Working Paper Series _______________________________________________________________________________________________________________________ National Centre of Competence in Research Financial Valuation and Risk Management Working Paper No. 831 Market Belief Risk and the Cross-Section of Stock Returns Rajna Gibson Brandon Songtao Wang First version: November 2012 Current version: November 2012 This research has been carried out within the NCCR FINRISK project on “Credit Risk and Non-Standard Sources of Risk in Finance” ___________________________________________________________________________________________________________
Transcript
Working Paper
Series _______________________________________________________________________________________________________________________
National Centre of Competence in Research
Financial Valuation and Risk Management
Working Paper No. 831
Market Belief Risk and the Cross-Section of Stock
Returns
Rajna Gibson Brandon Songtao Wang
First version: November 2012
Current version: November 2012
This research has been carried out within the NCCR FINRISK project on
“Credit Risk and Non-Standard Sources of Risk in Finance”
⇤We thank Yakov Amihud, Michael Tang, and Je↵rey Wurgler for their helpful comments and suggestions.
The financial support of the Swiss National Science Foundation and the NCCR-Finrisk Project C1 “Credit
Risk and Non-Standard Sources of Risk in Finance” is greatly acknowledged. All errors are ours.†Rajna Gibson Brandon is the Swiss Finance Institute (SFI) Chaired Professor of Finance at the Geneva
Finance Research Institute, University of Geneva, Geneva, Switzerland. Email: [email protected].‡Songtao Wang is currently visiting scholar at the New York University Stern School of Business. Email:
Standard asset pricing models such as the capital asset pricing model (CAPM, Sharpe 1964
and Lintner 1969) and the consumption-based CAPM (CCAPM, Ingersoll 1987, Huang and
Litzenberger 1986, Du�e 1996) were all developed based on the representative agent para-
digm. Due to their empirical tractability, the representative agent models have generated ex-
tensive empirical tests and subsequent theoretical extensions. But, as mentioned by Williams
(1977), “di�culties remain, significant among which is the restrictive assumption of homo-
geneous agents”. Recent studies show that the heterogeneity of investors plays an important
role in the formation of asset prices and their dynamics and that models incorporating hetero-
geneous investors can better account for empirical patterns in trading volume and in return
volatility.
Investors may have heterogeneous beliefs, information, or preferences1. This study focuses
on the heterogeneity in investors’ beliefs and investigates the e↵ect of heterogeneous beliefs
on the cross-section of stock returns. Heterogeneity in investors’ beliefs captures the fact
that individual investors may interpret commonly observed information di↵erently. Investors
often receive common information, but the ways in which they interpret this information
are di↵erent, and each investor only believes in the validity of his or her own interpretation.
Kandel and Pearson (1995) document that identical interpretation of information seems in-
consistent with the empirical data. Harrison and Kreps (1978), Varian (1985, 1989), De Long
et al. (1990), Harris and Raviv (1993), Detemple and Murthy (1994), Zapatero (1998), Basak
(2000), Scheinkman and Xiong (2003), Buraschi and Jiltsov (2006), Li (2007), Pavlova and
1Prior works on the heterogeneity in preferences mainly include Basak and Cuoco (1998), Benninga andMayshar (2000), Bhamra and Uppal (2009), Chan and Kogan (2002), Civitanic and Malamud (2009), Dumas(1989), Gollier and Zeckhauser (2005), Gomes and Michaelides (2008), Guvenen (2005), Isaenko (2008), Koganet al. (2007), Longsta↵ and Wang (2009), Wang (1996), Weinbaum (2001), Xiouros and Zapatero (2009).
2
Rigobon (2007), and Xiong and Yan (2010) all study the e↵ects of heterogeneous beliefs on a
variety of issues, including equity and bond risk premia, asset price volatility, interest rates,
exchange rates, the option-implied volatility, etc.
The basic idea of this paper stems from the theoretical works of Calvet et al. (2001), Jouini
and Napp (2007), and Kurz and Motolese (2011). One common contribution of those papers
is that they show, by developing models with investors di↵ering in beliefs, that in equilibrium,
the price of an asset is positively correlated with the aggregate belief of investors: investors
are willing to pay a higher price for the asset when the aggregate belief is optimistic2. This
result suggests that an asset will earn a higher contemporaneous return when investors hold
optimistic beliefs3. We rely on the market wide aggregate belief (or simply market belief)
defined as the cross-sectional average of the aggregate beliefs for all stocks. A key conjecture
made in this study is that stock returns change with innovations in market belief and that the
response to innovations in market belief varies across stocks: some stocks are more sensitive
to innovations in the market belief than others. Let market belief risk denote the sensitivity
of each stock’s excess returns to innovations in market belief, we will empirically examine
whether higher market belief risk yields higher expected returns, or in other words, whether
market belief risk is a priced factor.
Based on prior findings that analysts’ forecasts are good proxies for investors’ opinions4, we
rely on the actual EPS and analyst EPS forecast data to construct the market belief. First, we
adopt the econometric model developed by Brown and Roze↵ (1979) to forecast each stock’s
2In those paper, an investor is defined to be optimistic if his forecast is higher than the one made with aneconometric model.
3This paper is also related to the works of Abel (2002) and Cecchetti et al. (2000) who, di↵erent fromCalvet et al. (2001), Jouini and Napp (2007), and Kurz and Motolese (2011), examine the e↵ect of distortedbeliefs on asset returns by developing models with a representative investor whose belief, by definition, issimilar to the aggregate belief of investors in models with multiple investors.
4See Goetzmann and Massa (2005) and Anderson et al. (2005). The terms “financial analysts” (or simply“analysts”) and “investors” will be interchangeably used in the rest of this paper.
3
EPS. The aggregate belief of investors for a stock is then computed as the mean analyst EPS
forecast provided by the I/B/E/S minus the forecast made with the Brown and Roze↵ (1979)
model, and market belief is the cross-sectional average of the aggregate normalized beliefs
for all the sample stocks. Finally, we form portfolios based on the sensitivity of excess stock
returns to innovations in the market belief (market belief risk).
Our main empirical findings can be summarized as follows: the average return on stocks
with high market belief risk is significantly higher than that for stocks with low market belief
risk, this positive relationship being particularly strong for large-cap stocks. A strategy that
is long in stocks with high market belief risk and short stocks with low market belief risk
generates a significant alpha (4.608%/year in the Fama and French (1993) case and 5.928%/
year in the Cahart (1997) case), suggesting that the three- and four-factor models could not
explain this pattern in stock average returns. These results are robust to: a) an alternative
EPS forecasting model; b) an orthogonalisation of the market belief with respect to a set of
macro variables; c) a winsorization of stock returns at the 98% level; d) a di↵erent portfolio
holding period; and e) subsample analysis. Our market belief measure is quite di↵erent from
another widely used investor belief measure, namely the Baker and Wurgler (2006) sentiment
index. Indeed, the correlation between these two variables is rather small and negative and
more importantly, there does not seem to be any systematic relationship between average
stock returns and sentiment risk. Finally, we examine the determinants of a stock’s exposure
to market belief risk and find that the sensitivity of excess stock returns to market belief
innovations increases with their market beta, volatility, turnover rate, and their sale-to-asset
ratio and decreases with their size, momentum, and analyst coverage.
This paper contributes to the growing literature on the e↵ect of investor behavior on asset
returns by showing that innovations in market belief are a priced source of risk distinct from
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sources of systematic risk accounted for in standard asset pricing models. Another contribu-
tion of this paper is that it may o↵er yet another explanation for the equity premium puzzle
first pointed out by Mehra and Prescott (1985): indeed, part of the excess equity premium
may represent a compensation for investors’ exposure to market belief risk.
Diether et al. (2002) and Doukas et al. (2006) also examine the impact of heterogeneity in
investors’ beliefs on stock returns. What distinguishes our work is that we rely on the aggre-
gate belief (i.e. the first moment of the distribution of investors’ heterogeneous beliefs) while
those authors instead explore the impact of divergence in investors’ opinions (i.e. the second
moment of the opinion distribution). Specifically, Diether et al. (2002) document a negative
cross-sectional relation between divergence in analysts’ earnings forecasts and future stock
returns, supporting Miller’s (1977) view that divergence of opinion is priced at a premium in
the presence of short-sale constraints. By contrast, Doukas et al. (2006), using the diversity in
analysts’ forecasts measure of BKLS (1998), find a significantly positive relationship between
divergence of opinion and future stock returns. This result is consistent with the predictions
of models by Williams (1977), Mayshar (1983), and Epstein and Wang (1994) who posit that
divergence of opinion is a source of risk.
Baker and Wurgler (2006) study how investor sentiment a↵ects the cross-section of stock
returns. In their study as well, investor sentiment is a measure of the aggregate belief of in-
vestors. Baker and Wurgler (2006) demonstrate that the cross-section of future stock returns
is conditional on beginning-of-period investor sentiment. When sentiment is estimated to be
high, stocks that are attractive to optimists and speculators and at the same time unattrac-
tive to arbitrageurs - younger stocks, small stocks, unprofitable stocks, non-dividend paying
stocks, high volatility stocks, extreme growth stocks, and distressed stocks - tend to earn rel-
atively low subsequent returns. Conditional on low sentiment, however, these cross-sectional
5
patterns decrease and tend to disappear. As already said, their measure of investor sentiment
is broader and distinct from ours and does not constitute a priced source of risk. Stambaugh
et al. (2012) explore the role of investor sentiment as a potential explanation for a broad set
of anomalies in the cross-section of stock returns.
Lee et al. (1991) is closely related to our work in that they as well document that stocks
and closed-end stock funds with high sensitivity to investor sentiment earn an extra return as
compensation for this extra risk. One di↵erence between these two works is that the results
of Lee et al. (1991) are obtained with a very di↵erent investor belief measure - small investor
sentiment measured by the change in the discount on closed-end equity funds. Also, the pri-
mary target of Lee et al. (1991) is to solve the closed-end fund puzzle which is not addressed
in our study. It is worthwhile mentioning that the findings by Elton et al. (1998) however
do not support small investor sentiment as a priced factor as stocks with higher sensitivity
to this factor do not o↵er a higher expected return.
The remainder of the paper is organized as follows: Section II describes the dataset used
in this study. Section III shows how we use the actual EPS and analyst EPS forecast data to
construct the market belief measure. Section IV presents the empirical results on the e↵ect
of market belief risk on the cross-section of stock returns, including various robustness tests.
Section V investigates two further issues: first, the di↵erences between Baker and Wurgler
measure of investor sentiment and our measure of market belief and second the determinants
of stocks’ market belief betas. Section VI concludes the paper.
II Data
We use financial analysts’ forecasts as a proxy for investors’ beliefs given the fact that data
on investors’ direct beliefs are di�cult to collect. Previous studies have shown that financial
6
analysts are able to e↵ectively record the sentiment di↵used in financial markets, and that
their forecasts are good proxies for investors’ opinions.
The analyst forecast data is downloaded from the Institutional Brokers’ Estimate System
(I/B/E/S) U.S. Summary History database that contains summary statistics on analysts’
EPS forecasts. This database also contains the revision date when the forecast was last con-
firmed to be accurate. This data is usually disclosed on the third Tuesday of each month5.
The I/B/E/S database collects two main categories of analyst forecasts data: one concerns
EPS (Earnings Per Share) and another concerns DPS (Dividend Per Share). DPS is sensitive
to a firm’s dividend payout policy whose impact is di�cult to control for in empirical studies.
More importantly, the analyst DPS forecast data only has a short history and the coverage
of financial analysts for DPS forecasts is rather low. Due to these constraints, we will use the
analyst EPS forecast data in the following empirical analysis6.
In order to construct market beliefs about stock future earnings, we also need the actual
EPS data that we also download from the I/B/E/S database. The actual EPS data provided
by the I/B/E/S are called the ‘Street’ EPS since they are tracked by analysts and priced by
investors. COMPUSTAT provides another type of actual EPS known as the GAAP EPS re-
ported in firms’ financial statements. Bradshaw and Sloan (2002) record that there exists a
large and growing gap between the ‘Street’ EPS data and the GAAP EPS data as the former
excludes cost items such as ‘non-recurring’ and ‘no-cash’ charges7.
The ‘Street’ EPS data are quantitatively consistent with analysts’ EPS forecasts. To con-
struct market beliefs, we use the ‘Street’ EPS data rather than the GAAP EPS data although
5Diether et al. (2002) have a detailed description of the I/B/E/S database6If we assume that the payout ratios of firms are stable over time, the empirical results obtained with
either the EPS or the DPS forecasts should be similar.7The di↵erence between the ‘Street’ and GAAP earnings has been also discussed in Ciccone (2002), Cote
and Qi (2005), and Zhang and Zheng (2011)
7
the GAAP EPS dataset has a longer history. The actual EPS and analyst EPS forecast data
provided by the I/B/E/S have di↵erent periodicities: quarterly, semi-annually, annually, etc.
In this study, we use the quarterly EPS data due to the following reasons: first, the coverage
by financial analysts is relatively higher for the quarterly EPS forecasts (hence the forecast
reflects the opinions of broader financial analysts community); second, in the accounting lite-
rature, the econometric models developed to forecast earnings are mainly for quarterly EPS.
Stocks used to construct market beliefs are those with fiscal quarters ending in the months
of March, June, September, and December since the majority of stocks in financial markets
belong to this category. To be included in the construction of market beliefs, stocks also need
to meet two other criteria: 1) have more than 30 consecutive observations of quarterly EPS
over the period March 1983 through September 2009; 2) have the analyst EPS forecast as
well as the model-derived EPS forecast for at least one quarter over the period August 1990
through November 2009.
Stock data (like prices, returns, trading volume, the number of outstanding shares, etc) are
collected from the Center for Research in Securities Prices (CRSP) Monthly Stocks Combined
File, which includes NYSE, AMEX, and Nasdaq stocks. Only ordinary common shares (with
CRSP share code 10 or 11) are considered in this study. Also, to be considered in the portfolio
performance analysis below, stocks should have over 24 quarters of return observations during
the period between February 1991 and November 20098. The accounting data of firms, includ-
ing book values of equity, asset values, debt values, dividends, and sales, are drawn from the
COMPUSTAT-CRSP merged database.
8We need a longer time series of actual EPS data to forecast the EPS with the two econometric models,and the model based forecasts of the EPS, together with the analyst EPS forecasts, are then used to constructthe market beliefs.
8
III Methodology
In this section, we first show how to construct market beliefs with the actual and analyst EPS
forecast data, we further compute innovations in the market belief and finally form portfolios
based on the sensitivity of each stock excess returns to innovations in the market belief.
A EPS Forecasting Econometric Models
Forecasting EPS has been an important issue in the accounting literature, and many methods
have been developed to undertake such forecasts. As Callen et al. (1996) has shown, complex
methods such as neural network models are not necessarily superior to simple linear time
series models in that their forecasting errors are large.With this in mind, we chose to use two
simple but fairly accurate time series models to forecast quarterly EPS.
The first model was developed by Brown and Roze↵ (1979) (henceforth BR), and can be
formulated as:
E(Qs) = � +Qs�4 + �(Qs�1 �Qs�5) + ✓✏s�4 (1)
where Qs�k is the actual EPS for quarter s�k, ✏s�4 is the EPS shock experienced at quarter
s�4, and in general, � > 0 and ✓ < 0. The BR model contains an autoregressive component
Qs�1 � Qs�5 which reflects the positive autocorrelations in seasonal quarterly di↵erences at
the first three lags and a moving average component ✏s�4 which is responsible for the negative
correlation in seasonal di↵erences at the fourth lag9.
An alternative model to forecast quarterly EPS is the seasonal random walk with a drift
model (henceforth SRWD) that takes the following form:
E(Qs) = � +Qs�4 (2)
9From Eq. (1), we have ✏s�4 = Qs�4 �Qs�8 � �� �(Qs�5 �Qs�9)� ✓✏s�8. The last two terms are smallcompared to Qs�4 and Qs�8 and � is constant, this suggests that Qs�4�Qs�8 can be considered as a reason-able proxy for ✏s�4.
9
where E(Qs) is the earnings forecast for quarter s, � is a (typically positive) trend term, and
Qs�4 is the actual earnings for quarter s � 4. One main advantage of the SRWD model is
that it captures the seasonality characteristics in the quarterly earnings data documented for
example, by Lorek (1979). Sadka (2006) uses this approach to estimate unexpected earnings
shocks.
For each stock, the forecast of one-quarter ahead EPS during the sample period between
1983 and 2009 is derived with the estimated coe�cients from either a regression of Qs on Qs�4
or a regression of seasonal change in the actual EPS for quarter s, Qs�Qs�4, on Qs�1�Qs�5
and ✏s�4 (depending on which time series model is used), and each regression is estimated
using 30 quarters of actual EPS data.
We will use the BR model in the main empirical analysis and further provide some ro-
bustness tests relying on the SRWD EPS forecasting model.
B Market Belief
We denote by Ei,jt (EPSs) investor j’s forecast of the EPS of stock i for quarter s conditional
on the information available up to time t and by Ei,mt (EPSs) the forecast derived from an
econometric model, where t can be any time after the EPS for quarter s � 1 is known and
before the EPS for quarter s is publicly disclosed. Investor j’s belief gi,jt about the EPS of
stock i for quarter s is defined as the di↵erence between Ei,jt (EPSs) and Ei,m
t (EPSs):
gi,jt = Ei,jt (EPSs)� Ei,m
t (EPSs) (3)
A positive gi,jt indicates that investor j is optimistic relative to the econometrician about the
EPS for stock i during quarter s. The average of individual beliefs across investors, denoted
10
by Zit , equals:
Zit =
1
M
MX
j=1
gi,jt =1
M
MX
j=1
⇥Ei,j
t (EPSs)� Ei,mt (EPSs)
⇤= E
i
t(EPSs)� Ei,mt (EPSs) (4)
where M is the number of investors for stock i and Ei
t (EPSs) is the average investor forecast.
Even if provided with the same information, investors may still form di↵erent beliefs about fu-
ture stock earnings as they treat the information in di↵erent ways: some are more pessimistic
while others are more optimistic, and Zit captures the aggregate belief of the M investors: the
higher Zit , the more optimistic the investors. Previous studies showed that financial analysts
can e↵ectively record the sentiment di↵used in financial markets, thus analysts’ forecasts can
be used as proxies for investors’ opinions. We will thus use the average of analysts’ EPS fore-
casts provided by the I/B/E/S as a proxy for Ei
t(EPSs). Ei,mt (EPSs) will be estimated using
one of the two time-series models presented in Section III.A.
Generally, for stocks with fiscal quarters ending in March, June, September, and De-
cember, the actual EPS is respectively revealed in the second half of April, July, October,
and January. As mentioned above, analysts’ EPS forecasts are generally disclosed in the
middle (the third Tuesday) of each month. For a stock, as time moves towards the date
when next quarter’s EPS is disclosed, analysts’ forecasts will gradually contain more infor-
mation about the realized EPS so that market beliefs estimated with those forecasts are more
likely to reflect objective information rather than analysts’ subjective judgements. Due to
this reason, we only use in this study the analyst EPS forecast data released in February,
May, August, and November, that is, when analysts possess the least information about next
quarter’s EPS10. This procedure shall enable us to focus on studying the impact of investors’
subjective opinions on stock returns.
10This implies that we can construct market beliefs only for February, May, August, and November duringeach sample year.
11
In order for Zit to be comparable across stocks, we standardize it by first subtracting its
mean from each of its estimated values and then by dividing these de-meaned values by the
standard deviation of the variable. We define market belief as the cross-sectional average
of the standardized aggregate beliefs for all the sample stocks:
Zmt =
1
N
NX
j=1
Zi,stdt (5)
where N is the number of stocks11 and Zi,stdt is the standardized aggregate belief for stock i.
The variable Zmt measures the time t aggregate belief of investors about the EPS delivered
by a representative stock at quarter s. To a certain extent, the EPS of a representative stock
could be used as a proxy for the earnings generated by the real economy12, and thus Zmt also
reflects investors’ aggregate belief about the economic activity, a positive Zmt meaning that
investors are optimistic about the economy over quarter s. It is worthwhile mentioning that
Zmt only measures investors’ beliefs about the short term profitability.
INSERT FIGURE 1
The left set of graphs in Fig. 1 plot a time series of quarterly market beliefs estimated from
using the BR and the SRWD models for the period between August 1990 and November 2009.
These graphs show that market beliefs fluctuate dramatically over time and decline sharply
during economic recession periods such as the Asian financial crisis, the LTCM debacle in
1997-1998, the dot.com bubble burst at the beginning of this century and finally during the
2007-2009 subprime mortgage crisis. Particularly, the expectations of investors for the short-
term stock earnings reached the lowest level during the subprime mortgage crisis period.
Table I reports the summary statistics of the market belief variable. It shows that investors
11The number of stocks used to calculate market beliefs varies from 602 to 1629, with a increasing trendover time during the sample period due to the fact that more stocks have been covered by analysts.
12Intuitively, EPS and the real economy should be positively correlated, a healthy economy implying higherEPS.
12
were optimistic for over half of the sample period and that the distribution of market belief
is left-skewed, meaning that investors can be very pessimistic, as suggested by Fig. 1
INSERT TABLE I
Table I shows that market beliefs are highly autocorrelated and thus partially predictable.
We therefore estimate the unpredictable component of the market belief as the residuals of
the following autoregressive model of order two (i.e. innovations in market belief):
Zmt = c+ �1Z
mt�1 + �2Z
mt�2 + "Z,t (6)
where "Z,t is a white noise process with zero mean and variance �2Z . The estimated coe�cients
�1 and �2 are reported in Table I13. Let Bt denote innovations in market belief estimated in
Eq. (6)14.
INSERT TABLE II
Table II presents a correlation matrix between the following risk factors: the market factor
defined as the excess market returns; the size factor defined as the excess returns of small-cap
stocks over large-cap stocks; the value factor defined as the excess returns of value stocks over
growth stocks; the momentum factor defined as the excess returns of previous month winning
stocks over the losing stocks and market belief innovations. The latter and the market factor
are positively correlated with moderate coe�cients, suggesting that a positive belief shock
implies a higher contemporary stock market excess return. The correlations between market
belief innovations and the other risk factors are very low.
13The empirical results obtained with innovations in market beliefs estimated from auto-regressive modelsof other orders remain similar.
14Similarly, in the liquidity risk literature, researchers use innovations in aggregate liquidity measures tostudy how liquidity risk a↵ects the cross-section of stock returns.
where ri,t is the return of stock i, rf,t is the one-month riskless interest rate,MKTt is the excess
market return, SMBt is the excess return of small-cap stocks over big-cap stocks, HMLt is the
excess return of value stocks over growth stocks, and Bt is the belief factor.
16Unlike FM, we use individual stocks instead of sorting stocks into portfolios in the cross-sectional testof asset pricing. As shown in Ang et al. (2010), using portfolios does not necessarily lead to more preciseestimates of factor risk premia.
18
We conjecture that stocks’ expected returns are cross-sectionally related to the risk factor
Stocks with higher systematic risks should yield higher expected returns. Our main interest
is to examine if systematic market belief risk is priced. If systematic market belief risk is
important for stock pricing, �4 should be significantly positive.
Betas are estimated over rolling prior 24-quarter periods for each stock and then normal-
ized and used in the cross-sectional regression over the following three months17. Table VII
reports the average market belief risk premium and its t-statistic (As a robustness test, the
estimated market belief risk premia obtained with the CAPM model are also reported in this
table).
INSERT TABLE VII
Market belief risk yields a positive risk premium: indeed, the market belief beta coe�cient
is positive and statistically significant at the 5% level. For instance, in the BR and FF case,
�4 is 0.202% with a t-statistic of 2.34, meaning that a one standard deviation above the cross-
sectional mean of market belief betas is associated with an increase in stocks’ excess return
by 20.2 basis points per month. This is an economically meaningful e↵ect.
B Robustness Tests
The results obtained above are strongly supportive of the hypothesis that stocks with higher
market belief risk earn higher returns. It is however possible that these results are just driven
17Note that we only have quarterly data for estimated belief betas �B. The same betas are used for thethree months subsequent to the month when betas are estimated.
19
by alternative explanations or model mis-specifications. In order to address these concerns
we next perform a series of robustness tests.
B.1 Alternative Market Beliefs
The previously obtained results may be specific to the BR EPS forecasting model. In order
to address this concern, we conduct the empirical analysis with market beliefs estimated from
using an alternative method - the SRWD EPS forecasting model. Table VIII presents the re-
sults derived with this newly estimated market belief.
INSERT TABLE VIII
It is clear that the e↵ect of market belief risk in the cross-section of stock returns does not
disappear. Indeed, the performance di↵erence between the high- and low-belief-beta portfo-
lios even increases in the SRWD case. For the belief-beta-sorted portfolios, the performance
di↵erence is 0.831%/month, 0.381% higher than the one obtained in the BR case. Also, in
all but the third size quintiles, the performance di↵erence is at least 1%/month larger than
that obtained in the BR case. Similarly as in the BR case, the risk-adjusted excess returns
of portfolios positively depend on market belief risk. In the Cahart case, the di↵erence be-
tween the high and the low market belief beta portfolios amounts to 0.685%/month with a
t-statistic of 1.67. The alpha pattern in the FF case is similar (albeit less significant). Hence,
the positive relationship between stock returns and market belief risk is not specific to the
choice of the BR EPS forecasting model.
B.2 Controlling for the E↵ect of Macroeconomic Variables
We also wish to consider the case, where besides using stock specific information (like histo-
rical EPS data), investors also rely on information about macroeconomic conditions when
they forecast stocks’ EPS. If that is the case, the market belief we construct may partly reflect
20
information about macroeconomic conditions. Thus we next examine whether the positive
relationship between future stock returns and market belief risk is possibly due to the cross-
sectional di↵erence in the sensitivity of stock excess returns to fluctuations in the information
about macroeconomic conditions embedded in the market beliefs.
To control for this e↵ect, we remove macro-related variations from the market belief by
regressing Zt on the following macroeconomic variables: industrial production index, consu-
mer price index, employment rate, federal funds rate, and the NBER (National Bureau of
Economic Research) recession dummy variable18. For the first three macroeconomic variables,
we use year-over-year percentage changes.
INSERT TABLE IX
The correlation matrix between Zt and these macroeconomic variables is shown in Table
IX. Increases in industrial production, consumer price, employment, and federal funds rate
are accompanied by positive market beliefs, and investors have more pessimistic beliefs when
the real economy is in a recession. The left plot in Fig. 2 also shows that the NBER recession
periods are featured with a sharp decline in investors’ average belief about a representative
stock’s earning in the near future.
Using the residual ut estimated from the following regression as a proxy for the market
belief that is orthogonal to macro-related variations, we once again compute innovations in
market belief and market belief betas for each stock, and the performance results of the port-
folios formed on the orthogonalized market belief betas are displayed in Table X.
18Baker and Wurgler (2006) use similar macro variables, a di↵erence is that we also use the federal fundsrate - a factor that has been shown to strongly influence the economy.
21
where IPIt is the growth rate in industrial production index, CPIt is the growth rate in con-
sumer price index, EMPt is the growth rate in employment, FEDt is the federal funds rate,
and DUMt is the NBER recession dummy variable that equals 1 when the economy is in a
recession or 0 otherwise. The numbers in parentheses below Eq. (12) are t-statistic. The beta
coe�cients of the growth rate in consumer price index and of the NBER recession dummy
variable are statistically significant at the 1% level.
INSERT TABLE X
For the belief-beta-sorted portfolios, removing macro-related variations from the market
belief has a significant impact on the cross-section of stock returns since the average monthly
return di↵erence between the high- and low-belief-beta portfolios declines to 0.364%/month
which is insignificant at the conventional level. The e↵ect of macroeconomic variables con-
ditional on the size factor are less strong, particularly for large-cap stocks. In the three top
size quintiles, portfolios with high market belief risk deliver average returns which are about
0.47%-0.58%/month higher than those of portfolios with low market belief risk, and the dif-
ference is statistically significant at the 5% level. We can conclude from these results that the
higher returns earned by the high market belief beta portfolios cannot be fully explained as
mere premia for bearing macroeconomic risk.
B.3 Winsorizing Stock Returns at the 98% Level
It is well known that stock returns are not normally distributed and influenced by outliers. In
order to control for the e↵ect of infrequent yet extreme events on our empirical results, stock
returns are winsorized at the 98% level. Precisely, for a stock, we set its returns below the 1st
percentile to the 1st percentile and returns above the 99th percentile to the 99th percentile19.
19Winsorizing stock returns at di↵erent levels (95% and 99%) leads to similar results.
22
Winsorised estimators are usually more robust to outliers than their more standard forms.
INSERT TABLE XI
Table XI shows the results obtained with winsorized stock returns. The average monthly
returns earned by all the ten portfolios formed on market belief betas decrease in varying de-
grees. Again, as in the non-winsorized case, portfolios composed of stocks with higher market
belief risk deliver higher returns, and the average monthly return di↵erence between the high-
and low-market belief-beta portfolios is statistically significant at the 10% level although it
decreases by one third to 0.30%. Furthermore, as shown in Panel B, winsorizing stock returns
does not change the positive relation between future stock returns and market belief risk even
after the size e↵ect is taken into account, particularly in the top three size quintiles.
B.4 Holding Portfolios for Di↵erent Periods
At the beginning of each month of June and December during the period between December
1996 and December 2009, stocks are assigned either into 10 portfolios based on their market
belief betas or into 5⇥5 portfolios based on their size and market belief betas. Portfolios are
held for six months instead of three months, and we calculate the monthly portfolio return as
the equal-weighted average of the returns of all the stocks in the portfolios. The performance
results of portfolios held during six months periods are reported in Table XII.
INSERT TABLE XII
Holding portfolios for a longer time usually reduces the outperformance of the high-market
belief-beta portfolio, no matter whether or not the size e↵ect is accounted for. For instance,
the average return di↵erence between the high- and low-market belief-beta portfolios is reduc-
ed from 0.45%/month to 0.397%/month for the portfolios formed on market belief betas and
23
tends to be insignificant for middle sized stocks. A stock’s exposure to market belief risk is
supposed to evolve over time, stocks remaining in a portfolio for a longer time may less likely
meet the portfolio selection criteria. Despite the drop in the outperformance over longer hold-
ing periods, portfolio returns are still positively and significantly related with the exposures
of their constituent stocks to market belief risk, suggesting that the exposures of stocks to
market belief risk are rather persistent.
B.5 Subsample Analysis
We calculate average monthly portfolio returns for two subsample periods: the first one from
December 1996 to July 2003 and the second one from August 2003 to February 2010.
INSERT TABLE XIII
Looking at Table XIII, we can see that the e↵ect of market belief risk in the cross-section
of stock returns is stronger during the second subperiod while portfolios usually earn higher
average returns in the first subperiod. In the second subperiod, the average monthly return
di↵erence between the high- and low-market belief-beta portfolios is 0.603% with a t-value
of 1.99, 0.147% higher than the one obtained with the full sample data, and the di↵erence in
the first subperiod is much smaller (0.298%/month) and statistically insignificant. The per-
formance results of the portfolios formed via the two-way cuts on size and market belief beta
are similar: the high-market belief-beta portfolio performs significantly better than the low-
market belief-beta portfolio in the two largest size quintiles during the second subperiod.
These results are likely driven by the fact that the market belief was more volatile during
the second subperiod.
24
V Further Discussions
A Baker and Wurgler (2006) Sentiment Index
Baker andWurgler (2006, henceforth BW) construct a composite index of investors’ sentiment
that is based on the common variation in six underlying proxies for sentiment: the closed-end
stock fund discount (the average di↵erence between the net asset values (NAV) of closed-end
stock fund shares and their market prices); the NYSE share turnover (the ratio of reported
share volume to average shares listed from the NYSE Fact Book); the number of IPOs; the
average first day returns on IPOs; the equity share in new issues; and the dividend premium
(the log di↵erence of the average market-to-book ratios of payers and nonpayers). Precisely,
they start by estimating the first principal component of the six proxies and their lags. This
generates a first-stage index with 12 loadings, one for each of the current and lagged proxies.
Then, they calculate the correlation between the first-stage index and the current and lagged
values of each of the proxies. Finally, they define the sentiment index as the first principal
component of the correlation matrix of six variables – each respective proxy’s lead or lag,
whichever has the higher correlation with the first-stage index – rescaling the coe�cients so
that the index has unit variance20.
INSERT FIGURE 3
Fig. 3 plots the BW sentiment index along with the market belief estimated from the BR
EPS forecasting model21. The BW sentiment index is obviously much more volatile. These
two measures sometimes coincide. For instance, they simultaneously decline during economic
recession periods such as the dot.com bubble burst and the U.S. subprime mortgage crisis.
20BW also construct another sentiment index which is orthogonal to macro-related variations. They removemacro-related variations from their sentiment index by regressing raw sentiment measures on six macrovariables: the growth in industrial production, the growth in durable, nondurable, and services consumption,the growth in employment, and a dummy variable for the National Bureau of Economic Research recessions.
21The BW sentiment index can be downloaded from the website http://people.stern.nyu.edu/jwurgler/.
25
However, it is more frequent that the BW sentiment index and the market belief diverge. The
correlation between these two variables is small and negative: -0.106 (or -0.121 in case of the
BW sentiment index independent of macro-related variations). Although the BW sentiment
index and market belief are both designed to measure investors’ subjective opinions, they
do not capture the same pattern in their beliefs. First, the variables used to estimate these
two measures are di↵erent. We use the actual EPS and analyst EPS forecast data as well as
a forecasting model to construct market beliefs while the BW sentiment index is estimated
using a broad set of investor opinion-related variables that are purely market based. Second,
as suggested from the above description of the BW Index, the underlying estimation methods
are di↵erent. Given these di↵erences, it is not surprising that the BW sentiment index and
market belief are not strongly positively correlated.
INSERT TABLE XIV
The results obtained with the BW sentiment index are shown in Table XIV22. The relation
between future stock returns and sentiment risk is not monotonic any more: portfolios with
higher sentiment risk do not outperform those with lower sentiment risk, and the performance
di↵erence between the high- and low-sentiment-beta portfolios is negative and statistically
insignificant. This non-monotonic relation holds as well when the size e↵ect is controlled for.
These results imply that unlike market belief risk, sentiment risk is not priced in the cross-
section of stock excess returns, a claim already made by the authors in their original study
(Section IV.D).
22As above, we use innovations in the BW sentiment index in calculating stock sentiment beta.
26
B Understanding A Stock’s Exposure to Market Belief Risk
Finally, we are interested in determining how stock specific characteristics a↵ect their expo-
sure to market belief risk. For that purpose, we run panel data regressions with time fixed
e↵ect of individual market belief betas on the following lagged stock characteristics: the mar-
ket beta of stock returns estimated using the data over the period between 36 and 1 months
prior to t; the stock’s market capitalization in the month prior to t; the book-to-market ratio;
the accumulative return over the 11-month period between 12 and 2 months prior to t; the
stock return in the month prior to t; the annualized standard deviation of stock returns over
the 12-month period between 12 and 1 months prior to t; the average stock turnover rate
over the 12-month period between 12 and 1 months prior to t; the firm’s debt-to-book ratio;
the firm’s sale-to-asset ratio; the firm’s dividend-to-book ratio; the number of years between
the stock’s first appearance on the CRSP and t; the number of financial analysts covering the
stock in the month prior to t; the dispersion in beliefs of the financial analysts in the month
prior to t, scaled by the mean analyst forecast (the observations with zero mean forecast are
discarded). The accounting data from the fiscal year ending in year y�1 are matched to belief
betas from July of year y through June of year y + 1. These selected explanatory variables
reflect some important firm characteristics like their size, their maturity, their leverage, their
dividend policy, their growth opportunities, and most of them were also used in the studies
by Diether et al. (2002) and by Baker and Wurgler (2006). All the regressors are normalized
so that we can compare their powers in explaining the cross-sectional variations of stocks’
exposures to market belief risk.
INSERT TABLE XV
27
Table XV reports the panel regression results. The market belief beta and the market beta
are strongly positively correlated, this may be due to the fact that the excess market returns
and innovations in market belief are positively correlated (see Table II). The market belief
beta decreases with size and momentum and increases with volatility, meaning that smaller,
less performing and more volatile stocks face higher exposure to market belief risk. Similarly,
stocks with low analyst coverage have higher exposure to market belief risk. A high turnover
rate also increases a stock’s exposure to market belief risk: frequently traded stocks are not
surprisingly more sensitive to innovations in investors’ beliefs. The sale-to-asset ratio has a
strongly positive impact on the market belief beta making these large revenues generating
firms more sensitive to market belief risk regarding their future EPS. Finally, we observe that
a stock’s exposure to market belief risk is not related to the analyst forecast dispersion of
the stock, suggesting that the pattern in average stock returns documented in this study is
distinct from any pattern in stocks’ cross-sectional return di↵erences associated with their
analysts’ forecast dispersions.
VI Conclusion
This study shows that the average return on stocks with high sensitivities to market belief
innovations exceeds that of stocks with low sensitivities to market belief innovations by 5.4%
per annum, and this positive relationship is particularly strong among large-cap stocks. The
results are robust to: a) an alternative EPS forecasting model; b) an orthogonalisation of the
market belief with respect to a set of macro variables; c) a winsorization of stock returns at the
98% level; d) a di↵erent portfolio holding period; and e) subsample analysis. We also examine
the determinants of a stock’s exposure to market belief risk and find that the sensitivity of
excess stock returns to market belief risk increases with their market beta, volatility, turnover
28
rate, and their sale-to-asset ratio and decreases with their size, momentum, and analyst
coverage.
These findings jointly imply that market belief risk is priced and is an important behav-
ioral risk factor driving stock average and excess returns. Market belief risk is also distinct,
as we have seen, from another behavioral determinant of stock returns, namely investors’
market sentiment. Indeed, the results obtained with the BW sentiment index are not sup-
portive of sentiment risk being priced in the cross-section of stock returns. Although this
result is not surprising given the di↵erences between the BW sentiment index and our market
belief risk measure, it does raise an interesting question: why do these two opinion-related
variables have di↵erent e↵ects on stock prices? A potential explanation may be that market
belief risk reflects a systematic behavioral risk associated with investors forecasting stock
market fundamentals (meaning its capacity to generate earnings) whereas the BW index
rather proxies for investors’ general sentiment (perception) about stock market conditions.
A deeper insight into his question calls for further research on the importance of investors’
beliefs and of their embedded risk characteristics on asset pricing.
29
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Table
ISummary
StatisticsofQuarterlyM
ark
etBeliefs
Thistablereports
summarystatistics
ofqu
arterlymarketbeliefsestimated
from
usingtheBR(1979)
andSRW
Dmod
elsan
dof
innova-
tion
sin
marketbelief:minim
um,median,max
imum,mean,stan
darddeviation
,skew
ness,ku
rtosis,proportion
ofpositivemarketbelief
(PPMB),an
dau
tocorrelationcoe�
cients
ofthefirsttw
olags
(⇢1an
d⇢2).
Innovationsin
marketbeliefaretheestimated
residualsof
thefollow
ingau
toregressive
mod
elof
order
two:
Zm t=
c+�1Z
m t�1+�2Z
m t�2+" Z
t
whereZ
m tisthequ
artertmarketbelief.
Thesample
periodisAugu
st1990
through
Novem
ber
2009.
Min
Median
Max
Mean
Std
Skew
Kurt
PPMB
⇢1
⇢2
�1
�2
PanelA:TheBR
(1979)M
odel
MarketBeliefs
-0.472
0.0149
0.237
0.00035
0.125
-1.126
5.285
0.551
0.625
0.279
Innovationsin
MarketBelief
-0.265
0.0009
0.247
0.00032
0.095
-0.290
3.538
0.740
-0.184
PanelB:TheSRW
DM
odel
MarketBeliefs
-0.893
0.0479
0.264
0.00105
0.216
-2.191
8.764
0.679
0.767
0.431
Innovationsin
MarketBelief
-0.584
0.0223
0.428
0.00338
0.119
-1.178
11.040
1.062
-0.384
36
Table IICorrelation Matrix of Risk Factors
This table reports a correlation matrix of the following factors: market factor (MKT) definedas the excess market returns; size factor (SMB) defined as the excess returns of small-cap sto-cks over big-cap stocks; value factor (HML) defined as the excess returns of value stocks overgrowth stocks; momentum factor (UMD) defined as the excess returns of prior month winningstocks over losing stocks; innovations in market belief (BBR and BSRWD); and innovations inmarket belief, which are orthogonal to macro-related variations (B?
BR). The sample period isFebruary 1991 through November 2009, and the data frequency is quarterly.
Table IIISummary Statistics of Monthly Portfolio Returns
At the beginning of each month of March, June, September, and December during the periodDecember 1996 through December 2009, using prior 24 quarters of observations, we regressexcess stock returns on the excess market returns and innovations in market belief, and stocksare ranked into ten portfolios based on the sensitivities of their excess returns to innovationsin market belief (belief beta). Portfolios are held for three months, and portfolio returns areequal-weighted. This table reports summary statistics of monthly portfolio returns: minimum,median, maximum, mean, standard deviation, skewness, and kurtosis. The number in paren-thesis is t-statistic (Newey-West adjusted for autocorrelation).
Table IVMean Portfolio Returns by Size and Belief Beta
At the beginning of each month of March, June, September, and December during the periodbetween December 1996 and December 2009, stocks are ranked into five portfolios based ontheir market capitalizations at the end of previous month. In each size quintile, using prior24 quarters of observations, we regress excess stock returns on the excess market returns andinnovations in market belief, and stocks are then ranked into five further portfolios based onthe sensitivities of their excess returns to innovations in market belief (belief beta). Portfoliosare held for three months, and portfolio returns are equal-weighted. This table reports averagemonthly portfolio returns. The numbers in parentheses are t-statistic (Newey-West adjustedfor autocorrelation).
Table VMean Portfolio Returns by Book-to-Market Ratio and Belief Beta
At the beginning of each month of March, June, September, and December during the periodbetween December 1996 and December 2009, stocks are ranked into five portfolios based ontheir book-to-market ratios calculated as the book values of equity in the fiscal year endingin calender year t�1 for the month starting in July of year y divided by the market values ofequity at the end of previous month. Within each book-to-market ratio category, using prior24 quarters of observations, we regress excess stock returns on the excess market returns andinnovations in market belief, and stocks are then ranked into five further portfolios based onthe sensitivities of their excess returns to innovations in market belief (belief beta). Portfoliosare held for three months, and portfolio returns are equal-weighted. This table reports averagemonthly portfolio returns. The numbers in parentheses are t-statistic (Newey-West adjustedfor autocorrelation).
Belief beta is estimated with the Fama and French (1993, FF) or the CAPMmodel augmentedwith the belief factor B over rolling prior 24-quarter periods for each stock and then used inthe cross-sectional regression (normalized) in the following three months to estimate coe�-cients of belief beta. This table reports the time-series mean of coe�cients of belief beta (�4)with t-statistic in parentheses.
FF (1993) CAPM————– ———�4 (%) �4 (%)
The BR (1979) Model 0.202 0.178(2.34) (2.22)
The SRWD Model 0.297 0.309(2.51) (2.63)
42
Table VIIIMarket Beliefs Estimated with
the Seasonal Random Walk with a Drift Model
Panel A: At the beginning of each month of March, June, September, and December duringthe period December 1996 through December 2009, using prior 24 quarters of observations,we regress excess stock returns on the excess market returns and innovations in market belief,and stocks are ranked into ten portfolios based on the sensitivities of their excess returns toinnovations in market belief (belief beta).
Panel B: At the beginning of each month of March, June, September, and December duringthe period between December 1996 and December 2009, stocks are ranked into five portfoliosbased on their market capitalizations at the end of previous month. In each size quintile, usingprior 24 quarters of observations, we regress excess stock returns on the excess market returnsand innovations in market belief, and stocks are then ranked into five further portfolios basedon the sensitivities of their excess returns to innovations in market belief (belief beta).
Market beliefs are estimated with the SRWD model. Portfolios are held for three months, andportfolio returns are equal-weighted. This table reports average monthly portfolio returns andrisk-adjusted portfolio excess returns. The numbers in parentheses are t-statistic (Newey-West
adjusted for autocorrelation).
Panel A: Sorting by Belief Beta
Belief Beta———————————————————————————————————Low 2 3 4 5 6 7 8 9 High H - L
Table IXCorrelation Matrix of Market Belief and Macro Variables
This table reports a correlation matrix of the factors: market belief (ZBR); the growth rate inindustrial production index (IPI); the growth rate in consumer price index (CPI); the growthrate in employment (EMP); the federal funds rate (FED); and the NBER recession dummyvariable (DUM) that equals 1 when the economy is in a recession or 0 otherwise. The sampleperiod is February 1991 through November 2009, and the data frequency is quarterly.
Table XControlling for the E↵ect of Macro Variables
Using the estimated ut in the following regression as a proxy for market belief, we re-calculateinnovations in market belief as in Eq. (6) for each stock.
where IPIt is the growth rate in industrial production index, CPIt is the growth rate in consu-mer price index, EMPt is the growth rate in employment, FEDt is the federal funds rate, andDUMt is the NBER recession dummy variable which equals 1 when the economy is in a rece-ssion or 0 otherwise. The numbers in parentheses below the equation are t-statistic.
Panel A: At the beginning of each month of March, June, September, and December duringthe period December 1996 through December 2009, using prior 24 quarters of observations,we regress excess stock returns on the excess market returns and innovations in market belief,and stocks are ranked into ten portfolios based on the sensitivities of their excess returns toinnovations in market belief (belief beta).
Panel B: At the beginning of each month of March, June, September, and December duringthe period between December 1996 and December 2009, stocks are ranked into five portfoliosbased on their market capitalizations at the end of previous month. In each size quintile, usingprior 24 quarters of observations, we regress excess stock returns on the excess market returnsand innovation in market belief, and stocks are then ranked into five further portfolios basedon the sensitivities of their excess returns to innovations in market belief (belief beta).
Portfolios are held for three months, and portfolio returns are equal-weighted. This table pre-sents average monthly portfolio returns. The numbers in parentheses are t-statistic (Newey-West adjusted for autocorrelation).
Table XIWinsorizing Stock Returns at the 98% Level
Panel A: At the beginning of each month of March, June, September, and December duringthe period December 1996 through December 2009, using prior 24 quarters of observations,we regress excess stock returns on the excess market returns and innovations in market belief,and stocks are ranked into ten portfolios based on the sensitivities of their excess returns toinnovations in market belief (belief beta).
Panel B: At the beginning of each month of March, June, September, and December duringthe period between December 1996 and December 2009, stocks are ranked into five portfoliosbased on their market capitalizations at the end of previous month. In each size quintile, usingprior 24 quarters of observations, we regress excess stock returns on the excess market returnsand innovations in market belief, and stocks are then ranked into five further portfolios basedon the sensitivities of their excess returns to innovations in market belief (belief beta).
To control for the e↵ect of outliers, stock returns are winsorized at the 98% level. Portfolios areheld for three months, and portfolio returns are equal-weighted. This table presents averagemonthly portfolio returns. The numbers in parentheses are t-statistic (Newey-West adjustedfor autocorrelation).
Panel A: At the beginning of each month of June and December during the period betweenDecember 1996 and December 2009, using prior 24 quarters of observations, we regress excessstock returns on the excess market returns and innovations in market belief, and stocks areranked into ten portfolios based on the sensitivities of their excess returns to innovations inmarket belief (belief beta).
Panel B: At the beginning of each month of June and December during the period betweenDecember 1996 and December 2009, stocks are ranked into five portfolios based on their mar-ket capitalizations at the end of previous month. In each size quintile, using prior 24 quartersof observations, we regress excess stock returns on the excess market returns and innovationsin market belief, and stocks are then ranked into five further portfolios based on the sensiti-vities of their excess returns to innovations in market belief (belief beta).
Portfolios are held for six months, and portfolio returns are equal-weighted. This table reportsaverage monthly portfolio returns. The numbers in parentheses are t-statistic (Newey-Westadjusted for autocorrelation).
Figure 1: The left top graph plots a series of quarterly market beliefs estimated from usingthe Brown and Roze↵ (1979) model, and the right top graph plots innovations in market be-lief, which are the estimated residuals of an autoregressive model of order two for quarterlymarket beliefs. The graphs in the bottom panel plot quarterly market beliefs and innovationsin market belief, both estimated from using the seasonal random walk with a drift model.
53
1990 1995 2000 2005 2010
-0.5
0.0
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Market Beliefs and NBER Recession Indicator
Market BeliefsNBER Recession Indicator
1990 1995 2000 2005 2010
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0.1
0.2
Innovations in Market Belief
BRIndependent of Macro-Related Variations
Figure 2: The solid line in the left graph plots a series of quarterly market beliefs estimatedfromusing the Brown and Roze↵ (1979) model and the dashed line plots the evolution of NBER
recession indicator that equals 1 when the real economy is in a recession or 0 otherwise. In theright graph, the solid line plots innovations in market belief, which are the estimated residualsof an autoregressive model of order two for quarterly market beliefs, and the dashed line plotsinnovations in market belief, which are independent of the variations of the following macrovariables: industrial production index, consumer price index, employment, federal funds rate,and US recession indicator.
54
1990 1995 2000 2005 2010
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Market Beliefs and Sentiment Index
BRBW1BW2
Figure 3: The solid line plots a series of quarterly market beliefs estimated from using theBrown and Roze↵ (1979) model, the dashed line plots the Baker andWurgler (2006) sentimentindex, and the dotted line plots the Baker and Wurgler (2006) sentiment index orthogonal tomacro-related variations.
