1

 We thank George Evans, Ken Judd, Volker Wieland and participants in the SITE summer 2009 meeting at Stanford

University for helpful comments on an earlier draft. Howei Wu provided valuable technical assistance at various stage of the

research and her joint work with Motolese is interrelated with the results reported in Section 4. 

2 Department of Economics, Serra Street at Galvez, Stanford University, Stanford, CA. 94305-6072, USA

3 Istituto di Politica Economica, Università Cattolica di Milano, Via Necchi 5, 20123, Milano, Ital

Correspondence to: Mordecai Kurz, Email: mordecai@stanford.edu

1

The Role of Diverse Beliefs in Asset Pricing and 

 Equity Premia1

by Mordecai Kurz2

 and Maurizio Motolese3

A very preliminary draft: February 2, 2010

Summary:

Why do risk premia vary over time? We examine this problem theoretically and empirically by studying the effect of market belief

on risk premia. Individual belief is taken as a fundamental, primitive, state variable. Market belief which is the distribution of

individual beliefs is observable, it is central to the empirical evaluation and we show how to measure it. Our asset pricing model is

familiar from the noisy REE literature but we adapt it to an economy with diverse beliefs. We derive equilibrium asset prices and

implied risk premium. Our approach permits a closed form solution of prices hence we trace the exact effect of market belief on the

time variability of asset prices and risk premia. We test empirically the theoretical conclusions.

We study premia on long positions of S&P500 for investment periods of 6 -12 months and find that, on average, a change

in the mean market belief by 1 standard deviation changes equity premia by about 0.4 standard deviation of excess returns. We also

find that the introduction of belief variables reduces the effect of the traditional Fama and French (1989) variables used to forecast

excess returns. More specifically we find that given information about market belief the term premium, the default premium and

Lattau and Ludvigson’s (2001) CAY variables have no significant effect. As to the structure of the premium we show that when the

market holds abnormally favorable belief about the future payoff on the S&P500 the market views the long position as less risky

hence the equity risk premium declines. We also find that in a VAR model of asset and consumption growth, our market belief

variables have a significant effect. We thus conclude that market belief has significant independent effect on economic fluctuations

and risk premia. 

JEL classification: C53, D8, D84, E27, E4, G12, G14.

Keywords: Risk premium; heterogenous beliefs; market state of belief; asset pricing; Bayesian rationality;

Rational Beliefs.

0. Introduction

A large and growing literature in empirical finance has demonstrated that the predictability of

asset returns is increased by using observed measures of market belief or sentiment. A sample of work

using various measures includes Miller (1977), DeLong et al. (1990), Lee et al. (2002), Diether et al.

(2002), Johnson (2004), Park (2005), Baker and Wurgler (2006), Fan (2006), Karakatsani and Salmon

(2008), Campbell and Diebold (2009) and Kurz and Motolese (2011). Also, models with diverse

beliefs have become the basis for recent work in Behavioral Finance. All this work is in contrast with

the rational expectations view relating changes in risk premia only to changes in information about
2

exogenous fundamentals which correctly forecast changes in risky events, the most important of which

are business cycles. Volatility of asset prices is surely affected by real measures that forecast future

business conditions, but the work cited holds that market volatility is driven also by how reality is

perceived by investors. Leaving Fan (2006) and Kurz and Motolese (2011) aside for the moment, we

stress that the finance literature cited has searched primarily for measures that can be used to forecast

returns. From a conceptual perspective the literature takes the view that diverse beliefs reflect some

irrational behavior which can be exploited by smart investors who use public information about

irrational agents for better forecasting of future returns. The standard model inspiring this perspective

has been the Noise Trading model of DeLong et al. (1990).

An alternative view holds that all agents are rational and equally smart but hold diverse beliefs

due to their incomplete knowledge about the complex economy. This diversity is central since much of

asset returns’ volatility is caused by fluctuations in market belief which is an economy wide externality

with a predictable component. The theoretical work on this approach developed over the last two

decades. A sample of papers includes Harrison and Kreps (1978), Varian (1985), (1989), Harris and

Raviv (1993), Detemple and Murthy (1994), Kandel and Pearson (1995), Kurz (1974), (1994),

(1997), (2007), (2008) Kurz and Beltratti (1997), Kurz and Motolese (2001) (2011), Kurz and

Schneider (1996), Kurz et al (2005a) (2005b), Kurz and Wu (1996), Motolese (2001), (2003), Fan

(2006), Nakata (2007), Nielsen (1996), (2003) and Wu and Guo (2003), (2004). In particular, Kurz

and Beltratti (1997), Kurz and Motolese (2001) and Kurz et al. (2005a) demonstrate that models with

diverse beliefs can explain the equity premium. Most of these papers study market volatility via

simulations. In contrast, Kurz and Motolese (2011) examine determinants of risk premia by studying

analytical hypotheses which can then be tested directly with data on market belief.

The third line of research by Behavioral Finance, starts from the view that investors in financial

markets are not fully rational. Although this is a very large literature, one can generalize by observing

that the typical structure of a Behavioral Finance model assumes one population segment is fully

knowledgeable and rational while a second segment exhibits some form of bounded rationality (e.g.

Lee et al. (1991), Barberis et al. (1998), Daniel et al. (1998), Lee et al. (2002), Sheinkman and Xiong

( 2005)). But this is the structure of Noise Trading models. In such a world there are smart traders

who know the truth and not-so-smart agents who trade aimlessly. Equilibrium reflects both the truth
3

and the noise. Hence, a Behavioral Finance perspective is compatible with the empirical finance

literature cited above. From a perspective of modeling asset pricing the Noise Trading paradigm of

DeLong et al. (1990) can then be taken to be the underlying formal structure of this literature.

One may observe that we are yet to find market data on noise trading or discover agents who

provide data declaring themselves to be irrational noise traders. Hence, noise trading theory or

irrationality in markets cannot be falsified with direct data measuring actions taken by individuals in

the market who inform us about their irrationality. Standard empirical evidence such as orthogonality

tests which are common in the Behavioral Finance literature reject rational expectations but do not

prove irrational behavior. Rational Expectations is an extreme hypothesis which demands from agents

to know what they cannot know. We thus reject both rational expectations and irrational noise trading

as extreme hypotheses. We outline a middle ground theoretical framework for asset pricing under

diverse beliefs which are rationalized and derive the asset pricing theory developed under these

rationality restrictions. We then test the implications with data on market belief.

Briefly stated, our theoretical perspective proposes that all traders are smart but do not know

the true stochastic dynamics of the economy. The true dynamics is non-stationary in the sense that it

has a time varying structure reflected in changing products and technology, but the structure changes

faster than can be learned with precision from data. Since investors know only long term empirical

distributions deduced from data, equilibrium asset prices are determined only by what traders believe

and by what they know. The true dynamics is irrelevant to asset pricing except for the way it reveals

itself in the data agents know. In rejecting the false dichotomy of a noise trading paradigm between

smart and not-so-smart traders, we observe that lack of full knowledge of the truth is the foundation of

belief diversity but rational diversity is not noise; it has a structure and can be studied scientifically. 

The issue of rationality deserves a comment. It is obvious “rationality” of beliefs is different

from rationality of actions. The Rational Expectations hypothesis was skillfully marketed by attaching

the term “rational” to impossible requirements. This lead to a common view that one can either hold

Rational Expectations or be irrational. Indeed, virtually all “evidence” for irrationality presumes only

the false choice between these two norms. Rationalizing a belief is a complex problem and a theory

which permits diverse but rationalizable beliefs cannot be based on too tight a rationality principle that

would prevent diversity from arising in the first place. Belief diversity results from lack of some
4

essential knowledge. Hence, to rationalize a belief one starts from that ignorance of the truth and ends

by adopting principles of what is “reasonable” to believe rather than what is inevitable. In short, with

incomplete knowledge reasonable men can reach different conclusions using the same information!

An important dimension of the theory holds that agents’ beliefs are primary causes of economic

events and have their own independent dynamics. This view dictates some of the formal developments

below. It does not negate the role of real fundamental factors in asset pricing but insists market belief

is an independent force which affects market dynamics, economic performance and, ultimately, future

economic reality. This idea, which is central to our work has gradually had an impact on the study of

Economic History and Political Economy (e.g. see Mokyr (2010), Greif (2006) and Tabellini ( 2008)). 

In this paper we study the role of market belief in asset prices and risk premia. Beliefs are

diverse but rational in a sense to be discussed. An individual state of belief is an index which pins down

an agent’s conditional probability at a given date while a market state of belief is a distribution of

individual beliefs. Although individual beliefs are not observed, we show how to extract market belief

from forecast samples. In the analysis we focus on the first two moments of these distributions and

establish the relation between risk premia and market belief. We derive analytical results which are

then tested empirically with data on market belief. 

Observations on market belief are extracted from three sources. First, semi-annual data of the

Livingston Survey made available by the Philadelphia Federal Reserve. Second, monthly forecasts of

future interest rates and macro economic variables compiled by the Blue Chip Financial Forecasts

(BLUF) since 1983:1 and by the Blue Chip Economic Indicators (BLU) since 1980:1. Third, quarterly

forecasts of macro economic variables by the Survey of Professional Forecasters (SPF) conducted

since 1968:11 and made available by the Philadelphia Federal Reserve. Our analysis was conducted

using either the Livingston files or the combined Blue Chip and SPF files as explained later.

The present paper is a companion to Kurz and Motolese (2011) where we focus on risk premia

in debt markets: U.S. Treasuries and Federal Fund futures. This paper addresses the problem of the

stock market hence our results are more directly comparable to the empirical finance literature cited

earlier. Our results confirm that some traditional variables reported in Fama and French (1989) predict

excess returns, particularly the dividend yield. However, we also find that for the investment periods of

6-12 studied here the Default Premium, the Term Spread and the variable CAY developed by Lattau
5

and Ludvigson (2001) have no effect once belief variables are introduced . We develop a measure of

mean market belief and cross sectional standard deviation of beliefs and find that the mean market

belief contributes significantly to explaining the equity risk premium. In a simple VAR model of wealth

and consumption growth we also find that market belief has an independent effect on consumption

growth. In contrast with our study of the bond market where we show the cross sectional standard

deviation has a significant effect on risk premia, we cannot confirm this effect in the stock market. 

Apart from showing market belief has an independent effect on fluctuations and risk premia we

also show the effect on the risk premium takes a specific form which is predicted by the theory. We

thus prove first a theorem which asserts that when the market holds abnormally favorable belief about

future profits of an asset, the risk premium on long positions of that asset falls. More generally, market

optimism about future economic conditions lowers the risk premium while pessimism about future

economic conditions increases it. This is a strong result about market overshooting. We test this

conclusion empirically and find the data are compatible with the theoretical findings.

1. Asset Pricing Under Heterogenous Beliefs 

1.1 An Illustrative Decision Model

Consider an asset or a portfolio of assets whose market price is , paying an exogenous risky pt

sequence { } under a true and unknown probability which is non-stationary due to Dt

,t'1,2,... Πˆ

structural changes over time. Let be the riskless interest rate, and hence excess return rt Rt'1%rt

over the riskless rate is . The (1/p risk premium over the riskless rate is the t

)(pt%1%Dt%1&Rtpt)

conditional expectations of excess returns. Since it is a function of equilibrium prices, a risk premium -

as a function of state variables - is best deduced from equilibrium prices. With this in mind, the model

below is used to deduce a closed form solution of the asset price so as to enable a study of the factors

determining the risk premium. To obtain closed form solutions we use a model which is very common

in the literature on Noisy Rational Expectations Equilibrium (e.g. Brown and Jennings (1989), Grundy

and McNichols (1989), Wang (1994), He and Wang (1995), Allen, Morris and Shin (2006) and others

cited in Brunnermeier (2001)). Nevertheless, our key results are fully general and do not depend upon

the model used. We now address two issues. Our agents do not know the true probability and this Πˆ

raises two questions that are at the basis of our later development. First, why do agents not know ? Πˆ
4

 We always have finite data and cannot estimate with certainty the measure on sequences. However, if this mˆ

measure has a simple representation such as a Markov transition function, then with adequate data it can be approximated so

closely as to make the assumption in the text entirely reasonable. Estimation of with an epsilon error only increases belief mˆ

divergence as it reduces the scope of what is common knowledge and complicates the theory without adding much empirical

substance.

5 It would be more realistic to assume the values Dt

 grow and the growth rate of the values has a mean μ rather than

the values themselves. This added realism is useful when we motivate the empirical model later but is not essential for the

analytic development.

6

Second, what is the common knowledge basis of all agents in an economy with diverse beliefs? 

Starting with the second question, our answer is past data on observables. The economy has

observable variables and is one of them. Agents have a long history of the variables, allowing rich Dt

statistical analysis which leads all to compute the same relative frequencies of events or moments and

finite dimensional distributions of observed variables. Using standard extension of measures methods

they deduce from the data a unique empirical probability measure on infinite sequences denoted by . mˆ

It can be shown that is stationary (see Kurz (1994)) and we call it “the stationary measure.” This is mˆ

the empirical knowledge shared by all agents4

. We assume in this paper the data reveals that under the

measure , is a Markov process where is distributed conditionally normal with mˆ Dt

, t'1,2,... Dt%1

means and variance . It follows that is the unconditional mean5 μ %λ of . The unique d (Dt

&μ) σ

2

d μ Dt

probability is then known to all. To simplify define , hence the process { } is m d ˆ t'Dt&μ dt,t'1,2,...

zero mean Markov process with unknown true probability Π and known stationary empirical

probability m. We assume specifically that the dynamics of payoffs d deduced from the empirical t

frequencies is characterized by a first order Markov process with transition

(1) dt%1 ' λd dt

% ρ

d

t%1 , ρ

d

t%1 - N(0 , σ

2

d )

and since this characterizes the probability measure m, we write . E We often use the m[dt%1 |dt] ' λd dt

term “empirical distribution” to refer to the probability m on infinite sequences deduced from data.

Why is m not equal to Π? With this issue in mind we turn to the first question. 

Our economy is undergoing changes in technology and social organization. These are rapid

with major economic effects, making { } a non-stationary process. Although this means dt

,t'1,2,...

that the distributions of the ‘s are time dependent, it is more than viewing { } as a dt dt

,t'1,2,...

sequence of productivity “regimes.” It also means that, although we measure the in a single unit of dt

account, over time the nature of assets and commodities change. Such variability makes it impossible
7

to learn the unknown Π. The probability m is then merely an average over an infinite number of

regimes. Belief diversity starts from agents’ disagreement over the meaning of public information.

They believe Π is different from m and construct models to express the implications they see in the

data. Being common knowledge, m is then a basic point of reference for any concept of rationality.

Turning now to our infinite horizon model, at date t agent i buys shares of stock and θ

i

t

receives the payment for each unit of held. We assume the riskless rate is constant over dt%μ θ

i

t&1

time so there is a technology by which an agent can invest the amount at date t and receive with B i

t

certainty the amount at date t+1. The definition of consumption is then standard B i

t R

c . i

t ' θ

i

t&1 [pt

% dt% μ] % B i

t&1R & θ

i

t pt & B it

Equivalently, define wealth and derive the familiar transition of wealth W i

t 'c

i

t %θ

i

t

pt%B i

t

(2a) . W i

t%1'(W it &c

i

t )R%θ

i

t

Qt%1 , Qt%1'pt%1%(dt%1%μ )&Rpt

Q are excess returns. Given some initial values the agent maximizes the expected utility t (θ

i

0 , W i0 )

(2b) U ' E i

t

(θi

, c i)

[j

4

s ' 0

&βt%s&1e

&( 1

τ

c i

t%s

)

| Ht]

subject to a vector of state variables and their transitions, to be specified later. consists of all ψ

i

t Ht

past observable variables. We recognize the limitations of the exponential utility and use it as a vehicle

to explain the main ideas, hence the term “illustrative” in this Section’s title. After deducing the closed

form solution of equilibrium risk premium we show how to generalize the key results.

We assumed (1) is the dynamics of d deduced from the data but explained that it may or may t

not be the true data generating process. We assume later that the true data generating process is

(3) dt%1 ' λd dt% bt% g

d

t%1 , g

d

t%1 - N(0 , 1

β

)

where is an infinite sequence of unobserved “regime” parameters. One may ask under what bt

conditions can (1) be the empirical distribution of (3)? The answer is that for this computation one

considers to be a random variable and then require that the empirical distribution of bt bt% g

d

t%1

equals the distribution of . Three examples will explain the required conditions. ρ

d

t%1 - N(0 , σ

2

d )

Example 1: the sequence of is a realization of i.i.d. draws from a random variable . In bt Y - N(0,σ

2

y )

this case the required condition for (1) to be the empirical distribution of (3) is . σ

2

y

%

1

β ' σ

2

d

Example 2: are realizations of a stochastic process . The variance of b bt bt'λbbt&1%yt , yt- N( 0,σ

2

y)

is hence the conditions are σ

2

y /(1&λ

2

b)

 (i) .

σ

2

y

1&λ

2

b

%

1

β ' σ

2

d , (ii) cov(gt

, gt&1 ) ' & λb

σ

2

y

1&λ

2

b
8

That is, if the sequence is i.i.d and the exhibit serial correlation then for (1) to be the empirical ρ

d

t bt

distribution of (3) the sequence must exhibit serial correlation. Similarly, if the exhibit serial gt bt

correlation but the are i.i.d. then the empirical distribution will reveal serial correlation of the . gt ρ

d

t

Example 3: are realizations of a stochastic process where are bt bt'λbtbt&1%yt , yt- N(0,σ

2

y) λbt

derived from a tent map on (-1, +1) hence are uniformly distributed in the interval. In this case the bt

are not serially correlated, we can show that and the needed condition is . Var(b) ' 3σ

2

y 3σ

2

y

%

1

β

' σ

2

d

One can modify the example and have the change only slowly at random dates. λbt

The examples exhibit conditions on which rationalize (3) with respect to (1). Indeed, bt

suppose the truth was unknown and (3) was adopted as a belief by an agent. Then under the

restrictions on in each of the examples, (3) would have been a Rational Belief (see Kurz (1994)). bt

Since the truth is unknown what should agents do? Those who believe the economy is

stationary accept (1) as the truth. Such belief should be rational since there is no empirical evidence

against it. But the complexity of { } lead most to hold the view that (1) is not adequate to dt, t'1,2,...

forecast the future. All surveys of forecasters show that subjective judgment about the data contributes

more than 50% to the final forecast (e.g. Batchelor and Dua (1991)). Hence, agents form their own

beliefs about state variables. The structure of belief is then our next topic.

1.2 Modeling Diverse Beliefs I: Individual Belief as a State Variable and Rationality

As postulated earlier, we assume forecast distributions are public observations over time. This

fact points to a difference between markets with and without private information. With asymmetric

private information a market is secretive: agents do not reveal their forecasts since these provide real

information and such revelation eliminates the small advantage they have. If an agent’s forecast of

state variables is revealed in our market - without private information - others do not consider it

information. They view it as an expression of his opinion and do not update their own beliefs about

state variables. On the other hand the forecast of others is crucial information used to forecast

endogenous variables. This follows from the fact that future equilibrium prices depend upon future

market belief hence to forecast future prices one must forecast future market belief (i.e. the belief of

“others”). But then, how do we describe individual and market beliefs? Our response is to treat

individual beliefs as primitive state variables. They are basic cause of market change like any other

state variable. Analytically we use the approach of Kurz, Jin and Motolese (2005a), (2005b) and Kurz
9

and Motolese (2011) as adapted to the problem of this paper. This adaptation is outlined next. 

In our Markov economy a probability measure on sequences of state variables is represented by

a sequence of transition functions. Hence, an individual belief is described by a personal state of belief

which uniquely pins down the agent’s perceived transition functions of state variables. Hence, personal

state variables and the economy-wide state variables are not the same. A personal state of belief may

be thought of as analogous to an agent “type” at each date. However, at t he is not certain of his future

belief types which are then determined by a transition of his own personal state of belief. The

distribution of individual states of belief, which then defines “the market state of belief,” is an

economy-wide observable state variable. All moments of this distribution could matter in equilibrium,

but due to the exponential utility we use, equilibrium endogenous variables depend only on the mean

market states of belief. This is generalized in the empirical work reported later since we have already

noted that market belief is observable. In equilibrium, endogenous variables (e.g. prices) are functions

of the economy’s state variables, including market belief. In a large economy an agent’s anonymity

implies that a personal belief state has a negligible effect on prices and past personal states are not

observed. Finally, due to the effect of market belief on endogenous variables, to forecast future

endogenous variables an agent must forecast the beliefs of others. We thus turn now to discuss the

dynamics of individual beliefs and demonstrate that belief dynamics is dictated by rationality.

The theory of Rational Beliefs due to Kurz (1994), (1997) defines an agent to be rational if his

model cannot be falsified by observed data and if simulated, it reproduces the stationary probability m

deduced from past data. Here we do not use the detailed restrictions the theory imposes since we aim

to test Theorem 3 below and this theorem is true under less restrictive conditions. We thus accept the

idea that rationality should require a belief to be compatible with past data but translate it into three

rationality principles used in the analysis to follow. We now explain them. 

Rationality Principle 1: A belief cannot be a constant transition unless an agent believes the

stationary transition (1) is the truth. This is so since if one holds a constant transition as his

belief which is not (1) then the time average of his belief is different from (1). Since (1) is the

time average in the data, it proves the agent is irrational. Simply stated, an agent who is always

optimistic or always pessimistic relative to (1) is irrational since he holds beliefs which are

strongly rejected by the data. This leads to the observation that rationality implies that beliefs

must fluctuate and thus must have inherent dynamics which contributes to market volatility. 
10

Given the evidence for persistent diverse beliefs, one concludes that agents hold wrong beliefs

(since there is only a single truth). But being rational and “wrong” are not in conflict. Rational agents

hold wrong beliefs when there is no evidence against them. The term “wrong” is obviously used here

relative to an unknowable standard. We then turn to a second rationality principle. To state it note that

a belief index describes how an agent’s perceived a transition deviates from (1):

Rationality Principle 2: A belief does not deviate from (1) consistently and hence the belief index has

an unconditional mean of zero. 

 Agent i’s state of belief is denoted by It describes i’s perception by pinning down his g

i

t .

transition functions. Adding to “anonymity” we assume agent R knows his own and the market g

R

t

distribution of at t across i. In addition he observes past distributions of the for all g τ < t hence i

t g

i

τ

he knows past values of all moments of the distributions of . g

i

τ

How is used by the agents? If is agent i’s perception of t+1 payoff then pins down g

i

t d it%1 g

i

t

E by specifying the difference between his date t forecast of all state variables and the forecasts i

t d it%1

under the empirical distribution m. Agent i’s date t perceived distribution of is then defined by dt%1

(4) . d i

t%1 ' λd dt

% λ

g

d g

i

t % ρ

id

t%1 , ρ

id

t%1 - N(0 , ˆσ

2

d )

The assumption that is the same for all agents is made for simplicity. It follows that measures σˆ

2

d g

i

t

(5) . E i[d i

t%1 |Ht

,g

i

t ] & E m [dt%1 |Ht ]' λ

g

d g

i

t

Rationality Principle 2 requires to have a zero unconditional mean. But (5) also shows g how to i

t

measure in practice. For a state variable Xt, data on i’s forecasts of Xt+1 g (in (4) it is ) are i

t dt%1

measured by . To compute (5) we use standard econometric techniques to construct E i[X i

t%1 |Ht

,g

i

t ]

E . Such construction was used by Fan (2006), Kurz and Motolese (2010) and is later m [Xt%1 |Ht ]

explained. We have for an agent who believes m is the truth. Since a belief is about our g

i

t '0

changing society the reflect belief about different economies. For example, in 1900 the were g

i

t g

i

t

related to electricity and combustion engines, while in 2000 they reflected beliefs about information

technology. Success or failure of past tell you nothing about what present day should be. g

i

τ g

i

t

The two rationality principles we have adopted imply that in an economy with diverse beliefs

an agent’s belief (represented by his transition functions) must fluctuate and this means that the index

g must fluctuate over time. The third principle addresses the question of dynamics. i

t

Rationality Principle 3: The transition functions of are Markov, taking two possible forms which g

i

t

exhibit persistence 
11

(6a) g i

t%1 ' λZ g

i

t % ρ

ig

t%1 , ρ

ig

t%1 - N(0 , σ

2

g )

(6b) g i

t%1 ' λZ g

i

t % λ

d

Z dt

% kig

t%1 , kigt%1 - N(0 , σ

2

g )

where or are correlated across i reflecting correlation of beliefs across individuals. ρ

ig

t%1 kigt%1

We shall explain later why this correlation is a crucial component of the theory.

How can we justify (6a)-(6b) who play a central role in the model? Our first answer is that the

data supports this specification and we demonstrate this fact below. Next, in Section 1.3 below we

prove (6a)-(6b) as a result of Bayes rationality. Before doing so we comments on these conditions. 

In a rapidly changing environment it is useful to describe belief diversity so that equilibrium

analysis is tractable. As is shown below, the advantage of (6a)-(6b) is that they lead to a simple

description of equilibrium pricing with diverse beliefs. It does not entail extraction of information from

market prices, it needs each agent to have a distinct state space to describe his own uncertainty and

requires an endogenous expansion of the economy-wide state space for equilibrium pricing.

(6a) is the dynamics of Example 2, Section 1.1 above and under the two conditions in that

example it is a Rational Belief. We employed such dynamics in several papers (e.g. Kurz, Jin and

Motolese (2005a), (2005b) and Kurz and Motolese (2009)). Here we develop the theory under both

(6a) and (6b) and show that the main Theorem 3 below is true under either one. Persistence of beliefs

is common to (6a) - (6b). To see the difference between them recall our assumption that agents have

ample past data for learning (1). The question is how much they learn from recent data. Under (6a)

individual models are based on other factors than recent quantitative data while under (6b) recent data

contribute to their assessment. In the next section we deduce (6b) analytically from a Bayesian

learning procedure. Since (6a) is a special case of (6b) Theorem 1 below provides conditions for (6a)

or (6b) to be a rational dynamics of individual beliefs. We may note that for some readers the most

compelling justification for the persistent dynamics of (6a) or (6b) is the fact that it is supported by the

data. However, we now explore conditions for deducing (6a)-(6b) from Bayesian rationality.

1.3 Deducing (6b) from a Model of Bayesian Rationality

 In a standard Bayesian model an agent faces data generated under a stationary structure but

with an unknown fixed parameter. The agent starts with a prior on the parameter and uses Bayesian

inference for retrospective updating of his belief. The term “retrospective” stresses that inference is

made after data is observed. In real time an agent uses the prior to forecast future variables while
12

learning can only improve future forecasts of the variables. Under the true probability Π the value dt

has a sequence of transition functions of the form (4), i.e . The dt%1&λd dt

'bt%g

d

t%1 , g

d

t%1-N( 0, 1

β

)

fixed parameters are known as they are deduced from the empirical frequencies. We assume agents

know and λ β but not the “regimes” . The infinite number of time varying parameters express d bt bt

the non stationarity of the economy. Changes reflect technologies and social organizations that define

each era. In reality commodities change over time and represent different objects hence a single bt

commodity which is comparable over time is only a model simplification.

The structure of changing parameters requires us to supplement the standard Bayesian learning

process. To explain why note that at t-1 an agent has a prior belief about with which he forecasts bt&1

d . After observing he updates his prior into a sharper posterior estimate of which, t dt E i

t (bt&1 |dt

) bt&1

as a random variable, we denote by . But at date t he needs to forecast . For that he does bt&1 (dt

) dt%1

not need a posterior estimate of but rather, a new prior on bt b ! Agents do not know if and when a t&1

parameter changes. If they knew changes slowly or then an updated posterior of is a bt bt

'bt&1 bt&1

good prior of bt. Without such knowledge, they presume is possible and look for a new prior. bt&1...bt

They would seek additional information to arrive at an alternative subjective estimate of . Public bt

qualitative information is an important source which offers a route to such alternative estimate of . bt

1.3.1 Qualitative Information and Subjective Interpretation of Public Information

Quantitative data like arrive with d qualitative information about unusual conditions under t

which the data was generated. For example, if are profits of a firm then is a number in a financial dt dt

report which contains qualitative information about changing consumer fashion, new and competing

products, technology, joint ventures, research & development etc. If are profits of the S&P500 then dt

qualitative information includes business conditions, productivity trends, monetary policy, taxes and

other macroeconomic conditions. Qualitative information cannot, in general, be compared over time

and does not constitute conventional “data.” If a firm reports on research into something that did not

exist before, no past data is available for comparison. When a new product alters the nature of an

industry, it is a unique event. Financial markets pay a great deal of attention to qualitative

announcements which are often the focus of diverse opinions and investors’ activity.

There is little modeling of qualitative information. Saari (2006) uses qualitative information in a

competitive model of market shares. Toukan (2006) is a second example. Kandel and Pearson’s
13

(1995) model of diverse interpretation of public signals can be supported with an argument based on

qualitative information. Kurz (2009) formulates a model where, in addition to , date t qualitative dt

information lead agents to form subjective probabilities about unobserved events which can impact

d via . These effects are open to diverse assessments: they can be positive or negative and their t%1 bt

size and timing can be in doubt. A simple example will suffice. When a firm announced it also dt

announced that during the past year it completed a major project that includes an implementation of a

long term research effort of major significance. The conclusions of the project are now being put into

effect and these could have a big impact on next years profits. In this paper we avoid the problem of

assessing qualitative information. Instead we focus only on the fact that it leads agents to formulate an

alternate prior on which, as a random variable, is denoted by defined by . One bt Bt B i

t -N(Ψit , 1

γ

)

can say either that i “observes” and or that he assesses these values from qualitative public Ψi

t γ

information. The main question is how to reconcile with the posterior formulated given B i

t bt&1 (dt

)

the data . To do that we need to specify the updating process. dt

1.3.2 A Bayesian Inference: Beliefs are Markov State Variables with Transition (6b)

Agents believe the true transition of profits is (4), that β is known but are unknown. At date bt

t-1 (say t -1 = -1) an agent needs to forecast and uses for that a prior belief about described dt bt&1

by . Now we move to date t (here t = 0) and after observing the data (recall b i

t&1-N(b, 1

α

) dt

ρ ) the posterior on is updated to be d

t - N(0 , 1

β

) bt&1

(7) E i

t (bt&1 |dt

) ' αb % β[dt

&λd dt&1]

α % β , bt&1(dt

) - N[E i

t ( bt&1 |dt

) , 1

α % β

]

Now, using qualitative data, agent i makes the assessment independently of the random B i

t -N( Ψit , 1

γ

)

variable and we have two alternative priors. Our key assumption is: bt&1(dt)

Assumption (A): Agent i uses a subjective probability to form date t prior belief about as a μ bt

random variable defined by

(8a) bt

(dt,Ψi

t

) ' μ bt&1(dt) % (1 & μ )B i

t 0 < μ < 1

Generally, if is a posterior given data only, a revised prior given is given by bt(dt%1 ,Ψi

t

) dt%1 B i

t%1

(8b) , bt%1(dt%1 ,Ψi

t%1) ' μbt

( dt%1 ,Ψi

t) % (1&μ)B it%1
6 Note: we use a notation of for date t E prior belief about the parameter used to forecast . We i

t (bt

|dt,Ψi

t) bt dt%1

then use the notation for the posterior belief E about the same given the observation of but without i

t%1 (bt

|dt%1 ,Ψi

t) bt dt%1

changing the estimate of . Assumption (A) uses this posterior belief as a building block in revising the prior Ψi

t

E about the new parameter . i

t%1 (bt%1 |dt%1 , Ψit%1) bt%1

14

(8c) .

6 E i

t%1 ( bt%1 |dt%1 ,Ψit%1) ' μE it%1 ( bt

|dt%1 ,Ψi

t) % (1&μ)Ψit%1 0 < μ < 1

with the notation Γ(b . t

) ' Precision(bt|dt,Ψi

t

)

Theorem 1 : If Assumption (A) holds then converges for large t to a constant but the Bayes Γ(bt) Γ(

estimate fluctuates indefinitely. Let then this index is a E i

t (bt

|dt,Ψi

t) g it 'E it (bt

|dt,Ψi

t

)&[μ β/(Γ(%β)]dt

Markov state variable and (6b) holds with k : Assumption (A) implies (6b). ig

t%1 ' (1&μ)Ψit%1

Proof : See Appendix B.

The random component explaining both diversity and dynamics is . It arises kig

t%1'(1&μ)Ψit%1

from random arrival of qualitative information which is interpreted differently by different agent. 

There are two ways to justify (6a). From Theorem 1 it follows that (6a) is a special case of

(6b) when . Condition (B.5) in Appendix B shows that if . Second, λ

d

Z'0 λ

d

Z'0 μ ' λd (1%(β/Γ())

an alternate Bayesian learning can be developed in which persistence in the arrival of qualitative

information implies the Markov property in (6a). 

1.4 Modeling Diverse Beliefs II: Market Belief and the Central Role of Correlation

The analysis below and Appendix A are developed for the (6a) dynamics but can be easily

adapted to (6b). Hence we deduce the equilibrium and state Theorem 3 for both (6a) and (6b). 

Averaging (6a) we denote by the mean of the cross sectional distribution of and refer to it as “ Zt g

i

t

average market belief.” It is observable. Due to correlation across agents’ , the law of large ρ

ig

t

numbers does not apply and the average of over i does not vanish. We write it in the form ρ

ig

t

(9) . Zt%1 ' λZ Zt

% ρˆ

Z

t%1

The true distribution of is unknown. The random term in (9) is import and shows the dynamics of ρˆ

Z

t%1

Zt

 depends upon the correlation across agents’ beliefs rather than on their actual beliefs. If the random

terms or in (6a) or (6b) were independent, then by the law of large numbers we would have ρ

ig

t kigt

ρˆ . Correlation then ensures market belief does not degenerate into in case of (6a) or into a Z

t '0 Zt ' 0

deterministic relation for (6b). In either case correlation is why market belief does not Zt%1'λZZt%λ

d

Zdt

possess deterministic dynamics and becomes irrelevant to asset pricing. Since correlation is not
15

determined by individual rationality it becomes an important belief externality. 

Correlation may exhibit non stationarity inherited by the Zt

. Since they are observable, market

participants have data on { } hence they know the (d joint empirical distribution of t

, Zt),t ' 1,2,...

these variables. For simplicity we assume this distribution is described by the system of equations

(10a) dt%1 ' λd dt

% ρ

d

t%1 ρ

d

t%1

ρ

Z

t%1

- N

0

0 ,

σ

2

d, 0,

0, σ

2

Z

' Σ , i.i.d.

(10b) Zt%1 ' λZ Zt

% ρ

Z

t%1

Now, an agent who does not believe that (10a)-(10b) is the truth, formulates his own model\belief. We

have seen in (4) how agent i’s belief state pins down his forecast of . We now broaden this g

i

t d it%1

idea to an agent’s perception model of the two state variables . Keeping in mind that (d i

t%1 , Z it%1 )

before observing agent i knows , his belief takes the general form (dt%1,Zt%1 ) dt and Zt

(11a) d i

t%1 ' λd dt

% λ

g

d g

i

t % ρ

id

t%1

(11b) Z i

t%1 ' λZ Zt

% λ

g

Zg

i

t % ρ

iZ

t%1

ρ

id

t%1

ρ

iZ

t%1

ρ

ig

t%1

- N

0

0

0

,

σˆ

2

d , ˆσZd , ˆσgd

σˆ Zd , ˆσ

2

Z , ˆσZg

σˆ gd , ˆσZg , σ

2

g

, ' Σi

(11c) g

i

t%1 ' λZ g

i

t % ρ

ig

t%1

(11a)-(11b) show that, as required, pins down the transition of both state variables . g

i

t (d it%1 ,Z it%1 )

This simplicity ensures that one state variable pins down agent i’s subjective belief of how conditions

at date t are different from normal as reflected by the empirical distribution:

(12) E i

t

dt%1

Zt%1

& E m

t

dt%1

Zt%1

'

λ

g

d g

i

t

λ

g

Z g

i

t

.

Note: in (11a)-(11c) random terms are not required to be i.i.d. Also, adapting (10b), (11b)-(11c) to

the dynamics (6b) is simple. Instead of (10b) we write and instead of (11b)- Zt%1'λZZt

%λ

d

Z dt

%ρ

Z

t%1

(11c) we have and . Z i

t%1 ' λZ Zt

% λ

d

Zdt

% λ

g

Zg

i

t % ρ

iZ

t%1 g

i

t%1 ' λZ g

i

t % λ

d

Z dt

% ρ

ig

t%1

1.4.1 No Representative Agent: The Centrality of Belief Heterogeneity 

Is belief heterogeneity central to (11a)-(11c)? For exponential utility only average market

belief matters so why could the model not be reduced to a representative agent? For actually

heterogenous economy the model cannot be reduced to a representative agent for several reasons. 

(i) The first answer is in (11b). One element of the theory requires an agent to forecast which is Zt%1

the belief of "others” and for that he must perceive the market as different from himself. Without this
16

reality a representative agent identifies himself with the average market and there is nothing to

forecast: equation (11b) disappears. Formally it leads the average market belief to disappear as a Zt

state variable from the optimization of the agent. This fact is also seen when we define average market

expectation operator to be . From (11c) it is E ̄t(C)' 1

Nj

N

i

E i

t (C)

 (13) . E ̄t

dt%1

Zt%1

& E m

t

dt%1

Zt%1

'

λ

g

dZt

λ

g

ZZt In heterogenous market with a representative agent, the agent forecasts with . Such forecasts E ̄t

(C)

would hold for the dividend process but not for . One could formally replace (11b) with the mean Zt%1

of (11c) and put a 0 in the second component of (13) but this is rejected by reason 4 below.

(ii) Heterogeneity plays a key role in the dynamics of via which depends upon the Z correlation t ρ

d

t

across beliefs. Since the correlation is not determined by considerations of individual rationality,

aggregate dynamics is a consequence of the heterogeneity externality. Changes in heterogeneity

without changes in the mean matter but are missed by a representative agent. 

(iii) When utility is not exponential then the entire distribution matters. Many papers cited earlier (e.g.

Lee et al. (2002), Diether et al. (2002), Johnson (2004), Park (2005), Baker and Wurgler (2006),

Karakatsani and Salmon (2008), Kurz and Motolese (2011) find that fluctuations in the cross-sectional

variance of their measures of belief contributes to the predictability of returns. Hence, the effect of

cross sectional variance is a null hypothesis we also test below.

(iv) If the economy is heterogenous but there is a representative agent the belief of that agent would be

some average of the beliefs of individuals. But the average belief cannot even be a probability. Indeed,

the perception models (11a)-(11c) show that properties of conditional probabilities do not apply to the

market belief operator since it is E ̄ not a proper conditional expectation. To see why let t

(C) X'D×Z

be a space where take values and Gi (d be the space of . Since i conditions on , his t

, Zt) g i

t g

i

t

unconditional probability is a measure on the space where öi ((D×Z×G is a sigma field. The i)

4 ,öi

)

market conditional belief operator is an average over conditional probabilities, each conditioned on a

different state variable. Hence, this averaging does not even permit one to write a probability space for

the market belief. The market belief is neither a probability nor rational and the following result

summarizes this fact:
17

Theorem 2: The market belief operator violates iterated expectations: . E ̄t

(dt%2 ) ... E ̄tE ̄t%1( dt%2 )

Proof: Since it follows that E i

t (dt%2 ) ' λdE it ( dt%1 ) % λ

g

dE it (g it%1) ' λd [λd dt

%λ

g

d g

i

t ] % λ

g

d λZg

i

t

(14) . E ̄t(dt%2 ) ' λ

2

d dt

%λ

g

d ( λd % λZ) Zt

But we also have from (11a) that and using (11b) we can compute E ̄

t%1 ( dt%2 ) ' λd dt%1 % λ

g

dZt%1

E . i

t E ̄t%1 ( dt%2 ) ' λd [λd dt

% λ

g

d g

i

t ] % λ

g

d [λZ Zt

% λ

g

Zg

i

t ]

Aggregating we now conclude that

(15) . E ̄tE ̄t%1 (dt%2) ' λ

2

d dt

% λ

g

d (λd % λZ % λ

g

Z ) Zt

Comparison of (14) and (15) shows that . E ̄t

(dt%2 ) ... E ̄tE ̄t%1 ( dt%2 ) 

1.4.2 Market Belief, Information and Human Exuberance

We address two issues. First, the question of belief and Information. For an agent, is a state Zt

variable like others. News about are used to forecast prices in the same way data such as a leading Zt

indicator is used to assess the risk of a recession. How do agents update beliefs when they observe the

mean belief of others? In contrast with private information models, agents do not revise their beliefs Zt

about the state variable : (11a) d does not depend upon . They do not view as information t%1 Zt Zt

about since it is not a “signal” about unobserved private information they do not have. However, dt%1

Z is crucial “news” about what the market thinks about Hence, the importance of is it’s t dt%1! Zt

great value for forecasting future endogenous variables. Date t endogenous variables depend upon Zt

and future endogenous variables depend upon future Z’s. Since market belief exhibits persistence,

agents know that today’s market belief is useful for forecasting future prices.

Finally, in considering economies populated by fully rational agents who hold diverse beliefs, is

there a role for human “exuberance?” The answer is yes! If all agents are individually rational we still

have the mechanism of correlation which is not subject to any principle of individual rationality. We

have seen that with diverse beliefs such correlation plays a central role in market dynamics hence such

correlation is actually a belief externality, taken by all as given. In the model above there would be no

aggregate dynamics in (9) without correlation and such correlation can entail market herding, bubbles

and other forms of excess volatility which, in our opinion, are often offered incorrectly as evidence for

market irrationality. 
18

1.4.3 The Empirical Evidence Regarding Belief Dynamics (6a)-(6b)

We later use two data sources to extract time series on for six month horizon from GDP Zt

forecast data. We briefly examine now the question of which of the two models (6a) or (6b) fit the

data better. We test the following regression model for two data sources and different periods

(16) Zt%1 ' a0 % a1Zt

% a2yt% ε

Z

t%1

where yt is the annualized six month growth rate of GDP (final release). The results are reported in

Table 1. It is clear the persistence parameter a1 is always significant and takes values around 0.60, a

result which is virtually the same as the result in Kurz and Motolese (2011) where beliefs are derived

from of interest rate forecasts. As to a2 the results show it is small and statistically significant for some

periods. Hence, the results are inconclusive but favor the dynamic model of (6a). 

 Table 1: Estimated Coefficients of Market Belief Dynamics

Data Source and Period Coefficient Std. Error p-value R2

Livingston – 1968:S2-2007:S2 a0

a1

a2

-0.275

 0.842

 0.048

0.260

0.052

0.060

0.293

0.000

0.426

0.732

Livingston – 1988:S1-2007:S2 a0

a1

a2

-1.326

 0.515

 0.128

0.248

0.097

0.055

0.000

0.000

0.026

0.385

SPF – 1968:S2-2007:S2 a0

a1

a2

-0.173

 0.811

 0.040

0.283

0.054

0.074

0.543

0.000

0.586

0.684

SPF – 1988:S1-2007:S2 a0

a1

a2

-1.131

 0.587

 0.176

0.205

0.051

0.051

0.000

0.000

0.002

0.628

SPF/BLU – 1968:S2-2007:S2 a0

a1

a2

-0.005

 0.838

-0.008

0.321

0.051

0.084

0.988

0.000

0.929

0.697

SPF/BLU – 1988:S1-2007:S2 a0

a1

a2

-1.032

 0.633

 0.171

0.172

0.054

0.036

0.000

0.000

0.000

0.719

1.5 Combining the Elements: the Implied Asset Pricing Under Diverse Beliefs

We now derive equilibrium prices and the risk premium. We return to (2a)-(2b) where the state

variables are defined . We maximize (2b) subject to the budget constraint (2a) and ψ

i

t ' (1,dt

,Zt,g

i

t )

transitions (11a)-(11c). A full analysis of the problem is found in Appendix A. A simple way to

approach the problem is to specify an assumption and conjecture. First, assume the agent believes 

{ } are conditionally normal. Second, conjecture that equilibrium price is also dt

,t'1,2,... pt
19

conditionally normal. Theorem 3 will then confirm the conjectures. In Appendix A we show that for an

optimum of (2a)-(2b), there is a constant vector u so the stock demand function is

(17) θ . i

t

(pt) ' Rτ

r ˆσ

2

Q

[ E i

t ( Qt%1) % uψ

i

t

],u'( u0 , u1 , u2 , u3 ) , ψ

i

t ' (1,dt

, Zt,g

i

t )

σˆ is an adjusted conditional variance (see Appendix A for details) of excess stock returns, assumed 2

Q

constant and the same for all agents. The term is the intertemporal hedging demand which is linear uψ

i

t

in agent i’s state variables. 

For an equilibrium to exist we need some stability conditions. First we require the interest rate

r to be positive, R = 1 + r > 1 so that . Now we add: 0 < 1

R

< 1

(18) Stability Conditions: We require that . 0 < λd <1 , λZ <1 , 0< λZ % λ

g

Z < 1

The first requires {dt , t = 1, 2, ...} to be stable and have an empirical distribution. The second is a

stability of belief condition. It requires i to believe is stable. To see why, take expectations of (dt , Zt

)

(11b), average over the population and recall that Zt are market averages of the . This implies that g

i

t

E ̄ . t[Zt%1 ] ' (λZ % λ

g

Z )Zt

Theorem 3: Consider the model with heterogenous beliefs under the stability conditions specified

with supply of shares which equals N. Then either under (6a) or (6b) there is a unique equilibrium

price function which takes the form . pt ' ad dt

% aZ Zt% P0

Proof: Average (17), use the fact that the aggregate stock supply is N and rearrange to have

(19) .

r ˆσ

2

Q

Rτ ' [E ̄t

(pt%1%dt%1%μ ) & Rpt% (u0%u1dt%(u2%u3)Zt) ]

Now use the perception models (11a)-(11b) about the state variables, average them over the

population and use the definition of Zt to deduce the following relationships under (6a)

(20) E ̄t[dt%1 % μ ] ' λd dt% μ % λ

g

d Zt

(21) . E ̄t[Zt%1 ] ' (λZ%λ

g

Z)Zt

Using these to solve for date t price we deduce

(22) pt ' 1

R

[E ̄t(pt%1 )] %

1

R

[ (λd % u1)dt% (λ

g

d % u2 % u3)Zt

] %

1

R

[ μ % u0] & rˆσ

2

Q

R2τ
20

(22) shows that equilibrium price is the solution of a linear difference equation in the two state

variables . Hence, a standard argument (see Blanchard and Kahn(1980), (d Proposition 1, page t , Zt

)

1308) shows that the solution is 

 (23a) pt ' ad dt

% aZ Zt% P0

To match coefficients use (11a)-(11b) to insert (20) - (21) into (22) and conclude that

 (23b) , , . ad ' λd % u1

R & λd

aZ ' ( ad % 1)λ

g

d % ( u2 % u3 )

R&(λZ %λ

g

Z )

P0 ' (μ % u0)

r & σˆ

2

Q

Rτ

The stability conditions ensure that (23a) - (23b) is the unique solution as asserted.

Under the (6b) dynamics the conditions (20)-(21) are written as 

(20’) E ̄t[dt%1 % μ ] ' λd dt% μ % λ

g

d Zt

(21’) . E ̄t[Zt%1 ] ' (λZ%λ

g

Z)Zt

% λ

d

Z dt

Computing the corresponding stochastic difference equation we find the solution is the same as (23a)

with coefficients which are now defined by 

(23b’) , , . ad ' 

aZ λ

d

Z % (λd % u1 )

R & λd

aZ ' ( ad % 1) λ

g

d % (u2 % u3 )

R&(λZ %λ

g

Z )

P0 ' (μ % u0)

r & σˆ

2

V

Rτ

We do not have closed form solutions for hedging demand parameters and u'(u0, u1,u2, u3)

computed numerical Monte Carlo solutions instead. For all model parameters values given the relevant

case we find and . These are reasonable: (λ increases with higher and g

d>0, λ

g

Z>0) ad>0 aZ >0 pt d t

with higher - today’s market belief Z in unusually higher future dividends. t

1.6 Equilibrium Risk Premium Under Heterogenous Beliefs

1.6.1 The Main Equilibrium Results

Under heterogenous beliefs we have many risk premia and one chooses a concept which is

appropriate for an application. As a random variable the risk premium on a long position is

(24) . πt%1 ' pt%1 % dt%1 % μ & Rpt

pt (24) is actual excess returns of stocks over riskless bond. We need to measure the premium as a known

expected quantity, recognized by all. There are many such measures. One is a subjective premium of
21

agent i. We computed it by using the equilibrium map (23a) and perception model (11a) -(11c) to have

(25) 1

pt

E i

t (pt%1

%dt%1%μ&Rpt) ' 1

pt

[(ad%1)(λd dt%λ

g

d g

i

t )%aZ(λZZt

%λ

g

Zg

i

t )%μ %P0&Rpt

]

By taking the mean of (25) we can also compute the average market risk premium. We seek an

objective measure, computed by all agents who study the long term premium using to the empirical

measure m. By (23a) and the stationary transition (10a)-(10b) we can deduce that it is

(26) E . m

t [πt%1

] ' 1

pt

E m

t [pt%1%dt%1%μ &Rpt

] ' 1

pt

[(ad%1)( λd dt)%aZ (λZ )Zt%μ %P0&Rpt]

The concept in (26) is the way Econometricians and all researchers cited above have been measuring

the risk premium. For this reason we refer to it as “the” risk premium. 

We then arrive at two conclusions. First, the risk premium is different from the market

perceived premium when Z ...0 since 

(27) . 1

pt

E m

t (pt%1

%dt%1%μ&Rpt) & 1

pt

E ̄t

( pt%1%dt%1%μ &Rpt) ' & 1

pt

[(ad%1) λ

g

d% aZλ

g

Z] Zt

A second conclusion is derived by studying (26). From (23c) we can compute the condition

&(u and hence we can deduce the main result: 2%u3)'&aZ(R&λZ)%[(ad%1)λ

g

d%aZλ

g

Z]

Theorem 3: The equilibrium risk premium has the following analytical expression

(28a) 1

pt

E m

t (pt%1%dt%1%μ&Rpt

) ' 1

pt

[(

r ˆσ

2

Q

Rτ & u0 & u1dt

) & aZ (R & λZ) Zt]

Since az > 0, R > 1 and it follows that λZ < 1

(28b) the Risk Premium is decreasing in the mean market belief . E m

t [πt%1] Zt

Conclusions (28a) -(28b) are central to this paper. (28a) exhibits the endogenous component of the

risk premium which we call “The Market Belief Risk Premium.” It shows that market belief has a

complex effect on market risk premia. It consists of two parts

(I) The first is a direct effect on the permanent mean premium . It is shown in Appendix A

r ˆσ

2

Q

Rτ

that there exist weights such that (ω1 ,ω12,ω2 )

 . σˆ

2

Q ' Var it ((ω1(λd dt

% λ

g

d g

i

t %ω12ρ

id

t%1)%ω2 (λZ Zt

%λ

g

Zg

i

t %ω12ρ

iZ

t%1 ))

Volatility of belief contributes directly to the volatility of excess returns and increases

permanently the risk premium.

(II) The second is the effect of market belief on the time variability of the risk premium,

reflected in with a negative sign when Zt & a > 0. Z (R & λZ) Zt
22

To explain this second result we note that it says that if one runs a regression of excess returns on the

observable variables, the effect of market belief on long term excess return is negative. This sign is

surprising since when Zt

 > 0 the market expects above normal future dividends but in that case the risk

premium on the stock is lower. When Zt

 < 0 the market holds bearish belief about future dividend but

the risk premium is higher. Since we have data on Zt

 and on the distribution of belief the result will be

empirically tested. Before proceeding to the empirical test we discuss some ramifications of this result. 

1.6.2 The Market Belief Risk Premium is General

The main result (28b) was derived for the exponential utility function. We argue that this result

is more general and depends only on the positive coefficient az of in the price map. To show this, Zt

assume any additive utility function over consumption and a risky asset which pays a “dividend” or any

other random payoff . Denote the price map by . We are interested in the slope of dt pt ' Φ( dt , Zt

)

E with respect to . Focusing only on the numerator in (26), linearize the price around 0 and m

t [πt%1] Zt

write . The desired result depends only upon the condition that . It is pt ' Φddt% ΦZZt%Φ0 ΦZ > 0

reasonable as it requires current price to increases if the market is more optimistic about the asset’s

future payoffs. To prove the point note that 

E m

t [ pt%1%(dt%1%μ)&Rpt

] . E m

t [Φddt%1%ΦZZt%1%Φ0%(dt%1%μ)&R(Φddt

%ΦZZt%Φ0 )]

 . ' [(Φd%1)λd & RΦd]dt&ΦZ(R & λZ)Zt%[μ %Φ0 ( 1&R)]

The desired result follows from , R > 1 and . This result may be altered if the interest ΦZ > 0 λZ < 1

rate is not fixed and an endogenously determined Rt also changes with Zt in (28a).

1.6.3 Interpretation of the Market Belief Risk Premium

Why is the effect of Zt

 on the risk premium negative? Since this result applies to any asset with

risky payoffs, we offer a general interpretation. Our result shows that when the market holds

abnormally favorable belief about future asset payoffs the market views the long position as less risky

and consequently the risk premium on a long position falls. Hence, in the long run fluctuations in risk

premia are inversely related to degree of market optimism about future prospects of an asset’s payoff.

To explore the result, it is important to explain what it does not say. One could interpret it to

confirm a common claim that to maximize excess returns it is optimal to be a “contrarian” to the

market consensus. To understand why this is a false interpretation note that when an agent holds a
23

belief about future dividends, the market belief Zt

 does not offer him new information for altering his

belief about dividends. If the agent believes future dividends will be abnormally high but , the Zt < 0

agent does not change his forecast of . Zt d is an important input to forecasting returns since it is t%1

used to forecast future prices. Keep in mind that given available information and a probability belief,

denoted say by , an optimizing agent is already on his demand function. In response to a changed Z Γ t

i

he does not just abandon his demand by replacing with the empirical measure m. This argument is Γi

analogous to the one showing why it is not optimal to adopt log utility as your utility even though it

maximizes the growth rate of your wealth. Yes, it does that, but you dislike the sharp declines which

you expect to occur in the value of your assets if you follow the strategy called for by the log utility.

By analogy, following a “contrarian” policy implies a high long run average return in accord with m 

since this is what (28a) says. But if your subjective model disagrees with the probability m you will

dislike being short when your optimal position should be long. This argument explains why most

people do not systematically bet against the market, as a “contrarian” strategy (28a) would dictate. 

Taking a positive view, our results show fluctuations in market belief are important for the time

variability of risk premia and the market pricing of risk. Market optimism in bull markets or pessimism

in bear markets have added effects on market risk perception above and beyond any real information

about the economy. (28a) shows that in the long run market belief has an inverse effect on market risk

premia. To see how individually perceived premia are affected by market belief use (25) and (26) to

find . Hence, optimizing agents use in calculating their E i

t (πt%1

)'E m

t (πt%1

)%

1

pt

[(ad%1)λ

g

d%λ

g

Z]g it Zt

subjective premia in the same way they use any state variable which describes the stat of the economy.

We turn now to an empirical test of our theory. 

2. Testing the Time Variability of Stock Market Risk Premia: The Data

2.1 The Forecast Data

We use data on the distribution of commercial forecasts and take them as proxies for the

forecasts of future business conditions made by the general public. Several different sources of data are

employed: the Livingston survey (LIV), the Survey of Professional Forecasters (SPF), the Blue Chip

Financial Forecasts (BLUF) and the Blue Chip Economic Indicators (BLUE). In all cases forecast data

of future real gross domestic product (GDP) growth rates are taken as a proxy for expected business

conditions in the future. 
24

Livingston GDP Growth Rate Forecasts. The Livingston survey is conducted biannually in

June and December since June 1946. Our sample begins in December 1968 and continues until

December 2007. LIV provides forecast data on the level of nominal GDP in the following June and

December rather than forecast data of real GDP growth. Hence, we had to construct growth rate data

from individual expectated nominal GDP and consumer price index (CPI) levels. Also, LIV does not

ask participants to provide expected current levels for nominal GDP and CPI and this deprives us of a

common baseline data. Hence, like others who use LIV, we cannot compute individual 6-month-ahead

forecasts for real GDP growth rates. Instead, for all individuals and survey dates we compute 12-

month-ahead annualized real GDP growth rate forecasts based on the implied expected growth rate

between 6 months ahead and 12 months ahead. We then take the average forecast, obtaining a series

of 2 semester ahead forecasts spanning 1968:S2-2007:S2. 

Survey of Professional Forecasters GDP Growth Rate Forecasts. Among other data, SPF

provides forecasts of quarterly real GDP levels from which growth rate forecasts are derived. The

survey is currently conducted by the Federal Reserve Bank of Philadelphia, continuing the American

Statistical Association\NBER survey started in 1968. The survey is conducted quarterly and released

in the middle month of each quarter. The sample available extends from 1968:Q4 to 2007:Q4. Semiannual forecasts of real GDP growth rates which are comparable to those deduced from LIV are

computed by taking geometric averages of forecasts from SPF for two consecutive quarters.

Blue Chip Financial Forecasts. BLUF reports quarterly forecasts of U.S. interest rates and

quarterly forecasts of real GDP growth rates and inflation. Forecasts reported in BLUF are collected

on the 24th and 25th of each month and released to subscribers on the first day of the following month.

Since January 1983 BLUF has published individual and mean forecasts for each variable. The mean is

taken over forecasters participating in that month. The sample available extends from 1984:M7 to

2007:M12. As is the case for SPF data, to obtain semi-annual forecasts comparable to those from LIV

we compute geometric averages of forecasts of two consecutive quarters.

Blue Chip Economic Indicators. BLUE reports quarterly forecasts of U.S. economic growth,

inflation, interest rates, and other macroeconomic indicators of future business activity. Forecasts

reported in BLUE are collected on the 2nd and 3rd day of each month and released to subscribers on the

10th of the month. Since 1976 BLUE publishes, for each variable, individual and mean forecasts. The

mean is taken over all forecasters participating in that month. The sample available to us extends from
25

1980:M1 to 2007:M12. As is the case for SPF and BLUF, to obtain semi-annual forecasts comparable

with those from LIV we compute geometric averages of forecasts of two consecutive quarters.

Forecasts reported in the four surveys above are labeled by their release date. We assume these

forecasts are conditional on information available at the moment the forecasts were collected. In the

case of LIV forecasts are conditional on information available at the end of May and November

respectively for 1st and 2nd semester. For SPF, which does not follow a strict pattern of dates for

collection and release of forecasts, forecast are conditional on information available at the end of the

middle month of each quarter (May and November respectively for 1st and 2nd semester). Finally, for

BLUE and BLUF, forecasts are conditional on information available at the end of the month prior to

the month of release (June and December respectively for 1st and 2nd semester). In what follows we

take into account the above calendar dates so that the analysis is based on forecasts which are made

conditional on the same set of information. However, the reader must keep in mind that the underlying

data has a rather complex structure. The Livingston forecasts are for semester data based on

information provided one month prior to release; forecast data of SPF, BLUE and BLUF are given by

quarters into the future although much of it is released monthly based on information available at the

end of the month prior to release. Estimation of excess returns will later be done on semester basis but

some background procedures, such as the construction of belief data, employ models based on

monthly or quarterly data. All this points out to the reader that the time unit in our empirical

investigation is rather complex and a perfect notation could be very cumbersome. We have thus

selected our notation with care but regret in advance if we repeat too often the explanation of the

symbols employed. The basic principle we adopted is that all data is monthly based but models may

be estimated using semester or quarterly dates. Hence the index t refers to months. 

We study forecasts of GDP growth rates, mainly two semesters ahead of date t, from date t +6

to date t+12. Hence, we adopt a specialized notation for this specific variable. We denote by the yt,2

annualized growth rate of GDP over an interval of six months, 2 semesters ahead between the end of

t+6 and t +12. Similarly, denotes the same variable h semesters into the future. In Table 2 we list yt,h

our notation, time spans and frequencies of forecast data and market states of belief about future GDP

growth which are extracted from the survey data outlined above. We use the notation and E ̄ J

t {yt,2}

Z to indicate, respectively, the J source average expected six-month real GDP growth rate 2 J

t

semesters ahead, which is from date t+6 to date t +12 and the corresponding extracted market state of
26

belief. Note that we omit from the symbols y and 2 since these are fixed throughout the paper. All ZJ

t

forecasts and extracted market states of beliefs are annualized. A description of the procedure for

extracting data on beliefs follows below. 

Table 2: Notation for Forecasts and Extracted Market States of Belief

Notation Time Span Frequency

Average Forecast Market State of Belief

E ̄ LIV

t {yt,2}

E ̄ SPF

t {yt,2}

E ̄ BLU

t {yt,2}

E ̄ SPF/BLU

t {yt,2}

ZLIV

t

ZSPF

t

Z BLU

t

ZSPF/BLU

t

1968:S2-2007:S2

1968:S2-2007:S2

1980:S1-2007:S2

1968:S2-2007:S2

Semi-annual

Semi-annual

Semi-annual

Semi-annual

In Table 2 the superscript BLU indicates data obtained by combining the BLUE and BLUF files. The

resulting BLU data consist at each date of the weighted average of the average forecasts from BLUE

and BLUF where the weights are given by the number of individuals participating in each survey. The

superscript SPF/BLU refers to a combined BLU and SPF forecasts where SPF data is added before

the beginning date of the BLU data in order to extend the time span to 1968:S2-2007:S2.

Place Figure 1 Here

Figure 1 traces graphs of the average forecasts listed in Table 2. The Figure reveals a high

correlation among the three sources. The one between and for the entire time E ̄ LIV

t {y t,2} E ̄ SPFt {y t,2}

span of 1968:S2-2007:S2 is 0.75 while the correlations among the three sources for the common

period of time 1980:S1-2007:S2 is as reported in Table 3.

Table 3: Correlation among Market Forecasts

Note: the table reports correlation for the common sample 1979:S2-2007:S2, while the full

sample correlation between LIV and SPF is reported in the text.

E ̄ LIV

t {yt,2} E ̄ SPF

t {yt,2} E ̄ BLUt {yt,2}

E ̄ LIV

t {yt,2}

E ̄ SPF

t {yt,2}

E ̄ BLU

t {yt,2}

 1.00

 0.64

 0.93

 1.00

 0.76 1.00

Strong similarities among market forecasts from LIV, SPF and BLU can also be seen from Table 4

where we report some of their descriptive statistics.
7 In this general section, the mean belief is denoted by whereas in the specific empirical applications below the Zy,h

t

variable y and the forecast horizon h are fixed. Hence, we ignore these and define with respect to the data source J only. Z J

t

27

 Table 4: Descriptive Statistics of Market Forecasts

Note: the first three columns in the table report statistics of annualized growth rates for 1968:S2-2007:S2, while the

last one covers the sample 1979:S2-2007:S2.

E ̄ LIV

t {yt,2} E ̄ SPFt {yt,2} E ̄ SPF/BLUt {yt,2} E ̄ BLUt {yt,2}

Mean

Median

Max

Min

Std. Dev.

Skewness

Kurtosis

Autocorrelation

N. Obs.

3.024

2.897

5.912

-0.015

1.133

0.426

3.580

0.631

79

2.977

3.000

5.560

-0.889

1.155

-0.262

4.186

0.693

79

3.098

3.083

5.560

0.440

1.113

0.264

2.855

0.673

79

2.938

2.912

5.300

1.295

0.804

0.538

3.372

0.665

57

2.2 Extracting Mean Market States of Belief

We now explain how mean market belief is extracted from forecast data. Here we ignore the

identification of data sources. Let be the annualized growth rate of real GDP known h semesters yt,h

ahead of date t, by agent i’s conditional forecast of , and by the forecast under E i

t {yt,h} yt,h E mt {yt,h}

the stationary probability m. Agent i’s state of belief at date t about is then defined by yt,h

 (29) . Zy,h,i

t ' E it {yt,h}&E mt {yt,h}

This does not render m-orthogonal to date t information but it removes from the effect Zy,h,i

t E it {yt,h}

of all state variables, as measured by . Note that is a E deviation from the stationary m

t {yt,h} Zy,h,it

forecast: when agent i has an unusually favorable view of future GDP growth but this does Zy,h,i

t >0

not mean he believes output will go up. He believes output will grow faster than usual where “usual”

is defined by growth rate expected under m. The market state of belief is defined by

(30a) 7 Zy,h

t ' 1

N j

N

i'1

[ E i

t {yt,h} & E mt {yt,h}] ' E ̄t

{yt,h} & E m

t {yt,h}

and the cross sectional variance of beliefs is

(30b) (σt

(y,h))

2' 1

Nj

N

i'1

[E i

t {yt,h}&E mt {yt,h}]&[E ̄t

{yt,h}&E m

t {yt,h}] 2

' 1

Nj

N

i'1

E i

t {yt,h}&E ̄t

{yt,h} 2.

Since is the mean forecast it reflects the market’s view on abnormal conditions h semesters E ̄t

{yt,h}

into the future. These conditions are the reason why the market forecasts and not : if E ̄t

{yt,h} Em

t {yt,h}

Z it is “optimistic” about unusually high GDP growth in t+h. y,h

t > 0

To measure we need data on the two components which define it. The LIV, SPF, BLUF Zy,h

t

and BLUE files provide direct data on and as discussed. The problem is then the E i

t {yt,h} E ̄t

{yt,h}
28

construction of the forecasts . In Kurz and Motolese (2011) we employed the Stock and E m

t {yt,h}

Watson’s (2002) method of diffusion indices to compute stationary forecasts of interest rates. 

However, when forecasting future growth rate of GDP such a method leads to highly volatile

stationary forecasts. It is well known that real GDP growth is difficult to forecast (see, for example,

Romer and Romer (2000)). Kurz (2005) estimates correlation between the Fed’s Green book forecasts

1965-1995 and the realized growth rates of GDP. The table below shows that forecast accuracy falls

sharply after a horizon of six months: 

 Accuracy of Fed’s GDP Growth Forecasts

Horizon (Quarters) R 2

0 0.53

1 0.25

2 0.18

3 0.08

4 0.16

Models with many independent variables tend to generate unstable out of sample forecasts (see Stock

and Watson (2002)) hence the practice is to forecast GDP growth with models employing a small

number of independent variables. To compute the stationary forecasts we thus opt for a E m

t {yt,h}

simple model which is estimated with monthly data with the following four categories of variables:

Leading indicator expressed by the non-farm-payroll; state of monetary policy expressed by the

Federal Funds rate; inflation expressed by the consumer price index and general persistence of growth

expressed by the moving average of past GDP growth rates.

The information used to compute should consist of variables that E correspond to m

t {yt,h}

those which are available in real time to forecasters at date t, when forecasts are made. This requires

us to check on the dates for data release. The Bureau of Economic Analysis produces GDP quarterly

and releases the data towards the end of the last month of the following quarter. For example when

agents forecast future growth at the end of June they know the GDP of the first quarter of the year,

hence they can compute all six-month growth rates and moving averages up to that quarter. As for the

NFP and CPI data, they are compiled monthly by the Bureau of Labor Statistics and made available

during the following month. Thus, at the time the forecasts are made agents know NFP and CPI data

lagged by one month. Furthermore, NFP data are revised twice after the initial release and undergo an

annual benchmark revision every June. We therefore compiled vintage real-time NFP data available at

the Federal Reserve Bank of Philadelphia and used the first release of NFP for month t-1 and the
8

 Though we forecast semi-annual data at the end of the 1st and 2nd semester, in order to benefit from the information

flow in between we estimate the parameters in (31) on a monthly basis and then pick the forecasts of interest.

29

revised value of NFP for month t-13 to compute the year over year growth rate. To sum up, the

stationary model for computing is estimated with the following variables: E m

t {yt,h}

 ̄y a 4 quarters moving average of GDP growth rates known at date t, as explained above; t

NFPt-1 the monthly year over year growth rate of Non-Farm-Payroll known at the forecast month t

which is the one dated one month prior to t;

CPIt-1 the monthly year over year rise of Consumer Price Index in the month before the one in which

the forecast is made;

Ft the average Federal Funds rate during the month at the end of which the forecast is made.

Real Time vs. A Single Estimate. Had our data set been long, the stationary forecast could E m

t {yt,2}

have been constructed from any single, fixed long time interval and it would be time invariant. Since

our data set is short and we study excess returns, we decided not to use the parameters of a single

model estimated for the entire period 1948:M2 to 2007:M128

. Instead, all estimates of and E m

t {yt,2}

Z are made by using real time forecasts. For each semester date t we use monthly data from y,2

t

1948:M2 up to t in order to re-estimate parameters of a stationary model with which (ˆαt,βˆt, ˆδ

t

, ˆγt, ˆηt)

we compute and deduce the value . That is, at each semester date t we compute the E m

t {yt,2} Zy,2t

stationary forecast for 2 semesters in the future with a model of the form

E m

t {yt,2} ' αˆt % βˆt

 ̄yt&1 % ˆδ

t

NFPt&1 % γˆtFt% ηˆtCPIt&1

and to estimate the equation we use only data for {t, t-1, t-2, ..., 1}. We lose 2 semester observations

(or 12 months) since at t = 1 we use data for t = 1 on the right and for t = 13 on the left. Parameters 

αˆ are estimated at date t with an OLS regression in which the date τ runs from 1 to t: t, βˆt, ˆδ

t

, ˆγt, ˆηt

(31) . yτ,2 ' αt % βt ̄yτ&1 % δtNFPτ&1 % γtFτ % ηtCPIτ&1 % g

t

τ%12

In (31) the parameters vary with t but as t rises they stabilize. It is clear our estimates for the early part

of the sample have an added variance which arises from the time variability of the estimates. 

Figure 2 Place Here

Figure 2 traces graphs of mean market beliefs listed in Table 2. The most important fact to

stress is that the time series in Figure 1 and those in Figure 2 are different: mean forecast data and mean

market belief data are different time series. This is an important since one contribution of this paper is
30

the case made by the theory we developed earlier which stresses that the primitive variables to be used

in the analysis are the time series in Figure 2, not those in Figure 1. Here we show they are different.

High correlation among the different measures is also seen here. Correlation between and for ZLIV

t ZSPFt

the entire span 1968:S2-2007:S2 is 0.94 while the correlations among the three market states of belief

for the common span of 1980:S1-2007:S2 is reported in Table 5.

Table 5: Correlation among Mean Market Belief

Note: the table reports correlation for the common sample 1979:S2-2007:S2, while the full

sample correlation between LIV and SPF is reported in the text.

ZLIV

t ZSPFt Z BLUt

ZLIV

t

ZSPF

t

Z BLU

t

1.00

0.92

0.96

1.00

0.95 1.00

In Table 6 we report some descriptive statistics of the data series of extracted market states of belief

from different sources which also show significant similarities.

Table 6: Descriptive Statistics of Mean Market Belief

Note: the first three columns in the table report statistics for the period 1968:S2-2007:S2, while the last one covers

the sample 1979:S2-2007:S2.

ZLIV

t ZSPFt ZSPF/BLUt Z BLUt

Mean

Median

Max

Min

Std. Dev.

Skewness

Kurtosis

Autocorrelation

N. Obs.

-0.631

-1.209

4.615

-3.748

1.759

0.635

2.753

0.839

79

-0.121

-0.576

 4.438

-2.924

 1.751

 0.561

 2.431

 0.814 

79

 0.010

-0.549

 5.837

-2.714

 1.867

 0.919

 3.524

 0.826

79

-0.488

-1.043

 5.837

-2.714

 1.883

 1.721

 5.734

 0.836

57

As we noted, are not necessarily orthogonal to date t macroeconomic variables. To understand their ZJ

t

behavior we report in Table 7 their correlation with the semiannual growth rate of GDP known at date

t, the Federal Fund Rate at date t and the year over year growth rate of Non Farm Payroll at date t-1. It

is noteworthy to observe that are correlated with the Federal Funds rate. The same observation is ZJ

t

made in Kurz and Motolese (2011) although is extracted there from interest rate forecasts. The ZJ

t

correlation of with suggests that one cause of belief diversity is associated with monetary policy ZJ

t Ft

and disagreement among market participants about the effect and future course of monetary policy. 

This is actually not surprising and we return to this issue later. 
31

Table 7: Correlation of Mean Market Belief with Macro Variables

Note: the first three columns in the table report correlation for the period 1968:S2-2007:S2, while the last one covers

the sample 1979:S2-2007:S2.

ZLIV

t ZSPFt ZSPF/BLUt Z BLUt

yt-1

Ft

NFPt -1

DPt

DEFt

TERMt

CAYt

 0.146

 0.699

 0.284

 0.667

 0.422

-0.447

-0.334

 0.244

 0.627

 0.352

 0.591

 0.345

-0.411

-0.330

 0.133

 0.719

 0.253

 0.640

 0.452

-0.440

-0.330

 0.040

 0.880

 0.156

 0.679

 0.642

-0.466

-0.223

2.3 The Excess Returns Data

Stock Market Excess Returns are computed employing the Center for Research and Security

Prices (CRSP) data. In particular we compute Rt,t+6, the annualized 1-semester (6-month) return on the

CRSP value weighted index net of the return on a 90 day Treasury Bill between date t and t+6, and

Rt,t+12, the annualized 2-semester (12-month) return on the CRSP value weighted index net of the return

on a 90 day Treasury Bill between date t and t+12. The Excess Returns are of semiannual frequency

with starting dates on January and July 1st covering the sample period of 1968:S2-2007:S2. The graphs

of excess returns are reported in Figure 3.

Figure 3 Place Here

2.4 Baseline Model’s Financial and Macro Predictors

Predictability of excess returns was studied by Fama and French (1989) from which a large

literature emanated and hence our reference model is set to their specification. The key variables Fama

and French (1989) found useful for forecasting excess returns are (i) the dividend yield, (ii) the default

premium in bond pricing and (iii) the term premium. They explained that these variables predict excess

returns because they are leading indicators for recessions and future business conditions. We employ

the specification of Campbell and Diebold (2009) who argued that instead of variables which forecast

future business conditions we should use survey forecast data. Indeed, they show that by using the

mean forecasts of the Livingston survey they are able to offer better forecast functions of excess returns

and these render some of the Fama and French (1989) variables insignificant. Following Campbell and
32

Diebold (2009) we use financial and macro variables in our baseline model relative to which we test the

effect of market belief. The financial predictors known at semester date t which we use are the

annualized percentages of the following:

• DPt

 – the dividend yield, calculated for the CRSP value-weighted portfolio; 

• DEFt

 – the default premium, calculated as the yield difference between a broad corporate bond

portfolio and the AAA yield; 

• TERMt

 – the term premium, calculated as the yield difference between a 10-year Treasury bond

and a one-month Treasury bill. 

In our baseline model we also us a single macro predictor:

• CAYt the Lettau and Ludvigson’s (2001) consumption wealth ratio. 

All the variables above are of semiannual frequency and cover the sample period 1968:S2-2007:S2. 

3. Testing the Time Variability of Stock Risk Premia: The Estimated Functions

For both the 1-semester and the 2-semester stock excess returns we estimate linear functions of

the following general form

(32) Rt,t%h ' b hMt% c hZt% d hσ

Z

t % gt%h

where Mt is a vector of financial and macro predictors DPt, DEFt, TERMt, CAYt. Zt is one of the

extracted market state of belief indexes discussed in Section 2.2 and is the corresponding cross σ

Z

t

section standard deviation of individual belief states at date t. We estimate (32) for the three different

sources of market belief data LIV, SPF and SPF/BLU. Our testing procedure will take several steps.

3.1 Testing the Effect of vs. mean forecast and CAYt (ZJ

t ,σZ,Jt ) E ̄ J

t {y t,2}

Campbell and Diebold (2009) showed has predictive power in explaining stock E ̄ LIV

t {yt,2}

market excess returns. Here we show that, as put forward in the theoretical part of our paper, the

primitive variable to consider is , not . We employ the procedure of estimating equation (32) and ZJ

t E ̄ J

t

testing the two alternative specifications and . In addition, as shown by Lettau and ZJ

t E ̄ J

t {yt,2}

Ludvigson (2001) the variable CAY is also constructed to be a proxy for market belief and hence we

need to examine it in relation to our measures . To facilitate comparison of coefficient (ZJ

t ,σ

Z

t )

magnitudes we standardize both returns as well as predictors to have zero mean and unit variance in all
33

regression estimates of (32). Furthermore, we compare the results with the basic estimates in which no

belief variable is included. We report parameter estimates of (32) in Table 8A-8B.

Table 8A: Rt,t+6 - Excess Returns: Mean Forecast vs. Mean Belief

Note: regressions (32) to explain 1-semester ahead Stock Market Excess Returns including the cross-sectional standard deviation of the market state of belief. * and † denote

significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard R errors are reported in parenthesis. The sample is 2

over 1968:S2-2007:S2.

(1) LIV SPF SPF/BLU

E ̄t

{yt,2}

Z J

t

σZ,J

t

DPt

DEFt

TERMt

CAYt

---

---

---

0.10 (0.14)

0.13 (0.12)

0.11 (0.12)

0.24 (0.12)†

-0.24 (0.13)*

---

---

 0.16 (0.17)

 0.22 (0.11)†

 0.19 (0.13)

 0.16 (0.14)

---

-0.59 (0.17)†

-0.04 (0.14)

 0.52 (0.19)†

 0.11 (0.09)

-0.02 (0.10)

 0.01 (0.13)

-0.17 (0.10)*

---

---

 0.17 (0.17)

 0.15 (0.11)

 0.18 (0.13)

 0.15 (0.12)

---

-0.38 (0.13)†

-0.13 (0.15)

 0.43 (0.20)†

 0.09 (0.10)

 0.03 (0.11)

 0.07 (0.12)

-0.29 (0.15)*

---

---

 0.14 (0.16)

 0.24 (0.12)†

 0.21 (0.12)*

 0.11 (0.12)

---

-0.52 (0.15)†

-0.12 (0.14)

 0.48 (0.19)†

 0.16 (0.08)*

-0.03 (0.11)

 0.03 (0.12)

R ̄ 2

(%) 7.94 10.50 18.27 8.85 13.32 11.57 17.15

Table 8B: Rt,t+12 - Excess Returns: Mean Forecast vs. Mean Belief

Note: regressions (32) to explain 2-semester ahead Stock Market Excess Returns including the cross-sectional standard deviation of the market state of belief. * and † denote

significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard R errors are reported in parenthesis. The sample is 2

over 1968:S2-2007:S2.

(1) LIV SPF SPF/BLU

E ̄t

{yt,2}

Z J

t

σZ,J

t

DPt

DEFt

TERMt

CAYt

---

---

---

0.20 (0.16)

0.01 (0.14)

0.20 (0.14)

0.26 (0.14)*

-0.14 (0.15)

---

---

 0.28 (0.18)

 0.07 (0.14)

 0.22 (0.15)

 0.22 (0.17)

---

-0.40 (0.17)†

-0.09 (0.15)

 0.52 (0.19)†

 0.01 (0.12)

 0.09 (0.12)

 0.11 (0.16)

-0.14 (0.12)

---

---

 0.26 (0.17)

 0.03 (0.13)

 0.24 (0.15)*

 0.19 (0.16)

---

-0.28 (0.14)†

-0.12 (0.14)

 0.46 (0.17)†

-0.01 (0.12)

 0.13 (0.13)

 0.13 (0.16)

-0.27 (0.16)*

---

---

 0.24 (0.16)

 0.12 (0.13)

 0.29 (0.14)†

 0.15 (0.17)

---

-0.46 (0.15)†

-0.11 (0.12)

 0.53 (0.18)†

 0.04 (0.11)

 0.07 (0.12)

 0.08 (0.16)

R ̄ 2

(%) 12.69 13.22 17.28 12.69 15.09 15.48 19.58

It is clear from Tables 8A-8B that, as reported by Campbell and Diebold (2009), the mean forecast

E ̄ does improve the predictability of returns but our belief variables are much superior J

t {yt,2} (ZJ

t ,σ

Z

t )

both in terms of statistical significance and in terms of adjusted . This result is true for all data R2

sources and for sub-samples not reported here. Tables 8A-8B also show that when we use our

measures the approximate CAY variable is never significant. This remains true for all sub- (ZJ

t ,σZ,Jt )

samples that we examined. Our conclusions are compatible with Motolese and Wu’s (2009) results

which show that for all sub-samples after 1980 the CAY variable fails to predict changes of wealth.
34

Table 9A: Rt,t+6 - Excess Returns: Mean Forecast vs. Mean Belief (excluding CAY)

Note: regressions (32) without variable CAY to explain 1-semester ahead Stock Market Excess Returns and including the cross-sectional standard deviation of the market state of

belief. * and † denote significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newe R y-West robust standard errors are reported in 2

parenthesis. The sample is over 1968:S2-2007:S2.

(1) LIV SPF SPF/BLU

E ̄t

{yt,2}

Z J

t

σZ,J

t

DPt

DEFt

TERMt

---

---

---

0.21 (0.14)

0.02 (0.13)

0.19 (0.11)*

-0.30 (0.11)†

---

-0.08 (0.15)

 0.22 (0.18)

 0.18 (0.11)†

 0.26 (0.11)

---

-0.60 (0.14)†

-0.03 (0.14)

 0.53 (0.19)†

 0.11 (0.09)

-0.02 (0.10)

-0.23 (0.09)†

---

-0.16 (0.18)

 0.24 (0.17)

 0.11 (0.12)

 0.24 (0.11)

---

-0.41 (0.12)†

-0.14 (0.15)

 0.48 (0.18)†

 0.07 (0.10)

 0.04 (0.11)

-0.33 (0.12)†

---

-0.13 (0.17)

 0.20 (0.15)

 0.22 (0.12)*

 0.26 (0.10)†

---

-0.54 (0.14)†

-0.13 (0.14)

 0.51 (0.18)†

 0.15 (0.09)*

-0.03 (0.11)

R ̄ 2

(%) 4.47 9.78 19.37 8.54 14.22 11.87 18.20

Table 9B: Rt,t+12 - Excess Returns: Mean Forecast vs. Mean Belief (excluding CAY)

Note: regressions (32) without variable CAY to explain 2-semester ahead Stock Market Excess Returns and including the cross-sectional standard deviation of the market state of

belief. * and † denote significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newe R y-West robust standard errors are reported in 2

parenthesis. The sample is over 1968:S2-2007:S2.

(1) LIV SPF SPF/BLU

E ̄t

{yt,2}

Z J

t

σZ,J

t

DPt

DEFt

TERMt

---

---

---

 0.32 (0.15)

-0.11 (0.14)

 0.28 (0.12)†

-0.22 (0.12)*

---

-0.12 (0.17)

 0.37 (0.18)†

 0.01 (0.15)

 0.32 (0.11)†

---

-0.48 (0.15)†

-0.08 (0.15)

 0.60 (0.16)†

-0.03 (0.12)

 0.10 (0.12)

-0.21 (0.10)†

---

-0.15 (0.17)

 0.35 (0.17)†

-0.03 (0.14)

 0.32 (0.12)†

---

-0.35 (0.12)†

-0.12 (0.15)

 0.56 (0.14)†

-0.07 (0.12)

 0.15 (0.12)

-0.32 (0.12)†

---

-0.12 (0.16)

 0.31 (0.16)*

 0.08 (0.15)

 0.34 (0.11)†

---

-0.50 (0.13)†

-0.12 (0.13)

 0.60 (0.14)†

 0.02 (0.12)

 0.08 (0.12)

R ̄ 2

(%) 8.20 10.88 17.67 11.43 15.21 15.12 20.25

Sharpening the comparison between the effect of mean forecast and mean market belief we ZJ

t

exclude in Tables 9A-9B the CAY variable and while controlling for all financial variables we see that

the effect of mean forecast is small compared to the contribution of . We shall thus focus in the rest ZJ

t

of this paper on the effect of . (ZJ

t ,σZ,Jt )

Turning to the effect of the cross sectional standard deviation we note first that only in the σZ,J

t

case of we have a genuine measure of its correspondent . In the case of we use the ZLIV

t σZ,Jt ZSPFt

cross-sectional standard deviation of the second quarter among the two consecutive quarterly forecasts

used to compute semi-annual data. Furthermore, we use as proxy for where the σZ,SPF

t σZ,SPF/BLUt

computation of its cross-sectional standard deviation is quite cumbersome due to the fact that BLU

data is the result of the weighted average of the average forecasts from BLUE and BLUF. Regardless

of these fine issues Tables 8A-8B and 9A-9B also show that the cross-sectional standard deviation of

market belief does not add anything to the estimates of (32). The parameter estimates of are ZJ

t σZ,Jt

consistently negative, which is the sign we find in the bond market model of Kurz and Motolese (2009).

However, contrary to the results for the bond market, these parameters are never significant. Our best
35

explanation is that the effect of is complex. We know from the bond market model that for short σZ,J

t

investment horizons the sign of the parameter σ is either insignificant or is positive. It becomes Z,J

t

negative and very significant for investment horizons longer than 6-9 months. Hence it is not entirely

surprising that in the stock market model under consideration here these parameters are not statistically

significant for investment horizons of 6-12 months. 

3.2 Testing the Effect of other variable vs. (ZJ

t ,σZ,Jt )

We now turn to the correlation between the belief variables (Z and other variables as J

t ,σZ,Jt )

reported in Table 7 and test how robust the simple model (32) is. Our question is simple: is the effect of

(Z a reflection of other variables correlated with or does it represent an independent J

t ,σZ,Jt ) (ZJt ,σZ,Jt )

effect of belief? We study the variables DPt, DEFt, TERMt, NFPt-1, and Ft by using a baseline model and

then adding each of these variables one at a time. We also introduce them in pairs. The test is

conducted over the three data sources we have and for the two investment horizons of 6 and 12

months. The results are reported in the sequence of Tables 10A-10F. We report first the results for the

Livingston survey then for SPF and finally for the combination SPF\BLU.

To sum up our key result we find that although the parameters of change due to (ZJ

t ,σZ,Jt )

correlation, the conclusion remains the same and points to an independent effect of belief, represented

by , which is not explained by other variables. Moreover, the order of magnitude of the effect (ZJ

t ,σZ,Jt )

of remains essentially the same regardless of data source or investment period. (ZJ

t ,σZ,Jt )

Table 10A: Rt,t+6 - Stock Excess Returns (Livingston)

Note: testing the contribution of in regressions (33) to explain 1-semester ahead Stock Market Excess Returns by adding correla Z ted macro variables. * and † denote significance J

t

respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard errors are rep R orted in parenthesis. The sample is over 1968:S2- 2

2007:S2.

Livingston Zt

Zt

σt

DPt

Ft

NFPt-1

DEFt

TERMt

-0.56 (0.15)†

-0.03 (0.13)

 0.62 (0.19)†

-0.07 (0.13)

---

---

---

-0.57 (0.13)†

-0.03 (0.13)

 0.58 (0.18)†

---

-0.05 (0.09)

---

---

-0.59 (0.14)†

-0.03 (0.13)

 0.53 (0.19)†

---

---

 0.10 (0.10)

---

-0.58 (0.14)†

-0.03 (0.14)

 0.59 (0.18)†

---

---

---

 0.01 (0.10)

-0.54 (0.14)†

-0.03 (0.13)

 0.60 (0.18)†

-0.06 (0.12)

-0.04 (0.09)

---

---

-0.60 (0.13)†

-0.03 (0.13)

 0.52 (0.21)†

---

 0.02 (0.15)

 0.12 (0.17)

---

-0.57 (0.14)†

-0.03 (0.13)

 0.55 (0.21)†

-0.08 (0.11)

 0.04 (0.14)

 0.14 (0.17)

---

-0.61 (0.13)†

-0.03 (0.14)

 0.52 (0.21)†

---

 0.02 (0.15)

 0.13 (0.16)

-0.02 (0.10)

-0.59 (0.13)†

-0.03 (0.14)

 0.59 (0.22)†

-0.21 (0.16)

 0.05 (0.14)

 0.20 (0.16)

-0.13 (0.15)

R ̄ 2

(%) 19.90 19.91 20.41 19.71 19.00 19.36 18.54 18.29 18.05
36

Table 10B: Rt,t+12 - Stock Excess Returns (Livingston)

Note: testing the contribution of in regressions (33) to explain 2-semester ahead Stock Market Excess Returns by adding correla Z ted macro variables. * and † denote significance J

t

respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard errors are rep R orted in parenthesis. The sample is over 1968:S2- 2

2007:S2.

Livingston Zt

Zt

σt

DPt

Ft

NFPt-1

DEFt

TERMt

-0.50 (0.16)†

-0.08 (0.15)

 0.63 (0.17)†

-0.08 (0.14)

---

---

---

-0.52 (0.14)†

-0.08 (0.15)

 0.59 (0.15)†

---

-0.02 (0.12)

---

---

-0.53 (0.15)†

-0.08 (0.15)

 0.59 (0.16)†

---

---

 0.01 (0.12)

---

-0.48 (0.15)†

-0.08 (0.15)

 0.58 (0.15)†

---

---

---

 0.09 (0.11)

-0.49 (0.15)†

-0.08 (0.15)

 0.63 (0.17)†

-0.08 (0.14)

-0.02 (0.12)

---

---

-0.52 (0.14)†

-0.08 (0.15)

 0.60 (0.18)†

---

-0.04 (0.20)

-0.02 (0.20)

---

-0.49 (0.15)†

-0.08 (0.15)

 0.63 (0.18)†

-0.08 (0.15)

-0.02 (0.20)

-0.01 (0.21)

---

-0.47 (0.14)†

-0.08 (0.15)

 0.61 (0.18)†

---

-0.02 (0.19)

-0.05 (0.19)

 0.10 (0.12)

-0.47 (0.14)†

-0.08 (0.15)

 0.59 (0.19)†

 0.04 (0.23)

-0.03 (0.19)

-0.06 (0.21)

 0.12 (0.19)

R ̄ 2

(%) 18.28 18.03 17.99 18.71 17.20 16.94 16.07 16.57 15.44

Table 10C: Rt,t+6 - Stock Excess Returns (SPF)

Note: testing the contribution of in regressions (33) to explain 1-semester ahead Stock Market Excess Returns by adding correla Z ted macro variables. * and † denote significance J

t

respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard errors are rep R orted in parenthesis. The sample is over 1968:S2- 2

2007:S2.

SPF Zt

Zt

σt

DPt

Ft

NFPt-1

DEFt

TERMt

-0.37 (0.11)†

-0.15 (0.15)

 0.61 (0.19)†

-0.17 (0.12)

---

---

---

-0.40 (0.13)†

-0.15 (0.14)

 0.52 (0.17)†

---

-0.06 (0.10)

---

---

-0.43 (0.12)†

-0.15 (0.15)

 0.48 (0.18)†

---

---

 0.08 (0.10)

---

-0.40 (0.12)†

-0.13 (0.15)

 0.52 (0.17)†

---

---

---

 0.06 (0.11)

-0.35 (0.11)†

-0.16 (0.14)

 0.60 (0.19)†

-0.16 (0.12)

-0.04 (0.10)

---

---

-0.42 (0.13)†

-0.15 (0.15)

 0.48 (0.20)†

---

-0.02 (0.17)

 0.06 (0.17)

---

-0.37 (0.11)†

-0.16 (0.15)

 0.56 (0.21)†

-0.18 (0.11)

 0.02 (0.16)

 0.10 (0.18)

---

-0.41 (0.13)†

-0.14 (0.15)

 0.48 (0.20)†

---

-0.01 (0.17)

 0.06 (0.18)

 0.04 (0.11)

-0.38 (0.11)†

-0.18 (0.17)

 0.61 (0.24)†

-0.32 (0.19)

 0.02 (0.17)

 0.16 (0.19)

-0.14 (0.18)

R ̄ 2

(%) 16.07 15.09 15.23 15.11 15.12 14.10 14.31 13.05 13.80

Table 10D: Rt,t+12 - Stock Excess Returns (SPF)

Note: testing the contribution of in regressions (33) to explain 2-semester ahead Stock Market Excess Returns by adding correla Z ted macro variables. * and † denote significance J

t

respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard errors are rep R orted in parenthesis. The sample is over 1968:S2- 2

2007:S2.

SPF Zt

Zt

σt

DPt

Ft

NFPt-1

DEFt

TERMt

-0.35 (0.13)†

-0.17 (0.14)

 0.63 (0.17)†

-0.17 (0.14)

---

---

---

-0.40 (0.13)†

-0.17 (0.15)

 0.53 (0.13)†

---

-0.03 (0.13)

---

---

-0.41 (0.13)†

-0.17 (0.15)

 0.54 (0.14)†

---

---

 0.01 (0.12)

---

-0.37 (0.12)†

-0.15 (0.14)

 0.52 (0.13)†

---

---

---

 0.13 (0.11)

-0.34 (0.12)†

-0.18 (0.15)

 0.62 (0.16)†

-0.17 (0.15)

-0.02 (0.13)

---

---

-0.38 (0.12)†

-0.17 (0.15)

 0.58 (0.18)†

---

-0.07 (0.21)

-0.07 (0.20)

---

-0.34 (0.12)†

-0.18 (0.15)

 0.64 (0.19)†

-0.16 (0.15)

-0.04 (0.21)

-0.04 (0.21)

---

-0.33 (0.11)†

-0.15 (0.15)

 0.58 (0.18)†

---

-0.05 (0.21)

-0.10 (0.20)

 0.14 (0.13)

-0.33 (0.12)†

-0.15 (0.15)

 0.60 (0.21)†

-0.05 (0.25)

-0.04 (0.21)

-0.09 (0.22)

 0.11 (0.21)

R ̄ 2

(%) 15.97 14.72 14.66 16.08 14.86 13.73 13.75 14.15 13.00

As we have seen before, the accuracy of SPF data seems to be lower than that of Livingston or

BLU and we associate it with the fact that SPF forecasting dates have been less precise as to timing and

this introduces an error for which we cannot control.
37

Table 10E: Rt,t+6 - Stock Excess Returns (SPF/BLU)

Note: testing the contribution of in regressions (33) to explain 1-semester ahead Stock Market Excess Returns by adding correla Z ted macro variables. * and † denote significance J

t

respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard errors are rep R orted in parenthesis. The sample is over 1968:S2- 2

2007:S2.

SPF/BLU Zt

Zt

σt

DPt

Ft

NFPt-1

DEFt

TERMt

-0.47 (0.14)†

-0.13 (0.13)

 0.61 (0.19)†

-0.07 (0.13)

---

---

---

-0.48 (0.12)†

-0.13 (0.13)

 0.58 (0.17)†

---

-0.08 (0.09)

---

---

-0.52 (0.12)†

-0.13 (0.13)

 0.51 (0.18)†

---

---

 0.14 (0.09)

---

-0.50 (0.13)†

-0.12 (0.14)

 0.58 (0.18)†

---

---

---

 0.02 (0.11)

-0.45 (0.14)†

-0.14 (0.13)

 0.60 (0.18)†

-0.06 (0.12)

-0.08 (0.09)

---

---

-0.52 (0.12)†

-0.13 (0.13)

 0.51 (0.20)†

---

-0.00 (0.16)

 0.14 (0.17)

---

-0.49 (0.14)†

-0.13 (0.13)

 0.53 (0.20)†

-0.07 (0.11)

 0.01 (0.15)

 0.15 (0.17)

---

-0.53 (0.14)†

-0.13 (0.14)

 0.51 (0.20)†

---

-0.00 (0.16)

 0.15 (0.17)

-0.03 (0.11)

-0.49 (0.13)†

-0.16 (0.16)

 0.59 (0.22)†

-0.22 (0.18)

 0.01 (0.16)

 0.21 (0.18)

-0.15 (0.17)

R ̄ 2

(%) 18.14 18.60 19.24 17.98 17.66 18.15 17.24 17.07 16.84

Table 10F: Rt,t+12 - Stock Excess Returns (SPF/BLU)

Note: testing the contribution of in regressions (33) to explain 2-semester ahead Stock Market Excess Returns by adding correla Z ted macro variables. * and † denote significance J

t

respectively at the 10% and 5% level. All are adjusted for degrees of freedom and Newey-West robust standard errors are rep R orted in parenthesis. The sample is over 1968:S2- 2

2007:S2.

SPF/BLU Zt

Zt

σt

DPt

Ft

NFPt-1

DEFt

TERMt

-0.50 (0.13)†

-0.14 (0.13)

 0.65 (0.16)†

-0.08 (0.14)

---

---

---

-0.52 (0.11)†

-0.14 (0.12)

 0.62 (0.14)†

---

-0.04 (0.12)

---

---

-0.54 (0.12)†

-0.13 (0.13)

 0.59 (0.14)†

---

---

 0.05 (0.11)

---

-0.50 (0.12)†

-0.12 (0.13)

 0.60 (0.14)†

---

---

---

 0.08 (0.11)

-0.49 (0.13)†

-0.14 (0.13)

 0.64 (0.15)†

-0.07 (0.14)

-0.04 (0.12)

---

---

-0.53 (0.12)†

-0.14 (0.13)

 0.60 (0.18)†

---

-0.03 (0.20)

 0.02 (0.20)

---

-0.50 (0.13)†

-0.14 (0.13)

 0.63 (0.18)†

-0.08 (0.14)

-0.02 (0.20)

 0.03 (0.21)

---

-0.50 (0.12)†

-0.13 (0.13)

 0.61 (0.18)†

---

-0.02 (0.20)

-0.00 (0.20)

 0.07 (0.13)

-0.49 (0.13)†

-0.13 (0.14)

 0.61 (0.20)†

-0.00 (0.25)

-0.02 (0.20)

 0.00 (0.21)

 0.07 (0.22)

R ̄ 2

(%) 20.99 20.93 20.90 21.29 20.08 19.88 19.01 19.18 18.05

We now address the question of effect of the two Fama-French financial variables DEFt, TERMt

which do not seem to exhibit statistical significance. A careful examination of Fama and French (1989)

reveals that the effects of the variables DEFt

 and TERMt are significant only for longer investment

period of over a year, and mostly up to four years. Hence, we do not find it surprising that in the

shorter investment period of 6-12 months the standard errors of the estimates are too high to judge

their actual contribution to explaining risk premia. We thus do not draw any conclusions about the

efficacy of these variables in models with longer investment periods. 

3.3 Interaction of Z with DP

Table 7 reveals a correlation between and which we examine now. The hypothesis is ZJ

t DPt

that the effect of belief is not well approximated by a liner term since it may depend on or . ZJ

t DPt

To that end we first study the following model with the product variables ZJ

t ×DPt
38

(33) . Rt,t%h ' b hZJ

t % c h

σt% d hDPt% e h(ZJ

t ×DPt

) % gt%h

The results are reported in Tables 11A-11B and show a drastic increase in the standard errors of the

estimates due to the correlation between and . ZJ

t ZJt ×DPt

Table 11A: Rt,t+6 - Stock Market Excess Returns -- Interactions

Note: regressions (33) to explain 1-semester ahead Stock Market Excess Returns. * and † denote

significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom R2

and Newey-West robust standard errors are reported in parenthesis. The sample is over 1968:S2-

2007:S2.

LIV SPF SPF/BLU

Z J

t

σJ

t

 DPt

(Z J

t )

2

×DPt

-0.75 (0.48)

-0.03 (0.13)

 0.62 (0.21)†

 0.15 (0.40)

-0.30 (0.38)

-0.15 (0.15)

 0.52 (0.18)†

-0.13 (0.35)

-0.48 (0.39)

-0.12 (0.13)

 0.58 (0.18)†

-0.02 (0.35)

R ̄ 2

(%) 19.89 14.95 17.93

Table 11B: Rt,t+12 - Stock Market Excess Returns -- Interactions

Note: regressions (34) to explain 2-semester ahead Stock Market Excess Returns. * and † denote

significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom R2

and Newey-West robust standard errors are reported in parenthesis. The sample is over 1968:S2-

2007:S2.

LIV SPF SPF/BLU

Z J

t

σJ

t

DPt

(Z J

t )

2

×DPt

-1.18 (0.43)†

-0.08 (0.16)

 0.71 (0.16)†

 0.61 (0.38)

-0.77 (0.47)*

-0.15 (0.15)

 0.56 (0.14)†

 0.36 (0.47)

-1.00 (0.43)†

-0.12 (0.12)

 0.63 (0.14)†

 0.45 (0.42)

R ̄ 2

(%) 21.02 15.64 22.07

An alternative specification of the effect of is expressed in the model ZJ

t

(34) Rt,t%h ' b hZJ

t % c h

σt% d hDPt% e h((ZJ

t )

2

×DPt) % gt%h

and the results are reported in Tables 12A-12B. In these tables the standard errors remain small and the

non linear effect is significant.

Table 12A: Rt,t+6 - Stock Market Excess Returns (equation (34a))

Note: regressions (34) to explain 1-semester ahead Stock Market Excess Returns. * and † denote

significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom R2

and Newey-West robust standard errors are reported in parenthesis. The sample is over 1968:S2-

2007:S2.

LIV SPF SPF/BLU

Z J

t

σJ

t

 DPt

(Z J

t )

2

×DPt

-0.54 (0.13)†

-0.03 (0.13)

 0.59 (0.17)†

-0.13 (0.07)*

-0.29 (0.10)†

-0.18 (0.16)

 0.54 (0.17)†

-0.22 (0.09)†

-0.37 (0.14)†

-0.16 (0.14)

 0.57 (0.17)†

-0.16 (0.10)*

R ̄ 2

(%) 21.29 18.16 19.36
39

Table 12B: Rt,t+12 - Stock Market Excess Returns (equation (34a))

Note: regressions (34) to explain 2-semester ahead Stock Market Excess Returns. * and † denote

significance respectively at the 10% and 5% level. All are adjusted for degrees of freedom R2

and Newey-West robust standard errors are reported in parenthesis. The sample is over 1968:S2-

2007:S2.

LIV SPF SPF/BLU

Z J

t

σJ

t

DPt

(Z J

t )

2

×DPt

-0.52 (0.16)†

-0.08 (0.15)

 0.60 (0.15)†

-0.04 (0.10)

-0.33 (0.14)†

-0.18 (0.15)

 0.55 (0.14)†

-0.13 (0.13)

-0.46 (0.16)†

-0.15 (0.13)

 0.61 (0.14)†

-0.09 (0.11)

R ̄ 2

(%) 18.12 15.71 21.20

4. Effect of Mean Market Belief on Aggregate Consumption

In a paper which used the same belief data we used here, Motolese and Wu (2009) re- examined

the effect of the CAY variable (consumption\wealth ratio) proposed by Lattau and Ludvigson (2001) as

a proxy for market belief in order to explain the equity risk premium. They found that given the belief

variables, CAY is insignificant. This is not surprising in light of the results we presented above. 

Motolese and Wu (2009) also estimated the cointegrated VAR model of Lattau and Ludvigson (2001)

in which the growth rate of consumption and wealth are deduced from log linearization of the

optimizing agents budget constraints. A simple two variable version of their model is as follows 

(35a) Îat ' α0 % α1Îat&1 % α2Zt

% α3σ

Z

t % α4Ft

% εt

(35b) Îct ' β0 % β1Îct&1 % β2Îat&1 % β3Zt% β4σ

Z

t % β5Ft

% εt

where

Îa – annual growth rate of assets in quarter preceding semester date t t

Îc – annual growth rate of consumption in quarter preceding semester date t t

F – Fed funds rate at semester date t. t

(Z – measures of market belief. t , σ

Z

t )

In estimating (35a)-(35b) we used the data provided by Lattau and Ludvigson (2001) but the

definition of wealth we use here does not incorporate their estimated values of human capital. In

estimating these equations we have not standardized the variables and this facilitates the interpretation

of the parameters. Table 8 reports the estimated coefficients of (35a)-(35b) using the earlier semester

data 1968:S2-2007S2.
40

Table 8: VAR Model for Asset and Consumption Growth

Note: Estimated parameters of (35a) -(35b) to explain asset and consumption growth rates. * and † denote significance 

respectively at 10% and 5% levels. All are adjusted for degrees of freedom and Newey-West robust standard errors are in R2

parenthesis. The sample is over 1968:S2-2007:S2.

LIV SPF/BLU

 aat act aat act

Constant

aat&1

act&1

Z J

t

σZ,J

t

Ft 

 0.040 (0.012)†

 -0.094 (0.151)

 ----

 0.366 (0.172)†

- 0.622 (0.396)

- 0.211 (0.095)†

 0.008 (0.002)†

 0.313 (0.119)†

 0.051 (0.022)†

 0.103 (0.043)†

 -0.149 (0.071)†

-0.011 (0.021)

 0.028 (0.006)†

-0.095 (0.138)

 ----

 0.238 (0.133)*

- 0.074 (0.151)

- 0.257 (0.091)†

 0.007 (0.002)†

 0.228 (0.123)*

 0.056 (0.020)†

0.118 (0.035)†

-0.060 (0.036)*

-0.035 (0.018)*

R ̄ 9.16 26.17 4.47 25.42 2(%)

 It is seen the fit of the consumption equation is much better than of wealth, a well known result.

We also find that both mean market belief as well as the cross sectional standard deviation contribute

significantly to the explanation of consumption growth. Indeed, since all variables are in terms of

annualized growth rates, 1 percentage change in Z causes a 0.1 percentage change in the growth rate

of consumption. As seen in Table 6, a change of two standard deviations in Z would then cause a 0.35

percentage point change in the growth rate of consumption . The direction of change is also as one

should expect from the theory: consumer expectations of abnormally high growth rate of income in the

future would lead to an increased consumption. On the other hand an increase in dispersion of belief

causes a decrease in consumption. Finally, the mean market belief have a significant effect on the

growth rate of wealth and the effect is also in the direction predicted by the theory: a rise in the mean

market belief about better future business conditions increases the present prices of assets and the

growth rate of wealth. 

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APPENDIX A: Derivation of the Value Function

For simplicity we ignore in this Appendix the index i identifying the agent who carries out the optimization.

Hence, the dynamic programming problem is as follows. Given initial values , maximize (θ0 , W0 )

Ut ' Et

(θ , C)

[j

4

s ' 0

&βt%s&1e

&( 1

τ

Ct%s)

| Ht]

subject to the following definitions

Wt%1 ' (Wt & Ct

)R % θtQt%1

Qt%1 ' pt%1 % (dt%1 % μ) & ptR

ψt ' (1 , dt , zt , gt)

and stochastic transition functions
44

dt%1 ' λd dt % λ

d

g gt % g

d

t%1

Z , , . t%1 ' λzZt % λ

z

ggt % g

z

t%1 Λ

ψ '

1,0,0,0

0 , λd ,0, λ

d

g

0 ,0 , λz , λ

z

g

0 ,0 , 0 , λz

ˆ

gt'(1, g

d

t , g

z

t , g

g

t ), (g

d

t , g

z

t , g

g

t ) - N(0 , Σ)

gt%1 ' λzgt % g

g

t%1

Step 1: simplification. We thus define, for the unknown matrix V 

 , Λ '

λd ,0, λ

d

g

0 , λz , λ

z

g

0,0, λz

V '

v00 , v01 , v02 , v03

v01

v02 V11

v03

' v00 , ˆv T

0

vˆ0 , V11

We now have , where is a 4×4 matrix ψt%1 ' Λψ ψt % Λg

ˆ

gt%1 Λg '

0,0

0,I(3×3)

We assume that and verify it later when we solve for equilibrium. Using this price map pt ' ad dt % az zt % P0

we can compute excess return in terms of the state variables we have that 

Qt%1'(ad % 1)[λd dt%λ

d

g gt

%g

d

t%1]%az [λzZt

%λ

z

ggt%g

z

t%1]%P0 & [ad dt

%azZt%P0]R % μ

Hence

Qt%1'[(ad % 1) λd&Rad]dt % [azλz&Raz

]Zt%[(ad%1)λ

d

g % azλ

z

g]gt

%[P0 (1&R)%μ]%[(ad%1)g

d

t%1%az g

d

t%1]

Or, 

Qt%1 ' a T

ψt % bˆ Tgt%1 , hence Et[Qt%1] ' a Tψt

where

a . T'([P0 (1&R)%μ] , [(ad % 1) λd&Rad] , [azλz&Raz

] , [(ad%1) λ

d

g % azλ

z

g]) , bˆ T

'(0,(ad%1),az ,0)

Also, we shall use the notation . Now compute the expression b T'((ad%1),az ,0)

&αWt%1& 1

2

ψT

t%1Vψt%1'&α(Wt

&Ct)R&αθt[a Tψt%bˆ T ˆgt%1]& 1

2

ψT

t ΛT

ψVΛψψt

&ψT

t ΛT

ψVΛg

ˆ

gt%1& 1

2

ˆ

g

T

t%1ΛT

gVΛg

ˆ

gt%1

Algebra and simplification leads to the conclusion that we have 

&αWt%1& 1

2

ψT

t%1Vψt%1' &At & e Tt gt%1 & 1

2

g

T

t%1V11 gt%1

where 

At ' α(Wt&Ct)R % αθt a T

ψt % 1

2

ψT

t ΛT

ψVΛψψt

e (this is a 3 vector) where (3x4) matrix, T

t ' [αθt

b T % ψT

t ΛT0] ΛT

0' vˆ T

0

ΛTV11

Λ0' v0 , V11Λ

Step 2: The Bellman Equation. It is well known (see, for example, the Appendix of Wang (1994)) that the Bellman

Equation for this problem with is written in the form γ ' 1

τ

Jt ' Max for some parameter matrix V

(θt , Ct)

[&βt&1exp &γCt & βtEtexp &At & e T

t gt%1 & 1

2

g

T

t%1V11 gt%1

But we know that

E . t

exp &At & e T

t gt%1 & 1

2

g

T

t%1V11 gt%1 ' |1%ΣV11|

& 1

2 exp[ 1

2

e T

t (I % ΣV11 )&1

Σet & At]

Also 

 1

2

e T

t (1%ΣV)&1

Σet' 1

2

[αθt

b T%ψT

t ΛT0]

T

(I%ΣV11 )

&1

Σ[αθtb%Λ0ψt]

 ' . 1

2

α2 θ

2

t b TΩb%αθt

b TΩΛ0ψt% 1

2

ψT

t ΛT0ΩΛ0ψt , where Ω ' (I%ΣV11 )

&1

Σ
45

Hence, we have an expression for the expectations

. 1

2

e T

t (1 % ΣV11 )

&1

Σet&At ' &α(Wt&Ct)R&αθt[a T&b TΩΛ0]ψt% 1

2

α2 θ

2

t b TΩb& 1

2

ψT

t [ΛT

ψVΛψ&ΛT

0ΩΛ0]ψt

The first order conditions are then stated as follows. Equating the derivative with respect to θ to zero leads to 

& α[a T&b TΩΛ0]ψt % α2θtb TΩb ' 0

And this proves equation (11) in the text which we can write in the more explicit form (since ) Et

[Qt%1] ' a Tψt

θ . t ' 1

αb TΩb

[a T&b TΩΛ0]ψt / 1

αb TΩb

[Et(Qt%1)%u Tψt , u T ' &b TΩΛ0

This last equation determines the parameter vector u. It also shows that this vector is the same for all agents since the

assumption made in the text is that all agents are identically the same except for their belief states . The last equation gt

shows that the vector u depends only upon parameters of the stochastic structure. 

Step 3: The Adjusted Variance and Constants. We can also explain the “adjustment” to the variance in (11) since 

 ˆσ

2

Q ' b TΩb

which is the variance of the excess return function where the covariance matrix used is not G but rather Ω. 

We now have 

α . 2θ

2

t b TΩb ' 1

b TΩb

ψT

t [a T&b T

ΩΛ0]

T[a T&b T

ΩΛ0]ψt

Hence the optimized value of the exponent is simply

1

2

e T

t (1%ΣV11 )&1

Σet&At'&α(Wt&Ct)R & 1

2

ψT

t Mψt

Where

 . M ' 1

b T

Ωb

[a T

&b TΩΛ0]

T[a T

&b TΩΛ0]%[ΛT

ψVΛψ&ΛT

0ΩΛ0]

Now take the derivative with respect to C and equate to zero to obtain

γexp &γC . t ' αRβ|1%ΣV11|

& 1

2 exp &α(Wt

&Ct)R & 1

2

ψT

t Mψt , let G ' |1%ΣV11|

& 1

2

Hence the solution for C must satisfy (with log being the logarithm to base e)

γCt ' &log[ βαRG

γ

] % α(Wt&Ct)R % 1

2

ψT

t Mψt

hence we finally have

C . t ' & 1

γ % αR

log[ βαRG

γ

] % αR

γ % αR

Wt % 1

2(γ % αR) ψT

t Mψt

The final details of showing that the value function is indeed the solution of the Bellman Equation requires the insertion

of the optimal solutions into the Bellman Equation and deducing the unknown parameters. Doing this leads to the

following conclusion. First define the term . Then it is demonstration that the unknown parameter Gˆ ' 1

γR

log(rβG)

α and matrix V are determined by the conditions 

(i) . α ' r γ

R

(ii) M

R & V % 2[ γGˆ % log( rR

)]i11 ' 0

where is a 4×4 matrix with the (1,1) element being 1 and all others being 0. i

11
46

APPENDIX B: Proof of Theorem 1 

Proof : Using Assumption A we combine the two sources to have that 

b t

(dt,Ψi

t

) ' μ bt&1(dt) % (1 & μ )B i

t

with a mean of E i

t (bt

|dt,Ψi

t

) ' μE i

t (bt&1 |dt) % (1&μ)Ψit 0 < μ < 1

and conditional variance . Var(bt |dt,Ψi

t

) ' μ2

(α % β)

% (1&μ)

2

γ

Let and we write the precision of the distribution of this new posterior as ζ' 1

μ2

and ξ' 1

(1&μ)

2

Precision(bt |dt

,Ψi

t

)'Γ(bt)'1/[ 1

ζ(α%β)

% 1

ξγ ] ' ζ(α % β)ξγ

ζ(α % β)%ξγ

At date t+1 the agent observes . By (7) in the text it follows that updating the agent has dt%1 E i

t (bt

|dt,Ψi

t

)

E i

t%1(bt

|dt%1 ,Ψi

t)' Γ(bt

)E i

t (bt

|dt ,Ψi

t

)%β[dt%1&λd dt]

Γ(bt)%β , bt

(dt%1 ,Ψi

t) - N[E it (bt

|dt%1 ,Ψi

t), 1

Γ(bt)%β

].

After assessing the mean he formulates the new posterior which is Ψi

t%1

bt%1(dt%1 ,Ψi

t%1) ' μbt

(dt%1 ,Ψi

t) % (1&μ)B it%1

with mean

(B.1) E . i

t%1 (bt%1 |dt%1 ,Ψit%1) ' μE it%1 (bt |dt%1 ,Ψit) % (1&μ)Ψit%1 0 < μ < 1

conditional variance

(B.2) . Var(bt%1 |dt%1 ,Ψi

t%1) ' 1

ζ(Γ(bt) % β)

% 1

ξ γ

and precision

(B.3) . Γ(bt%1 ) ' 1/[ 1

ζ(Γ(bt) % β)

% 1

ξ γ] ' ζ (Γ(bt

) % β)ξγ

ζ (Γ(bt

) % β)%ξγ

We can now deduce the full symmetry of the process. For large t we then have

bt

(dt%1 ,Ψi

t

)-N[E i

t (bt

|dt%1,Ψi

t), 1

Γ(bt)%β

]

E . i

t (bt

|dt%1 ,Ψi

t

) ' Γ(bt

)E i

t (bt

|dt ,Ψi

t

)%β[dt%1&λddt]

Γ(bt)%β

After observing the new posterior is Ψi

t%1

b . t%1(dt%1 ,Ψi

t%1) ' μbt(dt%1 ,Ψit) % (1&μ)B it%1

The mean, conditional variance and precision are then as in (B1), (B.2) and (B.3) and hence we have an equation for the

precision 

 . Γt%1 ' ζ (Γt % β)ξγ

ζ (Γt % β)%ξγ

It is well defined for (i.e. ) and in that case it has the unique positive solution 1 < ζ < 4 0 < μ < 1

 , . ζ ' 1

μ2

and ξ ' 1

(1&μ )

2

Γ( '

(β%ξ γ(1& 1

ζ

))% (β%ξ γ(1& 1

ζ

))2%4β

2
47

The negative root has no economic meaning. When there is no solution, and diverges for large t, which ζ ' 1 , ξ ' 4 Γt

is the classical case. With we rewrite the equations above as Γ ' Γ(

 E i

t%1 (bt%1 |dt%1 ,Ψit%1) ' μE it%1 (bt |dt%1 ,Ψt

) % (1&μ)Ψi

t%1

 ' μ

Γ(E i

t (bt

|dt , Ψi

t

) % β[dt%1 & λd dt]

Γ( % β % (1&μ)Ψi

t%1

Hence,

(B.4) E i

t%1(bt%1 |dt%1,Ψit%1)& μβ

Γ(%β

dt%1' μΓ(

Γ(%β

[E i

t (bt

|dt,Ψi

t

)& μβ

Γ(%β

dt]% μβ

Γ(%β

[ μΓ(

Γ(%β

&λd

]dt%(1&μ)Ψi

t%1.

If we now define

(B.5) , , , , g

i

t%1'E it%1 (bt%1 |dt%1 ,Ψit%1)& μ β

Γ( %β

dt%1 kig

t%1 ' (1&μ)Ψit%1 λZ ' μ Γ(

Γ( %β

λ

d

Z' μ β

Γ(%β

[ μ Γ(

Γ(%β

&λd ]

then (B.4) is exactly (6b): g  i

t%1 ' λZ g

i

t % λ

d

Z dt % kigt%1
1

Figure 1: the 12-month-ahead forecasts from LIV, SPF and BLU

Figure 2: the 12-month-ahead Market States of Belief from LIV, SPF and BLU
2

Figure 3: The 6-month and 12-month Stock Market Excess Returns