Roth-schild (1974), a gambler chooses repeatedly between two slot machines in a casino, one with a known probability of payout and the other with an unknown probability.

environment where learning is costly and endogenous.

Even in the absence of endogenous learning, the eventual convergence of agents’ heterogeneous beliefs is not guaranteed. Kurz (1994) defined a belief to be rational if it generates the same long-run empirical fre-quencies as the data. In a stationary economic system there is a unique rational belief. In contrast, Kurz pointed out that if the system is not stationary there typically will be many rational beliefs.

Even in a stationary economic system, learning requires that agents know the conditional distribution of signals given the fundamental variable. Acemoglu, Chernozhukov, and Yildiz (2009) showed that when agents are uncertain about the signal distributions, even vanishingly small in-dividual uncertainty about the signal distributions can lead to substan-tial (non-vanishing) differences in asymptotic beliefs.” [Xiong, 2013, p. 18]

2.4

Overconfidence

It is important to remember that none of the previous hypotheses above can create bubbles unless the agents agree to disagree. For example, in the case of gradual information flow, usually investors cannotextract information from other agents trading behavior. If they could, the badly informed agents would be able to update their beliefs by observing the well informed investors’ buying or selling decisions, in which way they would be able to infer otherwise private information.

The hypothesis according to which investors are so confident in their estimates that they neglect the behavior of other investors can sound too strong at first sight. However, this behavior, called “overconfidence” is a well documented psycholog-ical phenomenon. Barberis and Thaler [2003] write that

"Extensive evidence shows that people are overconfident in their judg-ments. This appears in two guises. First, the confidence intervals people assign to their estimates of quantities – the level of the Dow in a year, say – are far too narrow. Their 98% confidence intervals, for example, include the true quantity only about 60% of the time [Alpert and Raiffa (1982)]. Second, people are poorly calibrated when estimating probabil-ities: events they think are certain to occur actually occur only around 80% of the time, and events they deem impossible occur approximately 20% of the time [Fischhoff, Slovic and Lichtenstein (1977)]. " [Barberis and Thaler, 2003, p. 1063]

when she begins to trade, however, as time passes by, she starts learning about her own abilities in a process that can create overconfidence. Here, when traders assess their ability they are influenced by another well documented bias: agents tend to overestimate the degree by which they are responsible for their own success.

Put in other words, traders who successfully forecast next period dividends improperly update their beliefs, over-weighting the probability that their success was due to superior ability. The most interesting feature of this paper is that at first sight, if we have rational and irrational traders, we may think that the overconfi-dent ones will be ruled out of the market by the most rational traders. However, although overconfidence does not create success per-se, the process of becoming successful generates overconfidence.

That is, the process of becoming rich involved, between other things, being lucky; but, due to the learning bias, the luckiest you got, the more overconfident you are going to be. Because of that, the most overconfident and non-rational traders are not the poorest traders. For any given level of learning bias and trad-ing experience, it is successful traders, though not necessarily the most successful traders, who are the most overconfident [see Gervais and Odean, 2001].

After that, it is not hard to imagine that if we live in a world where overconfi-dence is related to success, people are probably willing to act in an overconfident manner, so they signal success and thus receive some of the benefits of looking successful. Kahneman [2011] appears to agree that overconfident behavior can be encouraged by the environment where many professionals work. He writes

Acting on pretended knowledge is often the preferred solution.” [Kah-neman, 2011, Sec. 24 ref. no. 16]

2.5

Short-sale constraints

Until now, our hypotheses were able to generate trade but not bubbles. For dif-ferences of opinions between investor to generate overpricing we need to add a new hypothesis: short sale constraints. These refer to anything that makes it less attractive to establish a short position than a long one.

First of all, for short-selling a stock, the investor need to borrow this same stock, and for that he pays a fee.

Moreover, since there is no centralized market for borrowing, in order to bor-row a share, an investor needs to find an institution or individual willing to lend shares, and, consequently, finding shares can be difficult.

Furthermore, there can be legal constraints to short-selling. Many mutual fund managers are not allowed to short sell. According to Almazan et al. [2004] “73.3% of the 679 funds that filled Form N-SAR in 1994 reported that their investment policies formally restricted them from selling short”.

3

Nowcasting

Until now, we have showed how adding disagreement to economic models can provide interesting results. By allowing differences of opinion, Miller [1977] shows how bubbles can be inflated when pessimists are ruled out of the market due to short sales constraints. Mankiw and Reis [2001] suggest we should substitute sticky prices for sticky information; the authors create disagreement through dif-ferences in information sets available to agents. Here, the gradual difusion of infor-mation generates a more realistic model of inertial inflation as argued in subsection 2.1.

Real-time data is becoming ever easier to find, especially due to new nication technologies such as smartphones, broadband internet, wireless commu-nication and social media. On the other hand, many economic indicators are typi-cally only available with a lag of several weeks. Nowcasting is the idea of predict-ing the present. In other words, nowcastpredict-ing uses real-time high frequency data to predict this lagged indicators.

In this survey we are going to focus on a specific type of real-time data: num-ber of queries in Google. Several papers use other types of data, like Twitter or Facebook, but the main idea is the same.

Google Trends provide a time series of the volume of queries users enters into Google in a specific geographic area. Choi and Varian [2012] write

“The query index is based on query share: the total query volume for the search term in question within a particular geographic region divided by the total number of queries in that region during the time period be-ing examined. The maximum query share in the time period specified is normalised to be 100, and the query share at the initial date being ex-amined is normalised to be zero. The queries are ‘broad matched’ in the sense that queries such as [used automobiles] are counted in the calcu-lation of the query index for [automobile]. The data go back to January 1, 2004. Note that Google Trends data is computed using a sampling method, and the results therefore vary a few per cent from day to day. Furthermore, due to privacy considerations, only queries with a mean-ingful volume are tracked.”

Below, you are going to find several examples of how to use this data to make better predictions. The idea is simple, but powerful. Take for instance the example of car sales. Buying a car is an important decision, and because of that, it involves planning. That usually means accessing the internet to search for information be-fore actually buying the car. Consequently, an increase in the number of queries for “car insurance” or “SUV” should mean a near future increase at car sales.

Another well known example relates to flu spreading. Governments want to detect as early as possible disease activity, since a rapid response can potentially re-duce the impact of seasonal and pandemic influenza [Ferguson et al., 2005, Longini et al., 2005]. That’s why the U.S. Center for Disease Control and Prevention (CDC) spends time and money gathering virologic and clinical data, including influenza-like illness (ILI) physician visits. Needless to say that making predictions based on Google search index is not only cheaper but also faster than using regular CDC approach.

at Google (or other search engine) before going to the doctor.

In their paper “Detecting Influenza like epidemics using search engine query data”, Ginsberg et al. [2009] show that

“Google web search queries can be used to accurately estimate influenza-like illness percentages in each of the nine public health regions of the United States. Because search queries can be processed quickly, the resulting ILI estimates were consistently 1-2 weeks ahead of CDC ILI surveillance reports. The early detection provided by this approach may become an important line of defense against future influenza epidemics in the United States, and perhaps eventually in international settings. Up-to-date influenza estimates may enable public health officials and health professionals to better respond to seasonal epidemics. If a region experiences an early, sharp increase in ILI physician visits, it may be possible to focus additional resources on that region to identify the eti-ology of the outbreak, providing extra vaccine capacity or raising local media awareness as necessary”

The picture below shows Google estimates for dengue in Brazil in blue and the actual data in orange. The correlation is clear.

swimming pools now work as water storage, no swimming allowed! Remember thatAedes Aegypti, the mosquito that spreads dengue fever, only breeds in contain-ers that hold fresh water.

Other examples of nowcasting using Google Trends include: number of tourists flying to Hong Kong and initial claims for unemployment insurance both dis-cussed by Choi and Varian [2012].

Until now, all the examples above took advantage of the almost real time avail-ability of Google search index. But there is a even more interesting characteristic of Google Trends: people don’t lie to their computers. For the last six years, there have been more searches for the word “porn” than for the word “weather”.

From how to join the Islamic State to learning how to kiss, Google seems to be the right guy to ask. Needless to say, most people wouldn’t make the same ques-tion to their best friends. We would like also to inform our readers that it took us a long time to find this screenshots. At this date, the most obvious sentences like “is it wrong to...” are followed by unprintable autocomplete suggestions. Although these suggestions are based on likelihood and thus can change from time to time, we don’t expect this scenario to change. 7

Maybe the most interesting papers using Google Trends take advantage of this truthful relationship between users and search engines. Stephens-Davidowitz [2013] writes

“A key challenge in forecasting elections is estimating who will show up to vote. People misreport likelihood of voting to polls. Rogers and Aida (2012) find that 67 % of individuals who will not vote tell pollsters they are almost certain to vote. Recent research casts doubt on the re-liability of pollsters’ tools to screen “likely voters.” Ansolabehere and Hersh (2011) find that even the first screen used by polls, registration status, is subject to large deception. More than 60 % of non-registered voters falsely claim that they are registered.”

7_{autocomplete suggestions are based not only on search frequency but also in personal information}

In a innovative paper, Stephens-Davidowitz [2013] uses Google Trends to pre-dict area level turnout. The intuition is simple. Electors may lie to pollsters about their true intentions of vote. However, people willing to vote tend to Google ex-pressions like “where to vote” or “how to vote”. That means it should be possible to use number of searches for terms like “vote” or “voting” to predict turnout. The author finds compelling evidence that this is the case. The predictive power is also little affected by controlling for other common proxies such as registration rates and early voting rates.

The great contribution of the paper however is predicting the demographic makeup of the electorate. This is a key issue due to the polarization of the Ameri-can voters. According to exit polls, roughly 88% of AfriAmeri-can-AmeriAmeri-can voters sup-ported Democrat John Kerry in 2004. This number increased to 95% when Obama first run for president in 2008. White evangelical voters also show a strong ten-dency to support Republican candidates. [Stephens-Davidowitz, 2013]

That means a one percent point increase in African-American turnout stands for approximately one percent point increase at the Democratic candidate’s voter share. Stephens-Davidowitz [2013] compare demographic data and area voting interest (based on Google Trends data) to predict whether any demographic can be expected to turnout in unusually high numbers.

In 2008, African-American turnout in unusually high rates to support the first black candidate running for president. Stephens-Davidowitz [2013] shows that Google data would have correctly predicted a substantial increase in African-American turnout. The author also takes one step forward, and makes out-of-sample pre-dictions of 2012 electors demographics. His prepre-dictions based on Google Trends search index “correctly forecast elevated Mormon turnout. It correctly forecast, contrary to some pollsters’ predictions, that African-American, Hispanic, and youth turnout rates would remain at 2008, rather than 2004, levels.”

GSO8is calculated based on the equation:

GSOh(x) = log

_{hits(x&hawkish)}

hits(x&dovish)

+constant (3)

Where x represents a word or expression, e.g. “Pressures on inflation have picked up”

Carvalho et al. [2013] use this same methodology to investigate if the Central Bank of Brazil has lost credibility in the last few years. This paper is not related to disagreement, but, in my view, it brings interesting questions on how investors process information. To be more specific, it indicates that investors can start ignor-ing (useful) information when central banks lose their credibility. Although GSO for COPOM 9 _{statements Granger-causes changes in policy rates, investors seem}

to have started ignoring this information in recent years (after Tombini’s tenure). In other words, changes in Google Semantic Orientation score can help predict changes in policy rates; there is evidence that investors (used to) take this in con-sideration and we can see statistical significant changes in yields at medium ma-turities following changes in COPOM language. However, this result only holds prior to Tombini’s tenure. That could indicate monetary authorities lost their cred-ibility in recent years.

These semantic classification techniques open a new world of possibilities to the literature. In a recent paper Giannini et al. [2014] create a new measure of investors disagreement. Here, instead of plotting expressions in a hawkish versus dovish semantic axis, the authors classify “twits” and news in positive or negative senti-ments. Since sentiment analysis is beyond the scope of this text we will not spend too much time talking about the differences in semantic classification techniques employed by the cited papers, the general idea is the same. This new disagreement measure allows authors to investigate convergence of opinion around earnings re-lease and their effects on returns. Supporting Miller [1977] theory, they found evi-dence that when new information resolves disagreement returns are lower. How-ever, earnings releases not necessarily diminish disagreement, it may even create it.

The second important point connecting nowcasting to the rest of the survey is the following. If limited attention and gradual information flow are key in creating disagreement we must understand how financial information is flowing through the net.

In a paper with the suggestive name of “In search of Attention”, Da et al. [2011] , create a new measure for investor attention using Google Trends. The intuition

8_{The constant omitted in equation (3) is a function of}

hits(hawkish) hits(dovish) .

It is not important for practical matters.

here is simple, if you search for a stock in Google, you are undoubtedly paying at-tention to it. Common proxies for investor atat-tention in the literature, like abnormal turnover, extreme returns, news and headlines have all the same problem: they are

nota direct measure of attention. Even having a news article in the New York times does not guarantee attention unless investors actually read it. This is well exem-plified by the Entred Med case, cited in section 2.1. Using Google Trends allow the authors to create adirectmeasure of investor attention.

Following this idea, Barbosa and Pereira [2015] use Google trends as a proxy for investor attention, investigating if more information create convergence of agents opinions.

In other words, does more information makes investors agree? According to Miller [1977] disagreement tends to reduce as more information becomes available to agents. He writes: "uncertainty is reduced as the company acquires a history of earnings or lack of them, and the market indicates how it will value these earn-ings".

It seems intuitive that gradual information flow and limited attention agree with Miller’s intuition. If suddenly everybody starts to talk about a company, the cost of acquiring firm related information gets smaller. If no one is talking about the Higgs Boson, a scientist would have a smaller cost of acquiring this informa-tion then the average person. In January 2013 must people didn’t think Higgs Boson was a interesting topic. Most books and webpages talking about it looked like ancient Greek to the average person. Then a few months later, CERN scientists announced they have finally found a Higgs Boson. Suddenly even laypersons are talking about Higgs Boson and how amazing they are. Now you can find infor-mation at every magazine, internet, and – of course – this inforinfor-mation comes with different levels of complexity. It’s easy to find something that you can understand now.

On the other hand, Kondor in a paper named “The more we know on the fun-damental, the less we agree on the price” shows how even public information can generate disagreement in a common prior environment.This effect is generated be-cause differences in private information lead agents to interpret differently a public signal. The key point of his model is that some of the investors are trying to guess other investors’ valuations. After the arrival of news, those “speculative” investors disagree more in their second-order beliefs, which leads to an increase in trading activity. See Kondor [2012].

Models of differences of interpretation tend to follow Kondor’s conjecture, where more information leads to even more disagreement, as discussed in subsection 2.3.

differences of interpretation might be even more important than limited attention and gradual information flow in understanding investor disagreement.

4

Conclusion

Differences of opinion between specialists is a well know fact. President Truman famously asked for a one-armed economist. He was tired of hearing “on the other hand”. But more important than closely representing reality, models that allow dis-agreement between agents have provided important insights for economic theory. For example, we have shown how introducing differences of opinion can provide better explanations of how trade can increase after release of public information. On the empirical side, social media and search engines data are promising tools in the quest for a better understanding of how agents form expectations.

References

Daron Acemoglu, Victor Chernozhukov, and Muhamet Yildiz. Learning and dis-agreement in an uncertain world. Technical report, National Bureau of Economic Research, 2006.

Andres Almazan, Keith C Brown, Murray Carlson, and David A Chapman. Why constrain your mutual fund manager? Journal of Financial Economics, 73(2):289– 321, 2004.

Nicholas Barberis and Richard Thaler. A survey of behavioral finance. Handbook of the Economics of Finance, 1:1053–1128, 2003.

Fernando Barbosa and Gustavo Pereira. Investor disagreement: a nowcasting ap-proach. Working paper, 2015.

Carlos Carvalho, Fernando Cordeiro, and Juliana Vargas. Just words?: A quantita-tive analysis of the communication of the central bank of brazil.Revista Brasileira de Economia, 67(4):443–455, 2013.

Hyunyoung Choi and Hal Varian. Predicting the present with google trends. Eco-nomic Record, 88(s1):2–9, 2012.

John H Cochrane. Asset pricing, 2001, 2001.

Zhi Da, Joseph Engelberg, and Pengjie Gao. In search of attention. The Journal of Finance, 66(5):1461–1499, 2011.

J Bradford DeLong, Andrei Shleifer, Lawrence H Summers, and Robert J Wald-mann. The economic consequences of noise traders, 1987.

Eugene F Fama and Kenneth R French. Permanent and temporary components of stock prices. The Journal of Political Economy, pages 246–273, 1988.

Neil M Ferguson, Derek AT Cummings, Simon Cauchemez, Christophe Fraser, Steven Riley, Aronrag Meeyai, Sopon Iamsirithaworn, and Donald S Burke. Strategies for containing an emerging influenza pandemic in southeast asia. Na-ture, 437(7056):209–214, 2005.

Simon Gervais and Terrance Odean. Learning to be overconfident.Review of Finan-cial studies, 14(1):1–27, 2001.

Robert Charles Giannini, Paul J Irvine, and Tao Shu. Do local investors know more? a direct examination of individual investors’ information set. In A Direct Ex-amination of Individual Investors’ Information Set (August 8, 2014). Asian Finance Association (AsFA) 2013 Conference, 2014.

Jeremy Ginsberg, Matthew H Mohebbi, Rajan S Patel, Lynnette Brammer, Mark S Smolinski, and Larry Brilliant. Detecting influenza epidemics using search en-gine query data. Nature, 457(7232):1012–1014, 2009.

Milton Harris and Artur Raviv. Differences of opinion make a horse race. Review of Financial studies, 6(3):473–506, 1993.

J Michael Harrison and David M Kreps. Speculative investor behavior in a stock market with heterogeneous expectations. The Quarterly Journal of Economics, 92 (2):323–336, 1978.

Harrison Hong and Jeremy C Stein. Disagreement and the stock market. The Jour-nal of Economic Perspectives, pages 109–128, 2007.

Gur Huberman and Tomer Regev. Contagious speculation and a cure for cancer: A nonevent that made stock prices soar. The Journal of Finance, 56(1):387–396, 2001.

D. Kahneman. Thinking, Fast and Slow. Farrar, Straus and Giroux, 2011. ISBN 9781429969352.

D. Kahneman and A. Tversky. Choices, Values, and Frames. Choices, Values and Frames. Cambridge University Press, 2000. ISBN 9780521627498.

Peter Klibanoff, Owen Lamont, and Thierry A Wizman. Investor reaction to salient news in closed-end country funds. The Journal of Finance, 53(2):673–699, 1998.

Ira M Longini, Azhar Nizam, Shufu Xu, Kumnuan Ungchusak, Wanna Han-shaoworakul, Derek AT Cummings, and M Elizabeth Halloran. Containing pan-demic influenza at the source. Science, 309(5737):1083–1087, 2005.

David O Lucca and Francesco Trebbi. Measuring central bank communication: an automated approach with application to fomc statements. Technical report, National Bureau of Economic Research, 2009.

N Gregory Mankiw and Ricardo Reis. Sticky information versus sticky prices: a proposal to replace the new keynesian phillips curve. Technical report, National Bureau of Economic Research, 2001.

N Gregory Mankiw, Ricardo Reis, and Justin Wolfers. Disagreement about inflation expectations. Technical report, National Bureau of Economic Research, 2003.

Edward M Miller. Risk, uncertainty, and divergence of opinion. The Journal of Finance, 32(4):1151–1168, 1977.

Andrei Shleifer and Robert W Vishny. The limits of arbitrage.The Journal of Finance, 52(1):35–55, 1997.

Christopher A Sims. Implications of rational inattention. Journal of monetary Eco-nomics, 50(3):665–690, 2003.

Christopher A Sims. Rational inattention: Beyond the linear-quadratic case. The American economic review, pages 158–163, 2006.

Paul Slovic, Baruch Fischhoff, and Sarah Lichtenstein. Facts and fears: Under-standing perceived risk. InSocietal risk assessment, pages 181–216. Springer, 1980.

Seth Stephens-Davidowitz. Who will vote? ask google. 2013.

Mirko Wiederholt. Rational inattention. The New Palgrave Dictionary of Economics (Online Edition ed.), 2010.