Expectation Formation in Financial Markets:

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Expectation Formation in Financial Markets:

Heterogeneity and Sentiment

Bart Frijns

^∗

Thanh Huynh

^†

Remco C.J. Zwinkels

^‡

November 2019

Abstract

We set up an endowment based asset pricing model in which agents have heterogeneous expectations about the future price level. Expectations are a function of fundamentals, trends, and sentiment. Agents are allowed to switch between expectation formation functions based on past performance as well as sentiment. Estimation results on the S&P500 reveal that there is heterogeneity between agents, with substantial switching between groups. We find that sentiment has a direct effect on expectations.

Specifically, heterogeneity is increasing in sentiment, and sentiment reduces the fre- quency of switching between functions.

∗Auckland University of Technology; E: [email protected]

†Monash University; E: [email protected]

‡Vrije Universiteit Amsterdam and Tinbergen Institute; E: [email protected]

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1 Introduction

Expectations about future risk and return take center stage in the asset pricing literature (Sharpe, 1964; Lintner, 1965; Cochrane, 2005). While the traditional approach is to assume that investors hold rational expectations (Muth, 1961), more recent empirical work indicates that individual expectations are not rational (Cavaglia et al., 1994; MacDonald, 2000;

Bloomfield and Hales, 2002; Hommes et al., 2004). An open question is how individuals form expectations if not rational. Several alternatives have been proposed, typically separate from each other. For example, the investor sentiment literature (De Long et al., 1990; Baker and Wurgler, 2007, 2006) suggests that investors have stochastic but, on average, positive biases in their expectations. The heterogeneous agent literature assumes that investors use different heuristics in forming expectations and switch between those heuristics over time (Hommes and LeBaron, 2018). In an attempt to come closer to a unifying expectation formation framework, this paper develops an asset pricing model combining heterogeneity with sentiment. Estimation results for the S&P500 reveal that both sentiment and heterogeneity play an important role in the price formation process, and we document interesting interactions between sentiment and heterogeneity.

Noise trader risk, as introduced by Shleifer and Vishny (1997), is modelled by De Long et al. (1990) as a stochastic bias with positive mean added to an otherwise rational expectation. This definition makes it akin to an overoptimism bias (Huisman et al., 2012).

The most natural way to capture noise trader risk is through investor sentiment. The investor sentiment literature shows that sentiment, broadly defined, is positively associated with contemporaneous stock returns and negatively with lagged returns (Brown and Cliff, 2004; Baker and Wurgler, 2006; Chung et al., 2012).

Agent based models, or more specifically heterogeneous agent models (HAMs), assume that financial markets are populated by agents that use different heuristics to form expectations (Zeeman, 1974; Brock and Hommes, 1997; Barberis et al., 1998). Typically, these models include two types of agents: fundamentalists and chartists. Fundamentalists expect prices to revert to a (perceived) fundamental value. As such, they are closely related to the typical rational investor. Chartists, on the contrary, do not consider economic fundamen-

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tals but base expectations on recent price trends. This is motivated by the ample, mainly experimental, evidence that people have a tendency to extrapolate trends when forming expectations (Bloomfield and Hales, 2002; Greenwood and Shleifer, 2014; Landier et al., 2018).

An additional feature of these models is that agents are able to switch between heuristics.

Given the relatively simple nature of the heuristics, this can occur at no costs. Typically, the switching between heuristics occurs based on past performance. A host of studies have found empirical support for this approach to expectation formation in all sorts of asset classes; see Lux and Zwinkels (2018) for a recent overview.

In this paper, we develop a HAM in which investor sentiment directly affects the expectation formation processes of the fundamentalists and chartists. We extend the model of Brock and Hommes (1998) by integrating sentiment in the expectation formation process.

Specifically, we allow the fundamentalist mean reversion coefficient and chartist extrapolation coefficient to be time-varying and conditional on investor sentiment. We argue that in times of high sentiment, investors become more optimistic about their beliefs, which suggests that fundamentalism (mean-reversion towards the fundamental value) and chartism (extrapolation based on past price trends) beliefs become more extreme. As a consequence, high sentiment leads to more heterogeneity in investor beliefs. In addition to these more extreme beliefs, sentiment can affect an agent’s sensitivity to performance differences between heuristics. Since sentiment increases an investor’s overconfidence, which can result in a stronger self-attribution bias, we expect that an agent’s sensitivity to past performance decreases in times of high sentiment, as investors are more confident that the belief they follow is correct, and pay less attention to market signals.¹

In the empirical part of the paper, we estimate our model on S&P500 returns. We measure sentiment using Thomson Reuters News Analytics (TRNA), which quantifies the tone of each news article, and can be aggregated to a daily measure of investor sentiment. We find strong support for the joint importance of both heterogeneity and sentiment. Specifi- cally, the empirical results are consistent with a model with time-varying heterogeneity, in

1A related study to ours is Kukacka and Barunik (2013). They evaluate the impact of behavioral breaks in heterogeneous agent models. Specifically, they address the question of whether HAMs can capture behavioral biases, such as herding, overconfidence and market sentiment. Through a numerical analysis, they demonstrate that HAMs can indeed capture dynamics generated as a consequence of behavioral biases. Their work forms an important motivation for our empirical study.

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which fundamentalists and chartists co-exist in dynamic proportions. Furthermore, investor sentiment interacts with the expectation formation process as expected, generating more heterogeneity in times of high sentiment (high sentiment results in stronger mean-reversion for the fundamentalist, and stronger price extrapolation for the chartist). In addition, agents switch less between the various trading rules in times of high sentiment, in line with the notion that investors become overconfident in times of high sentiment and thus rely more on their private beliefs rather than performance signals they get from the market. The results are robust to the exact specification of the model, the definition of the fundamental value, and the exact definition of sentiment.

There are two channels through which investor sentiment can affect investor heterogeneity: optimism and overconfidence. With regard to the belief formation, (over)optimism could result in an exaggerated view that stock prices will evolve according to a specific belief that investors may hold. For instance, in times of high sentiment a fundamentalist may become more optimistic about the mean-reversion of the stock price towards a fundamental value.

Likewise, a chartist may become more optimistic about price-continuation and thus become more extreme in his/her belief, expecting a more extreme price continuation than he/she would in case of low sentiment. Hence, we expect that (over)optimism, resulting from high sentiment, leads to more extreme beliefs of the different agents, thus resulting in more heterogeneity in beliefs. At the same time, we expect sentiment to have an impact on the switching behavior of agents. Since the optimism induced by high sentiment can lead to overconfidence, agents may become less sensitive to past performance of a specific trading strategy. This is because overconfidence can lead to an overreliance on private beliefs and an underreliance on public signals. This results in agents holding on to the belief that they have formed (either fundamentalist or chartist) and paying less attention to public signals (Barberis et al., 1998), in this case the past performance of a specific trading strategy. Hence, our expectation is that overconfidence, induced by high sentiment leads to a reduction in the switching behavior of agents.

The remainder of the paper is organized as follows. In Section 2 we present the heterogeneous agent model with sentiment and Section 3 contains the description of the data and empirical methods. Section 4 presents the results and Section 5 concludes.

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2 Model

We use a modified version of the Brock and Hommes (1998) model as the basis of our model.

Assume an economy with a single risky asset with price p_t that pays a stochastic dividend y_t. Wealth then evolves according to:

W_t+1 =RW_t+ (p_t+1+y_t+1−Rp_t)z_t, (1)

where R is the risk free rate and zt the demand for the risky asset.

Investors are mean-variance optimizers such that their demand for the risky asset solves:

M ax_z

E_ht(W_t+1)−(a/2)σ²_ht(W_t+1) , (2)

in which E_ht(W_t+1) is the expectation that investors of type h = {1, . . . , H} at time t has about the about W_t+1, a is the risk aversion parameter and σ²_ht(W_t+1) the variance of wealth. We assume that the variance of wealth is time invariant and homogenous across investor types, i.e. σ²_ht(W_t+1) = σ². Solving yields the optimal demand for the risky asset, z_ht, i.e.,

z_ht =E_ht(p_t+1+y_t+1−Rp_t)/aσ². (3) Given that there areHgroups of investors, we definen_htas the fraction of typehinvestors in period t. Total demand for the risky asset is then given by:

H

X

h=1

n_ht

E_ht(p_t+1+y_t+1−Rp_t)/aσ² . (4)

Without loss of generality, we can set the outside supply of the risky asset to zero, such that:

H

X

h=1

n_ht

E_ht(p_t+1+y_t+1−Rp_t)/aσ² = 0, (5) and:

Rp_t=

H

X

h=1

n_htE_ht(p_t+1+y_t+1). (6)

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Now assume that a fundamental price is given by p^∗_t. It is then convenient to write the model in terms of deviations from the fundamental,x_t =p_t−p^∗_t. Under the assumption that all beliefs of the groups in H are of the form:

E_ht(p_t+1+y_t+1) =E_t(p^∗_t+1+y_t+1) +f_h(xt−1, ..., xt−L), (7)

Equation (6) can be written as:

Rx_t=X

n_htf_h(xt−1, ..., xt−L). (8)

Consistent with the literature on heterogeneous agents, we assume two types of traders with different beliefs, fundamentalists and chartists. Up to this point, our model is consistent with the original Brock and Hommes (1998) model. We now deviate from this model by allowing both fundamentalist and chartist beliefs to be conditional on sentiment. The fundamentalists expect the price level to converge to the fundamental value, and thus x_t to converge to zero. The speed of convergence, however, is conditional on market sentiment. Let f_F = (φ¹_F +φ²_FS_t)xt−1 be the extent of mean-reversion expected by fundamentalists, where St is market sentiment. In accordance with a fundamentalist belief, we expect 0 < φ¹_F <1, i.e., a deviation from the fundamental price will mean-revert towards zero in the subsequent period. In addition, we expect φ²_F <0, i.e., the higher the market sentiment, the stronger the belief of the fundamentalist in mean-reversion. This negative expectation for φ²_F is in line with (over)optimism resulting in a stronger fundamentalist belief. Chartists, on the other hand, are destabilizing and expect the deviation between price and fundamental to increase. Hence, fC = (φ¹_C +φ²_CSt)xt−1 with φ¹_C >1, i.e., according to the chartist belief, a price discrepancy observed in the past is expected to increase in subsequent periods. In line with the expectation that this belief will get stronger in periods of high sentiment (due to (over)optimism), we expect φ²_C >0. Given these functional forms, we can rewrite Equation (8) as:

Rx_t =n_{F t}(φ¹_F +φ²_FS_t)xt−1+n_Ct(φ¹_C +φ²_CS_t)xt−1. (9) Agents are able to switch between groups conditional on the relative performance of the

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groups. In other words, n_ht is endogenously determined by means of:

n_ht = exp

β_t π_ht π_{F t} +π_Ct

/Z_t, (10)

Z_t = X

h

exp

β_t π_ht π_{F t} +π_Ct

,

which simplifies to:

n_{F t} =

1 + exp

β_tπ_{F t}−π_Ct π_{F t}+π_Ct

⁻¹

, (11)

n_Ct = 1−n_{F t} =

1 + exp

β_tπ_Ct−π_{F t} π_{F t}+π_Ct

⁻¹ ,

in which π_{F t} and π_Ct is the performance of fundamentalists and chartists, respectively.

The intensity of choice parameter, β_t, varies over time conditional on sentiment. Specif- ically:

β_t= ¯β+ψS_t, (12)

where ¯βcaptures the degree of switching when sentiment is equal to zero, andψcaptures the impact of sentiment on switching behavior. A positive value ofβt implies that agents switch towards the investment belief that has yielded the highest performance in the recent past (i.e., they follow a positive feedback trading strategy), while a negative value of βt implies that agents switch towards the investment belief that has yielded the lowest performance in the recent past (i.e., they follow a negative feedback or contrarian trading strategy). When ψ takes on the same sign as ¯β, sentiment intensifies the switching behavior, while when ψ takes on the opposite sign of ¯β, sentiment mitigates the switching behavior. In line with the notion that high sentiment results in (over)optimism and overconfidence, we expectψ to take on the opposite sign of ¯β as agents would rely more on their private beliefs than public signals (which in this case is the past performance of the different trading strategies).

To complete the model, we need to define the performance measure πht. We assume that agents base their choice on the relative ability of the groups to forecast x_t over the previous I periods. Specifically:

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π_ht=

I

X

i=1

|xt−i−φ_hxt−1−i|, (13)

in which I is set to five in the benchmark case; sensitivity to this choice is checked later in the paper.

3 Data and Methods

To estimate our heterogenous agent model, we obtain daily data on dividend yield, price- earnings ratio, and the total return index on S&P500 from Thomson Reuters Datastream.

Our final sample contains 3,334 daily observations over the period from January 2003 to March 2016. Using these data, we first need to obtain a proxy the fundamental value p^∗. Similar to Boswijk et al. (2007) we compute the fundamental value based on the Gordon growth model:

p^∗ = 1 +g

r−gy_t, (14)

in which g is the growth rate of dividendsy_t and r the required return.

Assuming that the long-run growth in prices is equal to the long-run growth in dividends (Fama and French, 2002), we get that r−g =y/p. Hence:

p^∗ = 1 +g

E(y/p)yt, (15)

where E(y/p) is the expected dividend yield. In the benchmark setting, we assume that agents take a 100-day historical rolling average to calculate g and E(y/p).² Note that we take logs of all values. Figure 1 presents the evolution of p, p^∗, and x_t for the S&P500, and Table 1 the descriptive statistics.

INSERT FIGURE 1 HERE

Both the Figure 1 and Table 1 show that our measure of a fundamental value behaves as one would expect from a fundamental value estimate. On average, the fundamental value is

2In the empirical section we study the robustness of the results to this choice

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roughly equal to the market value. Furthermore, the market value is more volatile than the fundamental value, consistent with the findings of Shiller (1981).

To measure sentiment, we employ historical news data on S&P500 firms from Thomson Reuters News Analytics (TRNA). TRNA collects and analyzes firm-level news articles from major news outlets such as Down Jones Newswires, the Wall St Journal, Reuters, as well as local newspapers.³ To measure the information content of each news article, it employs a proprietary algorithm of textual analysis and produces three quantitative scores that represent the probabilities that the news item has positive, negative, and neutral sentiment (the three probabilities sum to one). TRNA also provides us with a score of how relevant the news item is for a given firm. For stories that mention multiple firms, the firm with the most mentions will have the highest relevance score. If the firm is mentioned in the headlines, then it is deemed a major player in the news article and hence the relevance score is 1. As in Hendershott et al. (2015), we match news articles to a firm using NYSE trading days and hours. News stories appearing after 4 pm Eastern time will be assigned to the following trading day. To allow for the fact that a news item can be more relevant to one firm than another, we follow Hendershott et al. (2015) and scale the tone score by its relevance to the firm as follows:

tonej =relevancej ×(posj −negj) (16) where relevancej ∈ [0,1] is the relevance score of the news item j to firm i; posj and negj

are the probabilities that the news article j has positive and negative tone, respectively.

For each firm, we compute the average tone of its news stories on a given day. Our measure of market-wide news sentiment on day t is then the simple average of firm-specific news tone across all firms in the market. Following Dzielinski and Hasseltoft (2014), we also compute the cross-sectional standard deviation of news tone scores across all firms on

3TRNA has increasingly become popular in academic research, as well as in the industry. For example, Hendershott et al. (2015) employ TRNA to examine whether financial institutions can predict the tone of firm-specific news. Heston and Sinha (2014) study the return predictability of TRNAs news sentiment in the U.S. market, while Smales (2016) examines the relation between TRNA’s news sentiment and bank credit risk. Li et al. (2015) show that the coverage of TRNA is comprehensive and covers over 92% of Compustats earnings announcements (the most common source of this information for the U.S. market). Mitra and Mitra (2011), Hendershott et al. (2015), Heston and Sinha (2014), and Li et al. (2015) provide detailed description of TRNA.

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day t and use this measure as a proxy for investor disagreement. Dzielinski and Hasseltoft (2014) show that this news dispersion measure can successfully capture disagreement among investors about the future market prospect as it is positively associated with higher market volatility and trading activities.

INSERT FIGURE 2 HERE INSERT TABLE 1 HERE

The descriptive statistics in Table 1 show that news items tend to be more positive than negative, with an average difference between positive and negative probability of approxi- mately 20%. There is a distinct dip in the sentiment measure around period 1500, which corresponds to the second half of 2008 reflecting the Lehman bankruptcy and subsequent global financial crisis.

We estimate the model given by Equation (9) by means of maximum likelihood. We build up the analysis by first estimating a restricted model with no switching (β = 0) and no sentiment interaction (φ²_h = 0). We then build towards the full model by first introducing switching, and then introducing sentiment. At each step, we use the estimated coefficients of the previous step as starting values.

4 Results

In this section, we present the estimation results for the model developed in Section 2. We present the results for various parameterizations of the model and document the results for several robustness tests.

INSERT TABLE 2 HERE

In Table 2 we present the estimation results. We start with the estimation of a homogenous agent model, where we do not allow for heterogeneous beliefs and do not consider the impact of sentiment. The estimation results for the homogeneous agent model (1) in Table 4 indicate that x_t is mean-reverting, with a highly significant AR(1) coefficient 0.947.⁴ Model

4Indeed, the Augmented Dickey-Fuller unit root test rejects the null of a unit root inxtwith p = 0.000;

results not shown.

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(2) includes sentiment in the homogeneous agent model and reveals that the behavior of the homogeneous agent model in (1) is conditional on sentiment. Specifically, in this model agents expect mean reversion to be increasing in sentiment: φ²_F < 0. Hence, with high (low) sentiment mean reversion is stronger (weaker). The LLR-1 statistic indicates that the model fit increases marginally after introducing sentiment to the homogeneous agent model.

Model (3) introduces heterogeneity, without sentiment. Results show that there is significant heterogeneity in S&P500 returns: fundamentalists expect mean reversion (φ¹_F <1) whereas chartists expect the mispricing to increase (φ¹_C >1). The switching parameter ¯β is negative, meaning that agents switch away from the strategy that performed well in the previous five periods, i.e. they follow a negative feedback trading strategy. The LLR-1 statistic indicates that allowing for heterogeneity significantly increases the fit of the homogeneous agent model (1).

Model (4) introduces sentiment in the switching function. The effect of sentiment on switching behavior is positive ψ > 0, implying that sentiment mitigates the switching behavior, and thus that agents switch less in reaction to positive sentiment. The effect, though, is not significant judging by the t-statistic of ψ as well as the LLR-3 statistic of model (4).

Model (5) introduces sentiment in the fundamentalist and chartist expectation formation functions. Sentiment has a direct effect on both the chartist and the fundamentalist expectation formation. Regarding chartists, we find that φ²_C > 0, implying that chartists extrapolate (even) stronger in times of high sentiment. In other words, they become ’more chartist’ in reaction to sentiment. Fundamentalists, on the other hand, expect greater mean reversion in times of high sentiment (φ²_F < 0). These findings are in line with agents be- coming more optimistic about their specific trading strategy in times of high sentiment, and indicate that sentiment is positively related to heterogeneity. The switching coefficient, ¯β, remains negative and significant. The LLR-3 statistic for model (5) shows that sentiment in the expectation formation functions adds to the explanatory power of the model. Fi- nally, model (6) is the full model with heterogeneity, switching, and sentiment in both the switching and the expectation formation functions. The findings of model (5) remain. Fur- thermore, sentiment has a negative effect on the switching tendency; ψ < 0. This implies that investors tend to stick to their investment strategy more (less) in terms of high (low)

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sentiment. This makes intuitive sense as sentiment is related to overconfidence. Different from model (4), now we do observe that introducing sentiment in the switching function adds to the explanatory power of the model; the LLR-5 statistic for model (6) is positive and significant. This suggests that sentiment has a significant moderating role on switching behavior, which is in line with the notion that agents become overconfident in times of high sentiment and therefore rely more on their private beliefs rather than public signals.

To demonstrate the impact of sentiment on switching behavior over time, Figure 3 presents the evolution of β_t over time, as well as the varying average AR(1) coefficient from Equation (9), given by n_{F t}∗(φ¹_F +φ²_F ∗S_t) +n_Ct ∗(φ¹_C +φ²_C ∗S_t). The time-varying intensity of choice β_t is negative, on average, indicating that investors tend to switch away from the better performing strategy. This could be driven by expectations of mean reversion in strategy performance. Otherwise, the movements of β_t follow those of sentiment as seen in Figure 2. The average AR(1) coefficient n_{F t}∗φ_F +n_Ct∗φ_C is below unity, on average, implying a globally stable system. Hence, the market price reverts to its fundamental value in the long-run; i.e.,x_tgoes to zero. There are, however, periods in which chartists dominate, pushing the AR(1) coefficient above unity, causing a locally unstable market environment.

We next assess the robustness of our results by separately examining the positive and negative tone components of our sentiment. To compute marketwide positive (negative) sentiment, we repeat the procedure used to calculate overall sentiment but replace the negative (positive) score in Equation (16) with zero. Table 3 reports the estimation results. The effect of negative sentiment appears to be stronger than the effect of positive sentiment. This result is consistent with the media literature (Tetlock, 2007; Tetlock et al., 2008; Garcia, 2013) that media pessimism has a stronger influence on stock prices than media optimism.

It is also in line with the notion of loss aversion as put forth by Kahneman and Tversky (1979).

INSERT TABLE 3 HERE

Figure 4 presents the sensitivity of the results to the lookback period in the switching

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function,I. Specifically, the figure presents the estimated coefficients as well as the likelihood values for the model with I varying from 1 to 100.

Results are largely robust to the choice of lookback period in the switching function in the sense that the signs of the coefficients do not change, and the likelihood ratio remains highly significant. There are, however, clear patterns when moving towards higher values of I. For the fundamentalist coefficients φ¹_F and φ²_F in the first row of Figure 4 we observe that the former increases in magnitude whereas the latter decreases in magnitude. For the chartist coefficients φ¹_C and φ²_C in the second row of Figure 4 we observe a decrease of both.

In other words, with higher values of I the effect of sentiment on expectation formation decreases, and the fundamentalist and chartist expectation formation rules become more similar. Hence, higher values of I imply less heterogeneity. This might be explained by the fact that higher values of I also means more similar performance because performance is taken over a longer period.

The third row displays the evolution of β and ψ for values of I. The former does not show a particular trend across the range although it remains negative throughout. The latter shows in increasing trend meaning that the impact of sentiment on switching increases with I. The fourth row shows the likelihood for different values of I. Interestingly, the line is virtually flat with a spike for I <15. Hence, the model performs best with a relatively low value ofI.

Figure 5 presents the sensitivity of the results to the exact configuration of the fundamental value proxy. Specifically, we increase the length of the moving average window from 10 to 200 days.

Results are largely robust to the choice of fundamental value as all coefficients remain of the same sign and significant across the range. Apart from the drop around 60, the fundamentalist coefficients φ¹_F and φ²_F in the first row of Figure 5 increase and decrease, respectively. The increase in φ¹_F is explained by the fact that the persistence inx increases

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as the moving average window increases. With a longer window, the effect of sentiment decreases, explaining the decreasing impact of sentiment. We observe the same patterns for the chartist coefficients in row two. Both the direct effect and the effect of sentiment decreases.

The switching parameters β and ψ become higher (in absolute value) as the moving average windows increases. Hence, switching becomes more intense with a smoother fundamental value estimate. The likelihood, finally, decreases with increases moving average window. This cannot be interpreted directly because the models are not nested, and the dependent variable varies over the different treatments.

5 Conclusion

This paper integrates the literature in investor sentiment with that on heterogeneous expectations. We adapt a relatively standard heterogeneous agent model by making both the expectation formation rules and the switching rule conditional on market sentiment. Esti- mation results on the S&P500 reveal that sentiment has an important confounding effect on the expectation formation process of fundamentalists and chartists in an agent-based setting. Specifically, fundamentalists’ expectation of the speed of mean reversion is increasing in sentiment. Chartists’ degree of trend extrapolation is increasing in sentiment. Hence, heterogeneity is increasing in sentiment. Furthermore, switching is decreasing in sentiment; agents become less sensitive to profit differences as sentiment increases. This set of results points towards the general conclusion that agents become more certain about their expectations as sentiment is higher.

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References

Baker, M. and J. Wurgler (2006). Investor sentiment and the cross-section of stock returns.

The Journal of Finance 61(4), 1645–1680.

Baker, M. and J. Wurgler (2007, June). Investor sentiment in the stock market. Journal of Economic Perspectives 21(2), 129–152.

Barberis, N., A. Shleifer, and R. Vishny (1998). A model of investor sentiment. Journal of Financial Economics 49(3), 307–343.

Bloomfield, R. and J. Hales (2002). Predicting the next step of a random walk: Experimental evidence of regime-shifting beliefs. Journal of Financial Economics 65(3), 397–414.

Boswijk, H. P., C. H. Hommes, and S. Manzan (2007). Behavioral heterogeneity in stock prices. Journal of Economic Dynamics and Control 31(6), 1938–1970.

Brock, W. A. and C. H. Hommes (1997). A rational route to randomness. Economet- rica 65(5), 1059–1095.

Brock, W. A. and C. H. Hommes (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics and Control 22(8-9), 1235–

1274.

Brown, G. W. and M. T. Cliff (2004). Investor sentiment and the near-term stock market.

Journal of Empirical Finance 11, 1–27.

Cavaglia, S. M. F. G., W. F. C. Verschoor, and C. C. P. Wolff (1994). On the Biased- ness of Forward Foreign Exchange Rates: Irrationality or Risk Premia? The Journal of Business 67(3), 321.

Chung, S.-L., C.-H. Hung, and C.-Y. Yeh (2012). When does investor sentiment predict stock returns? Journal of Empirical Finance 19, 217–240.

Cochrane, J. H. (2005). Asset Pricing. Princeton University Press.

(16)

De Long, J. B., A. Shleifer, L. H. Summers, and R. J. Waldmann (1990). Noise trader risk in financial markets. Journal of Political Economy 98(4).

Dzielinski, M. and H. Hasseltoft (2014). Why do investors disagree? The role of a dispersed news flow. Western Finance Association 2014 Meeting.

Fama, E. F. and K. R. French (2002). The equity premium. The Journal of Finance 57(2), 637–659.

Garcia, D. (2013). Sentiment during recessions. The Journal of Finance 68, 1267–1300.

Greenwood, R. and A. Shleifer (2014). Expectations of returns and expected returns. Review of Financial Studies 27(3), 714–746.

Hendershott, T., D. Livdan, and N. Schrhoff (2015). Are institutions informed about news?

Journal of Financial Economics 117(2), 249–287. cited By 35.

Heston, S. L. and N. R. Sinha (2014). News versus sentiment: Comparing textual processing approaches for predicting stock returns. Robert H. Smith School Research Paper.

Hommes, C., J. Sonnemans, J. Tuinstra, and H. van de Velden (2004, 08). Coordination of Expectations in Asset Pricing Experiments. The Review of Financial Studies 18(3), 955–980.

Hommes, C. H. and B. LeBaron (2018). Handbook of Computational Economics, Volume 4.

Elsevier.

Huisman, R., N. L. van der Sar, and R. C. J. Zwinkels (2012). A new measurement method of investor overconfidence. Economics Letters 114, 69–71.

Kahneman, D. and A. Tversky (1979). Prospect theory: An analysis of decision under risk.

Econometrica 47(2), 263–91.

Kukacka, J. and J. Barunik (2013). Behavioural breaks in the heterogeneous agent model:

The impact of herding, overconfidence and market sentiment. Physica A 392, 5920–5938.

(17)

Landier, A., Y. Ma, and D. Thesmar (2018). New experimental evidence on expectations formation. Working paper.

Li, E., K. Ramesh, M. Shen, and J. Wu (2015). Do analyst stock recommendations piggyback on recent corporate news? an analysis of regular-hour and after-hours revisions. Journal of Accounting Research 53(4), 821–861.

Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. The Review of Economics and Statistics 47(1), 13–37.

Lux, T. and R. C. Zwinkels (2018). Empirical validation of agent-based models, volume 4. In B. LeBaron and C.H. Hommes (Eds.), Handbook of Computational Economics, Heterogeneous Agent Models forthcoming.

MacDonald, R. (2000). Expectations formation and risk in three financial markets: Surveying what the surveys say. Journal of Economic Surveys 14(1), 69–100.

Mitra, G. and L. Mitra (Eds.) (2011). The Handbook of News Analytics in Finance. John Wiley & Sons Ltd.

Muth, J. (1961). Rational expectations and the theory of price movements. Economet- rica 29(3), 315–335.

Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk*. The Journal of Finance 19(3), 425–442.

Shiller, R. J. (1981). Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review 71(3), 421 – 435.

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

Smales, L. A. (2016). News sentiment and bank credit risk. Journal of Empirical Finance 38, 37–61.

Tetlock, P. (2007). Giving content to investor sentiment: The role of media in the stock market. The Journal of Finance LXII, 1139–69.

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Tetlock, P., M. Saar-Tsechansky, and S. Macskassy (2008). More than words: Quantifying language to measure firms’ fundamentals. The Journal of Finance LXIII, 1437–69.

Zeeman, E. (1974). On the unstable behaviour of stock exchanges. Journal of Mathematical Economics 1, 39–49.

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Tables and Figures

Figure 1: Price, Fundamental Price, and Deviation

6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8

-.4 -.3 -.2 -.1 .0 .1 .2

03 04 05 06 07 08 09 10 11 12 13 14 15 16

P* P X (right axis)

Notes: This figure presents the log fundamental value estimatep^∗based on Equation 15 using 100 days moving average, the log market pricep, and the deviation of the market price from fundamental,x=p−p^∗.

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Figure 2: Investor Sentiment

-.4 -.3 -.2 -.1 .0 .1 .2 .3

03 04 05 06 07 08 09 10 11 12 13 14 15 16

Notes: This figure presents the sentiment measureS, which is the equal weighted probability of positive news minus the probability of negative news. In the regressions, we take the de-meaned value ofS, and remove the outlier.

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Figure 3: Time-Varying Intensity of Choice

-8 -6 -4 -2 0 2 4

0.4 0.6 0.8 1.0 1.2 1.4

03 04 05 06 07 08 09 10 11 12 13 14 15 16

Beta_t (left scale) Phi_t (right scale)

Notes: This figure presents the time-varying intensity of choice βt = β +ψ∗ St together with the aggregate time-varying mean reversion parameterφt=nF,t(φ¹_F+φ²_FSt) +nC,t(φ¹_C+φ²_CSt).

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Figure 4: Sensitivity to Lookback Period in Switching Function

.50 .55 .60 .65 .70 .75

10 20 30 40 50 60 70 80 90 100

Phi_F^1

-6 -5 -4 -3 -2 -1 0

10 20 30 40 50 60 70 80 90 100

Phi_F^2

1.15 1.20 1.25 1.30 1.35 1.40

10 20 30 40 50 60 70 80 90 100

Phi_C^1

-1 0 1 2 3 4 5 6

10 20 30 40 50 60 70 80 90 100

Phi_C^2

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5

10 20 30 40 50 60 70 80 90 100

Beta

0 20 40 60 80 100 120

10 20 30 40 50 60 70 80 90 100

Psi

8,750 8,755 8,760 8,765 8,770 8,775 8,780

10 20 30 40 50 60 70 80 90 100

LL

Notes: This figure presents the sensitivity of the estimation results to the lookback period in the switching functionI in Equation (13). The x-axis representsI, ranging from 1 to 100 days. In the benchmark case, we setI= 5.

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Figure 5: Sensitivity to Fundamental Value Definition

.3 .4 .5 .6 .7

25 50 75 100 125 150 175

Phi_F^1

-6 -5 -4 -3 -2

25 50 75 100 125 150 175

Phi_F^2

1.20 1.25 1.30 1.35 1.40 1.45

25 50 75 100 125 150 175

Phi_C^1

1 2 3 4 5 6

25 50 75 100 125 150 175

Phi_C^2

-3.2 -2.8 -2.4 -2.0 -1.6 -1.2 -0.8 -0.4

25 50 75 100 125 150 175

Beta

10 20 30 40 50 60

25 50 75 100 125 150 175

Psi

7,000 7,500 8,000 8,500 9,000 9,500 10,000

25 50 75 100 125 150 175

LL

Notes: This figure presents the sensitivity of the estimation results to the length of the moving average in Equation (15). The x-axis represents moving average period, ranging from 10 to 200 days. In the benchmark case, we take 100 days.

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Table 1: Descriptive Statistics

Sentiment Disagreement

p p^∗ x Tone Pos Neg Tone Pos Neg Mean 7.193 7.182 0.012 0.149 0.327 0.178 0.346 0.247 0.178 Median 7.166 7.158 0.019 0.156 0.335 0.175 0.347 0.250 0.176 Max. 7.664 7.650 0.137 0.287 0.439 0.556 0.505 0.284 0.330 Min. 6.517 6.712 -0.389 -0.379 0.134 0.076 0.161 0.148 0.107 Std.Dev. 0.244 0.241 0.056 0.049 0.043 0.024 0.031 0.017 0.020 Obs. 3,334 3,455 3,456 3,334 3,334 3,334 3,334 3,334 3,334

Notes: This table presents the descriptive statistics of the market and sentiment data. prepresents the log-S&P500 index;p^∗ the fundamental value based on Equation (15);x=p−p^∗. Sentiment is the average probability of positive (Pos) or negative (Neg) news; Tone is the difference between the probability of positive news and negative news. Disagree- ment is the standard deviation of sentiment across firms in the sample.

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Table 2: Benchmark Estimation Results

1 2 3 4 5 6

φ¹_F 0.947*** 0.941*** 0.758*** 0.807*** 0.496*** 0.525***

(286.3) (205.5) (17.60) (12.55) (8.425) (12.95)

φ²_F -0.192*** -6.148*** -5.498***

(-2.985) (-7.471) (-10.56)

φ¹_C 1.156*** 1.107*** 1.423*** 1.376***

(25.50) (16.84) (25.50) (37.97)

φ²_C 6.203*** 5.202***

(8.134) (11.17)

β -0.676*** -0.703*** -1.153*** -1.454***

(-4.256) (-2.324) (-7.729) (-9.108)

ψ 3.578 26.88***

(0.781) (4.516)

c 0.001* 0.001*** 0.001** 0.001* 0.001*** 0.001***

(1.827) (2.519) (1.988) (1.889) (2.490) (2.947)

LL 8739.9 8741.3 8743.6 8743.3 8759.9 8772.0

LLR-1 2.824* 7.326***

LLR-3 -0.438 32.72***

LLR-5 24.11***

Notes: This table presents the estimation results of the model given by Equation (9). LL is the likelihood; LLR-1, LLR-3, and LLR-5 represent outcomes of the likelihood ratio test with model 1, 3, and 5 as benchmark, respectively. Robust t-values in parentheses; *, **, *** represents significance at the 10, 5, and 1% level, respectively.

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Table 3: Sentiment Measures

Sentiment Disagreement

Tone Neg Pos Tone Neg Pos

φ¹_F 0.525*** 0.792*** 0.393*** 0.792*** 0.801*** 0.790***

(12.95) (32.54) (7.309) (27.13) (31.98) (40.38) φ²_F -5.498*** 1.888*** -6.931*** -2.940*** 5.713*** -6.606***

(-10.56) (2.632) (-10.537) (-5.289) (4.994) (-8.285) φ¹_C 1.376*** 1.108*** 1.526*** 1.131*** 1.116*** 1.124***

(37.97) (41.21) (30.26) (39.48) (40.01) (63.12) φ²_C 5.209*** 0.451 6.770*** 3.502*** -3.956*** 6.774***

(11.17) (0.672) (10.91) (6.379) (-3.724) (9.567) β -1.454*** -1.073*** -1.290*** -2.108*** -1.424*** -2.320***

(-9.108) (-4.863) (-8.788) (-3.807) (-5.025) (-5.194) ψ 26.88*** -89.30*** 17.36*** 26.17*** -78.85*** 127.78***

(4.524) (-3.659) (3.737) (2.183) (-3.660) (5.090) c 0.001*** 0.001*** 0.001*** 0.001** 0.001** 0.001***

(2.951) (2.515) (2.294) (2.264) (2.031) (2.364)

LL 8772.0 8756.2 8770.9 8754.4 8748.7 8766.0

Notes: This table presents the estimation results of the model given by Equation (9) for different version of the sentiment measure S. LL is the likelihood; LLR-1, LLR-3, and LLR-5 represent outcomes of the likelihood ratio test with model 1, 3, and 5 as benchmark, respectively.

Robust t-values in parentheses; *, **, *** represents significance at the 10, 5, and 1% level, respectively.