Asset Pricing with Return Extrapolationjin/aprx_JS_13_ds.pdf · Asset Pricing with Return...

Asset Pricing with Return Extrapolation∗

Lawrence J. Jin† and Pengfei Sui‡

September 18, 2019

ABSTRACT

We present a new model of asset prices, in which the agent has Epstein-Zin preferences and extrap-

olative beliefs about stock market returns. Unlike earlier return extrapolation models, our model

allows for a quantitative comparison with the data on expectations and asset prices. The agent’s

beliefs are consistent with empirical evidence on return expectations. Moreover, when the agent’s

beliefs are calibrated to match the persistence of the price-dividend ratio, the model generates ex-

cess volatility and predictability of stock market returns, a high equity premium, a low and stable

risk-free rate, and a low correlation between stock market returns and consumption growth.

JEL classification: G02, G12

Keywords: Expectations, Return Extrapolation, Stock Market Movements

∗We thank Nicholas Barberis, John Campbell, Alexander Chinco, Stefano Cassella, Ricardo De la O, MichaelEwens, Cary Frydman, Robin Greenwood, David Hirshleifer, Philip Hoffman, Jonathan Ingersoll, Dana Kiku, TheresaKuchler, Lars Lochstoer, Jiacui Li, Sean Myers, Cameron Peng, Andrei Shleifer, Jesscia Wachter, and seminarparticipants at Caltech, Maastricht University, Tilburg, the University of California, Irvine, the University of SouthernCalifornia, the Young Economists Symposium at Yale, the Caltech Junior Faculty Behavioral Finance Conference, theFinance Down Under Conference, CICF, the SFS Cavalcade North America meeting, the NBER Behavioral Financemeeting, the NBER Summer Institute Asset Pricing meeting, and the American Finance Association Annual Meetingfor helpful comments. Please send correspondence to Lawrence J. Jin, California Institute of Technology, 1200 E.California Blvd. MC 228-77, Pasadena, CA, 91125; telephone: 626-395-4558. E-mail: [email protected].†California Institute of Technology.‡The Chinese University of Hong Kong, Shenzhen.

mailto:[email protected]

I. Introduction

In financial economics, there is growing interest in “return extrapolation,” the idea that in-

vestors’ beliefs about an asset’s future return are a positive function of the asset’s recent past

returns. Models with return extrapolation have two appealing features. First, they are consistent

with survey evidence on the beliefs of real-world investors and with data on household portfolio

holdings.1 Second, they show promise in matching important asset pricing facts, such as volatil-

ity and predictability in the aggregate market, momentum and reversals in the cross-section, and

bubbles (Hong and Stein (1999); Barberis (2018); Barberis, Greenwood, Jin, and Shleifer (2015,

2018); Liao and Peng (2019)).

One limitation of existing models of return extrapolation, however, is that they can only be com-

pared to the data in a qualitative way. Early models, such as Cutler, Poterba, and Summers (1990)

and De Long, Shleifer, Summers, and Waldmann (1990), highlight the conceptual importance of

return extrapolation, but they are not designed to match asset pricing facts quantitatively. Barberis

et al. (2015) is a dynamic consumption-based model that tries to make sense of both survey ex-

pectations and aggregate stock market prices. However, the simplifying assumptions in the model

make it difficult to evaluate the model’s fit with the empirical facts. For instance, their model

adopts a framework with constant absolute risk aversion (CARA) preferences and a constant in-

terest rate. Under these assumptions, many ratio-based quantities that we study in asset pricing

(e.g., the price-dividend ratio) do not have well-defined distributions in the model and therefore do

not have properties that match what we observe in the data.

In this paper, we propose a new model of aggregate stock market prices based on return ex-

trapolation that overcomes this limitation. The goal of the paper is to see if the model can match

important facts about the aggregate stock market when the agent’s beliefs are calibrated to the

data, and to compare the model in a quantitative way to alternative models of the stock market.

We consider a Lucas economy in continuous time with a representative agent. The Lucas tree is a

1On the one hand, Vissing-Jorgensen (2004), Bacchetta, Mertens, and van Wincoop (2009), Amromin and Sharpe(2013), Greenwood and Shleifer (2014), and Kuchler and Zafar (2019) examine survey data and find that manyindividual and institutional investors have extrapolative expectations: they believe that an asset’s price will continuerising in value after a sequence of high past returns, and that it will continue falling in value after a sequence oflow past returns. On the other hand, Malmendier and Nagel (2011) show that non-survey data including bothhouseholds’ stock market participation and their fraction of liquid assets invested in stocks depend positively on aweighted average of past stock market returns, with more recent returns weighted more heavily; these findings arealso consistent with return extrapolation.

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claim to an aggregate consumption process which follows a geometric Brownian motion. Besides the

Lucas tree, there are two tradeable assets in the economy: the stock market and an instantaneous

riskless asset. The stock market is a claim to an aggregate dividend process whose growth rate

is positively correlated with consumption growth. The riskless asset is in zero net supply with

its interest rate determined in equilibrium. The representative agent has Epstein-Zin preferences

and extrapolative beliefs. She perceives that the expected growth rate of stock market prices is

governed by a switching process between two regimes. If recent price growth of the stock market

has been high, the agent thinks it is likely that a high-mean price growth regime is generating

prices and therefore forecasts high price growth in the future. Conversely, if recent price growth

has been low, the agent thinks that it is likely that a low mean-price growth regime is generating

prices and therefore forecasts low price growth in the future.

An important parameter in the model, χ, specifies the agent’s perceived intensity for the tran-

sitions between the two price growth regimes described above. This parameter determines the

relative weight the agent puts on recent versus distant past returns when forming expectations of

future returns. One approach to setting χ is to calibrate it to non-price data—to survey expec-

tations of returns or of other variables, or to portfolio holdings. For instance, we can take survey

expectations of future stock market returns, and look at how they depend on recent and distant past

returns. Alternatively, assuming that people use similar heuristics when forecasting other economic

variables, we can take survey expectations of inflation, and look at how they depend on recent and

distant past inflation. Finally, we can take household stock holdings, which presumably depend

closely on expectations of stock market returns, and look at how these stock holdings depend on

recent and distant past returns.

All of these non-price data point to extrapolative expectations: investors’ expectations of an

economic variable—either future stock market returns or future inflation—depend positively on

a weighted average of past realizations of this economic variable, with more weight associated

with more recent realizations. In addition, household stock holdings also depend positively on

past stock market returns. Nonetheless, these non-price data lead to quite different estimates of

χ. For instance, survey expectations about stock market returns depend primarily on recent past

returns, implying a high χ of approximately one (Greenwood and Shleifer (2014)). Meanwhile,

survey expectations of inflation depend on inflation outcomes further back in the past, implying a

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much lower χ of about 0.015 (Malmendier and Nagel (2016)). Finally, household stock holdings

depend on past returns even further back in the past, implying an even lower χ of about 0.0075

(Malmendier and Nagel (2011)).

Given these disparate estimates, we take a different approach: we set χ so that it matches a

price-based quantity—specifically, the persistence of the price-dividend ratio—and then examine

how the model fares in explaining empirical facts about prices. We find that the implied χ is

about 0.022. Reassuringly, this value of χ lies within the range suggested by the non-price data.2

Importantly, with the agent’s beliefs calibrated to match the persistence of the price-dividend ratio,

the model quantitatively matches important facts about the aggregate stock market: it generates

significant excess volatility and predictability of stock market returns, a high equity premium, a low

and stable interest rate, as well as a low correlation between stock market returns and consumption

growth.

We now explain the model’s implications, starting with excess volatility. The model generates

average return volatility of 27.4% for the stock market, while the volatility of dividend growth is set

to 11%. The excess volatility of stock market returns comes from the interaction between return

extrapolation and Epstein-Zin preferences.3 Suppose that the stock market has had high past

returns. In such a case, return extrapolation leads the agent to forecast high future returns. Under

Epstein-Zin preferences, the separation between the elasticity of intertemporal substitution and risk

aversion gives rise to a strong intertemporal substitution effect. Therefore, the agent’s forecast of

high future returns leads her to push up the current price significantly, generating excess volatility.4

The mechanism described above for generating excess volatility, together with a strong degree of

mean reversion in the agent’s expectations about stock market returns, produces the long-horizon

2It might seem that survey expectations of stock market returns offer the most direct approach to calibrating χ.However, the way the survey questions are asked may bias people’s responses. For instance, the questions typicallyask people for their expectations of returns over the next six months or one year, instead of over a longer horizon.This may lead people to base their responses on returns in the past six months or one year, thereby overstating therole these recent returns play in their actual portfolio decisions and therefore explaining why the implied χ is so high.

3As we further discuss in Section III, the model-implied return volatility decreases significantly from 27.4% to14.8%, when we reduce Epstein-Zin preferences to power utility while keeping all other parameter values fixed.

4A feedback loop emerges from this mechanism. If current returns are high, that makes the agent think thatfuture returns will also be high, which leads her to push up prices, increasing current returns further, and so on. Ingeneral, there is a danger that this feedback loop could “explode.” Nonetheless, in the model, we assume the agentbelieves that the expected growth rate of stock market prices tends to switch over time from one regime to the other;she therefore believes her optimism will decline in the future. As a result, the cumulative impact of the feedbackloop on investor expectations and asset prices is finite; the model remains stable. Models like Cutler et al. (1990)and Barberis et al. (2015) instead introduce fully rational investors in order to counteract the behavioral investorsand preserve equilibrium.

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predictability of stock market returns that we observe in the data. The agent’s beliefs mean-revert,

for two reasons. First, by assumption, the agent believes that the expected growth rate of stock

market prices tends to switch over time from one regime to the other: the agent perceives that her

expectations about stock market returns will mean-revert. Second, the agent’s return expectations

actually mean-revert faster than what she perceives: when the agent thinks that the future price

growth is high, future price growth tends to be low endogenously, causing her return expectations

to decrease at a pace that exceeds her anticipation. As a result, following periods with a high

price-dividend ratio—this is when the high past price growth of the stock market pushes up the

agent’s expectation about future returns and hence her demand for the stock market—the agent’s

return expectation tends to revert back to its mean, giving rise to low subsequent returns and hence

the predictability of stock market returns using the price-dividend ratio.

Next, we turn to the model’s implications for the equity premium. When measured as the

rational expectation of stock market returns in excess to the interest rate, the model generates an

average equity premium of 8.73%; when measured as the expectation of log excess returns, the

average equity premium is 4.94%. Moreover, the equity premium perceived by the agent is only

about half of the total equity premium, while the remaining half is the difference between the true

equity premium and the perceived one. Two factors contribute to the perceived equity premium.

First, the agent’s risk aversion causes her to demand a positive equity premium in the face of

excess return volatility. Second, return extrapolation gives rise to perceived persistence of the

aggregate dividend process, which, under Epstein-Zin preferences, is significantly priced, pushing

up the perceived equity premium.5 As important, the true equity premium is significantly higher

than the perceived one. To see why, recall that the agent’s beliefs mean-revert faster than what

she perceives. Given this, she underestimates short-term stock market fluctuations and hence the

risk associated with the stock market. In other words, if an infinitesimal rational agent, one that

has the same preferences as the behavioral agent but holds rational beliefs, enters our economy, she

would have demanded a higher equity premium.

Finally, the model generates low interest rate volatility and a low correlation between stock

market returns and consumption growth. In the model, the agent separately forms beliefs about

5Epstein-Zin preferences allow for the separation between the elasticity of intertemporal substitution and riskaversion, which in turn keeps the equilibrium interest rate low and hence helps to keep the equity premium high.

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the dividend growth of the stock market and about aggregate consumption growth. Here, we assume

that the bias in the agent’s beliefs about consumption growth derives only from the bias in her

beliefs about dividend growth. Given the low observed correlation between consumption growth

and dividend growth, the bias in the agent’s beliefs about consumption growth is small, consistent

with the fact that there is no direct empirical evidence for extrapolative beliefs about consumption

growth. The agent’s approximately correct beliefs about consumption growth allow the model to

generate low interest rate volatility. They also imply that the agent’s beliefs about stock market

returns—these beliefs co-move strongly with the agent’s beliefs about dividend growth—are not

significantly affected by fluctuations in consumption growth, giving rise to a low correlation between

stock market returns and consumption growth.

Besides the implications for asset prices, the model produces further implications for investor

expectations. The agent’s return expectations positively depend on past twelve-month returns in a

statistically significant way. This implication is consistent with the empirical findings of Greenwood

and Shleifer (2014). Moreover, our return extrapolation model yields direct implications for cash

flow expectations. When the past price growth of the stock market has been high, this has a

positive effect not only on the agent’s beliefs about future returns, but also on her beliefs about

future dividend growth; indeed, her expectations about dividend growth rise at a pace that exceeds

her expectations about future returns.6 Given this, a Campbell-Shiller decomposition using the

agent’s subjective expectations about stock market returns and dividend growth shows that changes

in subjective expectations about future dividend growth explain 109% of the variance of the price-

dividend ratio, while changes in subjective return expectations explain –7% of the variance of the

price-dividend ratio. This implication is consistent with the recent empirical findings of De la O

and Myers (2019): they find that changes in investors’ subjective expectations of future dividend

growth explain the majority of stock market movements. In this way, our model simultaneously

explains the empirical findings of Greenwood and Shleifer (2014) on return expectations and the

empirical findings of De la O and Myers (2019) on cash flow expectations.

After presenting the model, we compare it to alternative models of the aggregate stock market,

including models with rational expectations. As with the habit formation model of Campbell and

Cochrane (1999), the long-run risks models of Bansal and Yaron (2004) and Bansal, Kiku, and

6We provide a detailed explanation of this finding in Sections II and III.

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Yaron (2012), and the rare disasters models of Barro (2006), Gabaix (2012), and Wachter (2013),

our model is developed in a Lucas economy with a representative agent. This model specification

allows for a direct comparison between our model and models with rational expectations. We

document some different implications in Section IV.

Our paper adds to a new wave of theories that study the role of extrapolative beliefs in the dy-

namics of asset prices (Choi and Mertens (2013); Jin (2015); Hirshleifer, Li, and Yu (2015); Barberis

et al. (2015, 2018); Glaeser and Nathanson (2017); DeFusco, Nathanson, and Zwick (2018); Li and

Liu (2018); Greenwood, Hanson, and Jin (2019)).7 Different from these earlier theories, our model

is quantitatively realistic in matching both stock market prices and investors’ return expectations.

Our paper is also closely related to a recent model of Nagel and Xu (2018), in which the representa-

tive agent has fading memory. An important difference lies between these two models. Our model

directly imposes biased beliefs for the agent about future stock market returns. In contrast, Nagel

and Xu (2018) assume that the agent incorrectly forecasts dividend growth. As such, our model

is able to match the observed positive correlation between return expectations and one-year past

returns, whereas the model of Nagel and Xu (2018) does not deliver this result. Finally, Wachter

and Kahana (2019) develop a model of human memory and apply it to studying financial decision

making and asset prices. Their model can potentially generate novel predictions. At the same time,

the model has not been directly compared to the data.

The paper proceeds as follows. In Section II, we lay out the basic elements of the model and

characterize its solution. In Section III, we parameterize the model and examine its implications

in detail. Section IV compares our model with alternative models of the stock market. Section V

concludes and suggests directions for future research. Some technical details are in the Appendix.

7More broadly, the dynamics of asset prices have also been studied in models with natural expectations (Fuster,Hebert, and Laibson (2011)), neglected risks (Gennaioli, Shleifer, and Vishny (2012)), diagnostic expectations (Bor-dalo, Gennaioli, and Shleifer (2018)), experience effect (Collin-Dufresne, Johannes, and Lochstoer (2017); Malmendier,Pouzo, and Vanasco (2018); Ehling, Graniero, and Heyerdahl-Larsen (2018)), parameter learning (Collin-Dufresne,Johannes, and Lochstoer (2016)), model uncertainty and ambiguity aversion (Bidder and Dew-Becker (2016)),expectations-based reference-dependent preferences (Pagel (2016)), unknown mapping between fundamentals andprices (Adam, Marcet, and Beutel (2017)), and biased prior beliefs (Guo and Wachter (2019)).

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II. The Model

In this section, we first describe the model setup. We then characterize the model’s solution

and briefly discuss equilibrium quantities that are important for understanding the implications of

the model.

II.1. Model setup

Asset space.—We consider an infinite-horizon Lucas economy in continuous time with a repre-

sentative agent. The Lucas tree is a claim to an aggregate consumption process. We assume it is

a geometric Brownian motion

dCt/Ct = gCdt+ σCdωCt , (1)

and we denote the price of the Lucas tree at time t as PCt .

Besides the Lucas tree, there are two other tradeable assets in the economy; they are the main

focus of our analysis. The first asset is the stock market which is a claim to an aggregate dividend

process given by

dDt/Dt = gDdt+ σDdωDt ; (2)

we denote the price of the stock market at time t as PDt .8 Both ωDt and ωCt are standard Brownian

motions. We assume that the instantaneous correlation between dDt and dCt is ρ > 0: Et[dωDt ·

dωCt ] = ρdt. The second asset is an instantaneous riskless asset. This asset is in zero net supply,

and its interest rate rt is determined in equilibrium.

Agent’s preferences.—We follow Epstein and Zin (1989, 1991) and assume that the agent has a

recursive intertemporal utility

Ut =

[(1− e−δdt)C1−ψ

t dt+ e−δdt(Eet [U

1−γt+dt]

)(1−ψ)/(1−γ)]1/(1−ψ)

, (3)

8Since the aggregate consumption process in the model is exogenous, the dividend payment from the stock marketdoes not further affect consumption. As a result, we can think of the stock market as an asset in zero net supply witha shadow price determined in equilibrium. This is a common assumption adopted by some other consumption-basedmodels such as Campbell and Cochrane (1999) and Barberis, Huang, and Santos (2001).

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where δ is the subjective discount rate, γ > 0 is the coefficient of relative risk aversion, and ψ > 0 is

the reciprocal of the elasticity of intertemporal substitution. As we discussed in Section I, recursive

preferences, rather than power utility, allow the model to generate quantitatively realistic impli-

cations; when ψ equals γ, (3) reduces to power utility. The superscript “e” is an abbreviation for

“extrapolative” expectations: the certainty equivalence in (3) is computed under the representative

agent’s subjective beliefs, which, as we specify later, incorporate the notion of return extrapolation.

The subjective Euler equation, or the first-order condition, is

Eet

e−δ(1−γ)dt/(1−ψ)

(Ct+dtCt

)−ψ(1−γ)/(1−ψ)

M(ψ−γ)/(1−ψ)t+dt Rj,t+dt

= 1. (4)

Here Mt+dt is the gross return on the optimal portfolio held by the agent from time t to time t+dt.

In a Lucas economy with a representative agent, the optimal portfolio in equilibrium is the Lucas

tree itself, and therefore

Mt+dt =PCt+dt + Ctdt

PCt=PCt+dt + Ct+dtdt

PCt+ o(dt). (5)

On the other hand, Rj,t+dt is the gross return on any tradeable asset j in the market from time t

to time t+ dt; as mentioned above, the two tradeable assets we focus on are the stock market and

the riskless asset.

Agent’s beliefs.—We now turn to the key part of the model: the representative agent’s beliefs

about stock market returns. According to surveys, real-world investors form beliefs about future

stock market returns by extrapolating past returns (Vissing-Jorgensen (2004); Bacchetta et al.

(2009); Amromin and Sharpe (2013); Greenwood and Shleifer (2014); Kuchler and Zafar (2019)).

One natural way to capture this notion of return extrapolation is through a regime-switching model.

Specifically, we suppose that the agent believes that the expected growth rate of stock market prices

is governed by (1 − θ)gD + θµS,t, where the parameter θ (0 ≤ θ ≤ 1) controls the extent to which

the agent’s beliefs are extrapolative: setting θ to zero makes the agent’s beliefs fully rational. More

importantly, µS,t in the agent’s beliefs is a latent variable which switches between a high value µH

in a high-mean price growth regime H and a low value µL (µL < µH) in a low-mean price growth

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regime L with the following transition matrix9

µS,t+dt = µH µS,t+dt = µL

µS,t = µH 1− χdt χdt

µS,t = µL χdt 1− χdt

. (6)

Here χ is the intensity perceived by the agent for the transitions of regime between H and L.

Given this regime-switching model—this is a perceived model, not the true model—if recent

stock market price growth has been high, the agent subjectively believes it is likely that the high-

mean price growth regime is generating prices and therefore forecasts high price growth in the

future. Conversely, if recent price growth has been low, the agent believes it is likely that the

low-mean price growth regime is generating prices and therefore forecasts low price growth in the

future. Formally, at each point in time, the agent computes the expected value of the latent variable

µS,t given the history of past price growth: St ≡ E[µS,t|FPt ]. That is, she applies optimal filtering

theory (see, for instance, Lipster and Shiryaev (2001)) and obtains

dSt = χ [(µH − St) + (µL − St)] dt+ (σDP,t)−1θ(µH − St)(St − µL)dωet

≡ µeS(St)dt+ σS(St)dωet ,

(7)

where dωet ≡ [dPDt /PDt − (1 − θ)gDdt − θStdt]/σDP,t is a standard Brownian innovation term from

the agent’s perspective. As a result, she perceives the evolution of the stock market price PDt to be

dPDt /PDt = µD,eP (St)dt+ σDP (St)dω

et , (8)

where

µD,eP (St) = (1− θ)gD + θSt. (9)

The agent’s expectation about price growth µD,eP (St) is therefore a linear combination of a rational

component gD and a behavioral component St; hereafter we call St the “sentiment variable.”

In summary, the evolution of sentiment in (7) captures return extrapolation: high past price

9The models of Barberis, Shleifer, and Vishny (1998), Veronesi (1999), and Jin (2015) also adopt a regime-switchinglearning structure.

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growth dPDt /PDt pushes up the perceived shock dωet , which leads the agent to raise her expectation

of the sentiment variable St, causing her expectation about future price growth µD,eP (St) to rise.10

Although the subjective evolution of sentiment (7) is derived through optimal learning, the

representative agent, it should be emphasized, does not hold rational expectations. With rational

expectations, the agent will realize in the long run that the regime-switching model (6) is incorrect :

she can look at the historical distribution of dωet and realize that it does not fit a normal distribution

with a mean of 0 and a variance of dt. Instead, the agent in our behavioral model always believes

that the regime-switching model is correct. In reality, it is possible that investors in the market

learn over time that their mental model is incorrect. At the same time, new investors who hold

extrapolative beliefs may continuously enter the market. The stable belief system in (6) is an

analytically convenient way to capture these dynamics. Alternatively, if equations (6) and (7)

represent the true data generating process, then the agent does hold rational expectations. In that

case, the model becomes a fully rational model with incomplete information.11 We discuss the

predictions of such a model in Section IV.

So far we have been focusing on the agent’s beliefs about stock market prices. These beliefs also

have direct implications for the agent’s beliefs about dividend growth. If we write the perceived

dividend process as

dDt/Dt = geD(St)dt+ σDdωet , (10)

we can connect the agent’s expectation about dividend growth geD(St) to her expectation about

stock market price growth µD,eP (St). To formally make this connection, we first observe that all

the ratio-based quantities in our model (e.g., the price-dividend ratio of the stock market) are a

function of the sentiment variable St; we can define f(St) ≡ PDt /Dt. We then apply Ito’s lemma

on both sides of this equation f(St) = PDt /Dt and match terms to obtain

10There are many ways to specify the evolution of St in order to capture return extrapolation. We derive St from aregime-switching model, in part because such a learning model captures base rate neglect, an important consequenceof the representativeness heuristic (Tversky and Kahneman (1974)). To see this, note that the perceived regimesor states, H and L, are not part of the true states of the economy. As a result, assigning positive probabilities tothese regimes reflect the bias that the agent neglects the zero base rate associated with such regimes. Moreover, aregime-switching model allows us to bound St by a finite range (µL, µH), reducing the analytical difficulty of solvingthe model.

11Information is incomplete in the sense that the agent does not directly observe the latent variable µS,t in (6).

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geD(St) = (1− θ)gD + θSt︸︷︷︸expectation of price growth

−(f ′/f)µeS(St)︸︷︷︸expectation of sentiment evolution

+σ2D − σDP (St)σD − 1

2(f ′′/f)(σS(St))2︸︷︷︸

Ito correction terms

,

(11)

where

σDP (St) =σD +

√σ2D + 4θ(µH − St)(St − µL)(f ′/f)

2> σD. (12)

Equation (11) highlights an “expectations transmission mechanism:” it says that the agent’s expec-

tation about dividend growth equals the sum of her expectation about stock market price growth,

her expectation about how the price-dividend ratio evolves with respect to changes in sentiment,

and the Ito correction terms that are related to the agent’s risk aversion and the volatility of div-

idend growth, price growth, and changes in sentiment. In this way, the agent’s expectation about

price growth affects her expectation about dividend growth.

With the parameter values we specify later, equation (11) suggests that the agent’s expectation

about dividend growth is more responsive to changes in sentiment than her expectation about

price growth. Under Epstein-Zin preferences, the separation between the elasticity of intertemporal

substitution and risk aversion gives rise to a strong intertemporal substitution effect. As a result,

when the past price growth has been high, the agent’s forecast of high future price growth leads her

to push up the current price-dividend ratio, making it a positive function of sentiment. Furthermore,

under the regime-switching model, the agent perceives sentiment to be mean-reverting: µeS(St)

in (7) is a negative function of St. This suggests that the agent also perceives the price-dividend

ratio to be mean-reverting. Together, these two conditions—the price-dividend ratio is a positive

function of sentiment and is perceived to be mean-reverting—imply that the agent anticipates that

the price-dividend ratio will decline from a high value when she expects high future price growth.

That is, when the agent expects high future price growth, her expectation about dividend growth

rises at a pace that exceeds her expectation about future price growth.

To complete the description of the model, we need to further specify the agent’s beliefs about

consumption growth. To do this, first note that, with a local correlation of ρ between consumption

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growth and dividend growth, we can rewrite the true aggregate consumption process of (1) as

dCt/Ct = gCdt+ σC

(ρdωDt +

√1− ρ2dω⊥t

), (13)

where ω⊥t is a Brownian motion that is locally uncorrelated with ωDt , the Brownian shock on

dividends. We then assume that the agent perceives the consumption process as

dCt/Ct = geC(St)dt+ σC

(ρdωet +

√1− ρ2dω⊥t

). (14)

That is, we replace the true Brownian shock on dividends dωDt by the agent’s perceived Brownian

shock dωet and factor the difference between these two Brownian shocks into geC(St), the agent’s

subjective expectation about consumption growth. Conceptually, this amounts to assuming that

the bias in the agent’s beliefs about consumption growth comes only from the bias in her beliefs

about dividend growth.12 In doing so, we derive the agent’s expectation about consumption growth

as

geC(St) = gC + ρσCσ−1D (geD(St)− gD). (15)

Empirically, the correlation between consumption growth and dividend growth is low—ρ is

positive but low—and consumption growth is much less volatile than dividend growth—σC is

much smaller than σD. As a result, (15) implies that the bias in the agent’s expectation about

consumption growth—the difference between geC(St) and gC—is small. This is in keeping with

the lack of any evidence that investors have extrapolative beliefs about consumption growth.13

Moreover, the agent’s approximately correct beliefs about consumption growth allow the model to

generate low interest rate volatility and a low correlation between consumption growth and stock

market returns, both of which are consistent with the data (Campbell (2003); Hansen and Singleton

(1982, 1983)).

12For any alternative assumption, one needs to explain why the agent has incorrect beliefs about consumptionabove and beyond her incorrect beliefs about dividends.

13Consistent with the way we model the agent’s expectations about consumption growth, Kuchler and Zafar (2019)find that survey expectations are “asset-specific:” respondents who become pessimistic about their employmentsituation after experiencing unemployment are not pessimistic about other economic outcomes, such as stock prices orinterest rates. Similarly, Huang (2016) finds that investors who become optimistic about an industry’s future returnsafter having positive prior investment experience in the industry do not invest heavily in an unrelated industry.

12

II.2. Model solution and equilibrium quantities

The subjective Euler equation in (4) shows that, when pricing the stock market, both the

stock market return and the return from holding the Lucas tree are part of the pricing kernel.

This observation has two implications. First, both the price-dividend ratio f(St) = PDt /Dt and

the wealth-consumption ratio PCt /Ct are functions of the sentiment variable St; we can define

l(St) ≡ PCt /Ct. Second, the two functions f and l are interrelated through Euler equations, so they

need to be solved simultaneously. Specifically, by using the Euler equation to price the stock market

and the Lucas tree—that is, by setting Rj,t+dt in (4) to the gross return on the stock market and

to the gross return on the Lucas tree, respectively—we obtain two ordinary differential equations

that jointly determine functions f and l. In the Appendix, we provide a detailed derivation of these

two differential equations, as well as the numerical procedure that solves them.

With the model solution at hand, we derive the expressions of several equilibrium quantities

that are important for understanding the model’s implications. We start with the equilibrium

interest rate. By using the Euler equation in (4) to price the riskless asset—by setting Rj,t+dt to

the gross return on the riskless asset 1 + rtdt—we obtain

rt = 1−γ1−ψ δ + γgeC −

γ(γ+1)2 σ2

C −ψ−γ1−ψ ×

(µeS − γρσCσS)(l′/l) + 1

2σ2S(l′′/l)

+2ψ−γ−12(1−ψ) σ

2S(l′/l)2 + l−1

. (16)

The interest rate is linked to the agent’s time preferences, her subjective expectation about con-

sumption growth, precautionary saving, as well as how the wealth-consumption ratio PCt /Ct re-

sponds to changes in sentiment.14

Next, we compute the agent’s expectation about future stock market returns, a quantity that

is important for understanding the risk-return tradeoff in the model. From equations (8) and (9),

the log excess return on the stock market from time t to time t+ dt is

rD,et+dtdt ≡ `n(PDt+dt +Dt+dtdt)− `n(PDt )− rtdt

= [(1− θ)gD + θSt + f−1 − 12(σDP )2 − rt]dt+ σDP dω

et .

(17)

14When θ = 0, r = δ + ψgC − γ(ψ+1)2

σ2C .

13

Therefore, the agent’s subjective expectation about the log excess return is

Eet [rD,et+dt] = (1− θ)gD + θSt + f−1 − 1

2(σDP )2 − rt, (18)

and the subjective Sharpe ratio is [(1− θ)gD + θSt + f−1 − 12(σDP )2 − rt]/σDP .

To compute the (objectively measured) rational expectation about future stock market returns,

we compare (2) with (10) and obtain a relationship between the true and perceived Brownian shocks

dωet = dωDt − (geD(St)− gD)dt/σD. (19)

We then substitute (19) into (17) and derive

rD,et+dtdt = [(1− θ)gD + θSt + f−1 − σ−1D σDP (geD − gD)− 1

2(σDP )2 − rt]dt+ σDP dωDt . (20)

As a result, the rational expectation about the log excess return on the stock market is

Et[rD,et+dt] = (1− θ)gD + θSt + f−1 − σ−1D σDP (geD − gD)− 1

2(σDP )2 − rt, (21)

and the objectively measured Sharpe ratio of the stock market return is [(1− θ)gD + θSt + f−1 −

σ−1D σDP (geD − gD)− 1

2(σDP )2 − rt]/σDP .

Finally, we note that all the ratio-based quantities in this model such as the agent’s expectation

about stock market returns and the interest rate are a function of the sentiment variable St.

Therefore, a statistical assessment of the model’s fit with the empirical facts requires knowing

the steady-state distribution for the sentiment variable St as objectively measured by an outside

econometrician. We leave the derivation of this steady-state distribution to the Appendix.

III. Model Implications

In this section, we examine the implications of the model. We begin by setting the benchmark

values for the model parameters. In particular, we discuss how to calibrate the agent’s beliefs to

match certain aspects of the data. We then discuss the model’s implications for asset prices and

for investor expectations.

14

III.1. Model parameterization

There are three types of parameters: asset parameters, utility parameters, and belief parameters.

For the asset parameters, we set gC = 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2.

These values are consistent with those used in Campbell and Cochrane (1999), Barberis et al.

(2001), Bansal and Yaron (2004), and Beeler and Campbell (2012).15 For the utility parameters,

we set γ, the coefficient of relative risk aversion, to 10. As pointed out in Bansal et al. (2012)

and Bansal and Yaron (2004), the long-run risks literature—a literature that depends significantly

on the parameter values of Epstein-Zin preferences for the models’ implications—typically assigns

a value of 10 or below for γ. Bansal and Yaron (2004), for instance, set γ to either 10 or 7.5.16

For ψ, the reciprocal of the elasticity of intertemporal substitution, there exists a wide range of

estimates in the asset pricing literature. The majority of previous papers suggests that ψ should

be lower than one, but several other papers argue the opposite.17 Given this, we set ψ to 0.9, a

value that implies an elasticity of intertemporal substitution slightly above one.18 Finally, for δ,

the subjective discount rate, we assign a value of 2%.

We now turn to the belief parameters. We set µH and µL, the mean price growth in the high

and low regimes, to 12.5% and −12.5%, respectively. As we will discuss later in this section, the

probability of the agent’s price growth expectations approaching the boundaries of µH and µL is

approximately zero. As a result, the model’s implications are not very sensitive to the choice of µH

and µL. Next, we set θ to 0.5. This parameter controls the extent to which the representative agent

is behavioral; θ = 0.5 is a natural default value because it implies that the representative agent is

approximately an aggregation of rational and behavioral agents with equal population weights.

Finally, we discuss how we set the value for χ, which specifies the intensity perceived by the

agent between the high-mean price growth regime and the low-mean price growth regime. This

parameter determines the relative weight the agent puts on recent versus distant past returns when

forming beliefs about future returns. As a consequence, it directly affects the model’s implications

for asset prices, such as the predictability of stock market returns and the persistence of the price-

15The parameter values for gC and gD are set such that both `n(C) and `n(D) grow, on average, at an annual rateof 1.84%; this rate is also used in Barberis et al. (2001).

16An estimate of 10 for γ is also the maximum magnitude that Mehra and Prescott (1985) find reasonable.17See Bansal et al. (2012) for a discussion of this point.18Our model’s implications are quantitatively robust even with an elasticity of intertemporal substitution signifi-

cantly lower than one. We discuss this point further in Section IV.

15

dividend ratio. One possible approach to setting χ is to calibrate it to non-price data. In particular,

we examine three sets of non-price data. First, we take survey expectations of future stock market

returns, and look at how they depend on recent and distant past returns. Second, assuming that

people use similar heuristics when forecasting other economic variables, we take survey expectations

of inflation, and look at how they depend on recent and distant past inflation. Finally, we take

household stock holdings, which presumably depend closely on households’ expectations of stock

market returns, and look at how these stock holdings depend on recent and distant past returns.

Using survey expectations of stock market returns, Greenwood and Shleifer (2014) estimate the

following non-linear least squares regression

Expectationt = a+ b ·∑n

j=1wj(φ)RD(t−j∆t)→(t−(j−1)∆t) + εt, (22)

where Expectationt is the average time-t investor expectation of stock market return over the next

six months or twelve months, from each of the six surveys, RD(t−j∆t)→(t−(j−1)∆t) is the past stock

market return from time t−j∆t to t−(j−1)∆t, ∆t = 1/4 (one quarter), and wj(φ) ≡ φj/∑n

l=1 φl.19

In equation (22), each past realized return is assigned a weight. The weight decreases exponentially

the further back we go into the past, and the coefficient φ in weights {wj(φ)}nj=1 measures the

speed of this exponential decline. Greenwood and Shleifer (2014) report an average φ of 0.56 across

survey specifications, indicating that returns four quarters earlier (RD(t−5∆t)→(t−4∆t)) are about 10%

as important as returns in the past quarter (RD(t−∆t)→t) when investors form expectations about

future returns. To match this “decay” pattern from Greenwood and Shleifer (2014), we run the

same non-linear regression in (22), now using model simulations. Specifically, for a given parameter

value of χ, we look at the average estimate of φ across 100 trials, with each trial representing a

regression with simulated data over 10,000 years. We then search for the value of χ that leads to

an average estimate of φ around 0.56. We find that when χ = 1, the estimate of φ from the model

is 0.58.

We now look at another source of non-price data: individuals’ expectations of future inflation

over the next twelve months, documented in the Reuters/Michigan Survey of Consumers. Mal-

19The six surveys of investor expectations are: the Gallup survey, the Graham and Harvey survey, the AmericanAssociation of Individual Investors survey, the Investor Intelligence survey, the Shiller survey, and the Michigansurvey.

16

mendier and Nagel (2016) study these inflation expectations by estimating a non-linear regression

that, under some conditions, can be written as

Expectationt = a+ b ·∑n−1

j=1wj(λ)π(t−j∆t)→(t−(j−1)∆t) + εt. (23)

Here Expectationt is the time-t expectation of inflation over the next twelve months, averaged

across individual forecasters from the same age cohort, π(t−j∆t)→(t−(j−1)∆t) is the past inflation

rate from time t − j∆t to t − (j − 1)∆t, ∆t = 1/4, and wj(λ) ≡ (n − j)λ/∑n−1

l=1 (n− l)λ.20 The

parameter λ measures the relative weight individual forecasters put on recent versus distant past

inflation rates when forming expectations of future inflation. Malmendier and Nagel (2016) include

different control variables to the regression in (23) and report a range of estimates of λ; the average

estimate is around 2.5, indicating that inflation from a decade earlier (π(t−41∆t)→(t−40∆t)) is about

57% as important as inflation in the past quarter (π(t−∆t)→t) when a typical 50-year-old investor

forms expectations about future inflation.

If the way investors form expectations about stock market returns is similar to the way they

form expectations about future inflation, then we should expect that λ estimated from

Expectationt = a+ b ·∑n−1

j=1wj(λ)RD(t−j∆t)→(t−(j−1)∆t) + εt (24)

would be similar to the one estimated from (23). Compared to (23), equation (24) replaces the

average time-t expectation of future inflation by the average time-t expectation of future stock

market returns, and it also replaces past inflation rates by past stock market returns. The weights

{wj(λ)}n−1j=1 are kept the same. To calibrate our belief parameter χ to investor expectations about

future inflation, we run the non-linear regression in (24) using model simulations.21 We search for

the value of χ that gives rise to an estimate of λ around 2.5. We find that when χ = 0.015, the

estimate of λ from the model is 2.62.

Finally, we look at household stock holdings from the Survey of Consumer Finances. Malmendier

20Note that wj(φ) from equation (22) is an exponential function of j, whereas wj(λ) from equation (23) is apolynomial function of j.

21The regressions in (23) and (24) require specifying the age of the investor. Recall that in Malmendier and Nagel(2016) and Malmendier and Nagel (2011), the median age of the investors is about 50 years. Therefore, we set n inboth (23) and (24) to 200 (quarters).

17

and Nagel (2011) run the following regression

Yt = a+ b ·∑n−1

j=1wj(λ)RD(t−j∆t)→(t−(j−1)∆t) + εt, (25)

where Yt is the time-t fraction of liquid assets invested in stocks, averaged across households from the

same age cohort. With different control variables, Malmendier and Nagel (2011) provide a range

of estimates of λ; the average estimate is around 1.5. If we further assume that stock holdings

depend closely on households’ expectations of stock market returns, then we can calibrate χ to

stock holdings by searching for the value of χ that leads to an estimate of λ around 1.5. We find

that when χ = 0.0075, the estimate of λ is 1.57.

[Place Tables 1 and 2 about here]

Table 1 details the model simulations that calibrate χ to the three sets of non-price data

described above. Table 1 makes it clear that these non-price data lead to very different estimates of

χ, ranging from χ = 1—if we calibrate it to survey expectations about stock market returns studied

in Greenwood and Shleifer (2014)—to χ = 0.0075—if we calibrate it to household stock holdings

studied in Malmendier and Nagel (2011). Why do we get these disparate estimates? It remains

an open question. However, one possible explanation is: we do not elicit return expectations from

investors precisely at the moment when they trade the stock market. In other words, the current

non-price data—either survey expectations or stock holdings—are only a noisy proxy of investor

beliefs that drive asset prices. Given this limitation of the non-price data, we choose to set χ to

match a price-based quantity. Specifically, we calibrate χ to match the persistence of the price-

dividend ratio, because variation in χ has a direct impact on the persistence of asset prices. We

find that this χ is about 0.022.22 Reassuringly, this value of χ lies within the range of [0.0075, 1]

suggested by the non-price data. We summarize the default parameter values in Table 2.

III.2. Model implications for asset prices

In this section, we study the model’s implications for asset prices. We begin with examining the

long-run properties of stock market prices and returns. Table 3 reports the model’s predictions for

22Later in this section, we present and discuss the model’s implications for the autocorrelations of the price-dividendratio.

18

six important moments, and compares them side by side with the empirical values. In general, the

model matches the facts: it generates significant excess volatility, a high equity premium, a Sharpe

ratio similar to the empirical value, an interest rate that has a low level and low volatility, and a

price-dividend ratio whose average level is close to the empirical one.

[Place Table 3 about here]

As explained in Section I, the model generates significant excess volatility from the interac-

tion between return extrapolation and Epstein-Zin preferences. This interaction is quantitatively

important. In the absence of return extrapolation, Epstein-Zin preferences with i.i.d. dividend

growth and consumption growth do not lead to any excess volatility. In the absence of Epstein-

Zin preferences—that is, setting both ψ and γ to 10 while keeping all the other parameter values

unchanged—return extrapolation alone leads to average return volatility of 14.8%, which implies

much less excess volatility compared to the data.23

The model also generates a significant equity premium: when measured as the rational expec-

tation of excess returns E[(dPDt +Dtdt)/(PDt dt)− rt], the average equity premium is 8.73%; when

measured as the rational expectation of log excess returns, it is 4.94%. To understand this model

implication, we first note that the model produces a substantial equity premium perceived by the

agent—this is the average equity premium that the agent thinks she will receive. When measured

as the subjective expectation of excess returns Ee[(dPDt +Dtdt)/(PDt dt)− rt], the perceived equity

premium is 5.71%; when measured as the subjective expectation of log excess returns, it is 1.93%.

Two factors contribute to the perceived equity premium. First and most intuitively, the agent’s

risk aversion causes her to demand a positive equity premium in the face of excess return volatil-

ity. Second, return extrapolation gives rise to perceived persistence of both the aggregate dividend

process and, to a lesser extent, the aggregate consumption process. Under Epstein-Zin preferences,

this perceived persistence is significantly priced, pushing up the perceived equity premium.24

With incorrect beliefs, the true equity premium in the model can be different from the perceived

23Moreover, when Epstein-Zin preferences reduce to power utility, the equity premium, measured as the rationalexpectation of log excess returns, reduces from 4.94% to 1.25%, a level much lower than the empirical one. Ingreater detail, Li and Liu (2018) show that when combined with power utility, return extrapolation does not lead toquantitatively realistic implications for asset prices.

24The agent is averse to persistent shocks when γ > ψ; with our choice of parameter values, this condition issatisfied.

19

one: we find that, on average, the true equity premium is significantly higher. To see why, recall that

the agent’s beliefs mean-revert faster than what she perceives, causing the agent to underestimate

short-term stock market fluctuations and hence the risk associated with the stock market. If an

infinitesimal rational agent, one that has the same preferences as the behavioral agent but holds

rational beliefs, enters our economy, she would have demanded a higher equity premium.

[Place Figure 1 about here]

Figure 1 plots the rational expectation and the agent’s subjective expectation about price

growth, each against sentiment S. We find that for S less than 7.9%, the rational expectation

about price growth is higher than the subjective expectation; for S greater than 7.9%, the opposite

is true. We also find that the distribution of sentiment has a mean of 5.4% and a standard deviation

of 3.9%. Given this distribution, the rational expectation about price growth is above the subjective

expectation 70.8% of the time. That is, the true price growth is more likely to be higher than the

perceived price growth. Overall, the model produces an average equity premium that is about twice

as high as the one perceived by the agent.

Furthermore, the model produces a countercyclical equity premium that turns negative dur-

ing high-sentiment periods: the equity premium averaged over the top quartile of the sentiment

distribution is −5.6%. In general, rational expectations models do not generate a negative equity

premium at any time. In our model, however, subjective expectations and objective expectations

about stock market returns differ significantly during high- or low-sentiment periods: when the

sentiment level is high, the agent expects high stock market returns moving forward, but pre-

cisely because of her incorrect beliefs, future stock market returns are low on average, generating a

negative equity premium. This model implication is consistent with the recent empirical findings

of Greenwood and Hanson (2013), Baron and Xiong (2015), Yang and Zhang (2016), and Cassella

and Gulen (2018): these papers document that the expected excess return turns negative during

high-sentiment periods.

Next, we examine the model’s implications for the predictability of stock market returns. Em-

pirically, Campbell and Shiller (1988) and Fama and French (1988) document that a regression of

future log excess returns on the current log price-dividend ratio gives a negative and significant

regression coefficient. Moreover, the predictive power of the price-dividend ratio is greater when

20

future returns are calculated over longer horizons.


Table 4 reports the regression coefficient βj and the R-squared for a regression of the log excess

return of the stock market from time t to time t+ j on the current log price-dividend ratio

rD,et→t+j = αj + βj`n(PDt /Dt) + εj,t, (26)

where the time horizon j = 1, 2, 3, 5, and 7 (years). For each j, we calculate the regression

coefficient βj and the R-squared using 10,000 years of monthly data simulated from the model, and

compare them side by side with the empirical values. Consistent with the data, βj is negative and

its magnitude increases as the time horizon j increases. When the stock market has had high past

price growth, the agent’s expectation about future price growth increases, pushing up the current

price-dividend ratio. Since the agent’s expectation—the sentiment variable—tends to revert back

to its mean, subsequent returns are low on average, giving rise to a negative regression coefficient

in (26).

The magnitudes of the regression coefficient and the R-squared generated from the model are

consistent with the empirical values. When χ, the perceived intensity between the high-mean price

growth regime and the low-mean price growth regime, is set to 0.022, the agent views the past

stock market return a decade earlier about 47% as important as the return in the past quarter

when forming her expectations about the future return. This level of persistence in the agent’s

beliefs—and the implied mean reversion in sentiment—leads to the predictability of stock market

returns that matches the data very well.


We now investigate the model’s implications for the autocorrelations of asset prices. As we

mentioned earlier in this section, we set χ = 0.022 to match the persistence of the price-dividend

ratio. Table 5 shows the precise comparison between the model-implied autocorrelations of log

price-dividend ratios and their empirical counterparts: the theoretical values match up closely with

the empirical values, up to a lag of five years. For the autocorrelations of log excess returns, the

21

model-implied values are also broadly consistent with the empirical ones.

In our model, the persistence of the agent’s beliefs determines the persistence of asset prices.

One potential source for this persistence of beliefs is the retrieval of prior contextual states from

human memory, as studied in Wachter and Kahana (2019): when investors observe a stock market

downturn, they associate the downturn with financial crises from the distant past—these are called

prior contexts in some memory theories—causing the investors to “jump back in time” and expe-

rience the crises again. This mechanism allows past financial crises to have a long-lasting impact

on investor beliefs. Moreover, both Cassella and Gulen (2018) and Wachter and Kahana (2019)

imply that the persistence of investor beliefs varies over time. Understanding the source for the

persistence of beliefs and its time variation is an important direction for future research.

Lastly, we look at the model’s implications for the correlation between stock market returns

and consumption growth. Empirically, Hansen and Singleton (1982, 1983) document that this

correlation is low. Nonetheless, many existing models have a consumption-based pricing kernel as

the only source of stock market movements, resulting in a high, if not perfect, correlation between

stock market returns and consumption growth.25


Table 6 reports both the model-implied values and the empirical values for the correlation

between consumption growth and stock market returns. Consistent with the data, the model

produces a low correlation: the correlation between annual log consumption growth and annual

log excess returns is 0.20 in the model, and similarly it is 0.09 in the data. In comparison, the

model of Campbell and Cochrane (1999) generates a correlation of 0.79, a much higher value.

In our model, we assume that the bias in the agent’s beliefs about consumption growth comes

only from the bias in her beliefs about dividend growth. Given the low correlation observed in

the data between consumption growth and dividend growth, the bias in the agent’s beliefs about

consumption growth is small. As a result, the agent’s beliefs about stock market returns—they co-

move strongly with her beliefs about dividend growth—are not significantly affected by fluctuations

25One exception is Barberis et al. (2001). They use “narrow framing”, the notion that investors may evaluatefinancial risks in isolation from consumption risks, to generate a low correlation between stock market returns andconsumption growth. Specifically, they use power utility as the agent’s preferences over consumption, but use prospecttheory developed by Kahneman and Tversky (1979) as the agent’s preferences over financial wealth.

22

in consumption growth, giving rise to the low observed correlation between stock market returns

and consumption growth.

Table 6 shows that the model also generates a small but negative correlation between the current

consumption growth and the stock market return in the subsequent period, an implication that

is consistent with the data. Recall that consumption growth and dividend growth are weakly but

positively correlated. If the current consumption growth is high, dividend growth is also high on

average, which leads the agent to push up the current price, increasing the current price growth

and the current level of sentiment. In the subsequent period, sentiment reverts towards its mean

value, giving rise to a low stock market return.

III.3. Model implications for investor expectations

When we assign parameter values, we set χ to 0.022. This value implies that the agent’s

return expectations are positively related to both recent and distant past returns, consistent with

the empirical findings of Malmendier and Nagel (2011). In this section, we further examine the

relationship between the agent’s return expectations and recent past returns alone.

Empirically, Greenwood and Shleifer (2014) regress investor expectations about future stock

market returns from six surveys on past twelve-month cumulative returns. They find that the

regression coefficient is positive and statistically significant. Using model simulations, Table 7 re-

ports the regression coefficient, its t-statistic, the intercept, and the R-squared for a regression of the

agent’s time-t expectation about future stock market return on the past twelve-month cumulative

return

Eet [(dPDt +Dtdt)/(PDt dt)] = a+ b ·RDt−12→t + εt, (27)

over a sample of 15 years or 50 years. The sample sizes are chosen to match various lengths of

survey data studied in Greenwood and Shleifer (2014).26


Table 7 shows that, for a 15-year simulated sample, the regression coefficient is 0.7% with a

Newey-West adjusted t-statistic of 3.6. Running the same regression for a 50-year simulated sample,

26For instance, their paper looks at the Gallup data over a 16-year period, from 1996 to 2011. It also looks at theInvestor Intelligence survey over a 49-year period, from 1963 to 2011.

23

the regression coefficient is 0.7% with a t-statistic of 4.4.27 These results suggest that, in our model,

the agent’s return expectations positively depend on past twelve-month returns in a statistically

significant way. Furthermore, the values of t-statistic fall within the range of empirical estimates

from Greenwood and Shleifer (2014). However, as we discussed earlier, survey expectations are not

collected from investors at the exact moment when they trade the stock market; as a comparison, the

agent’s expectations in our model precisely drive trading decisions and asset prices. As a result, we

do not expect the coefficient and the t-statistic estimated from the model for the regression in (27)

to perfectly match the empirical values.

Next, we turn to the model implications for cash flow expectations. Although our model is based

on return extrapolation, it yields direct implications for cash flow expectations. The expectations

transmission mechanism described by equation (11) suggests that the agent’s expectation about

dividend growth is more responsive to changes in sentiment than her expectation about price

growth. Moreover, the total return equals the sum of the price growth and the dividend yield,

with the latter decreasing as sentiment increases. Therefore, the agent’s expectation about price

growth is more responsive to changes in sentiment, compared to her expectation about stock market

returns.


Figure 2 plots the agent’s expectation about stock market returns, Ee[(dPDt + Dtdt)/(PDt dt)],

the agent’s expectation about price growth, Ee[dPDt /(PDt dt)], and the agent’s expectation about

dividend growth, Ee[dDt/(Dtdt)], each as a function of the sentiment variable St. Figure 2 suggests

that a one-standard deviation (3.9%) increase in sentiment from its mean (5.4%) pushes up the

agent’s expectation about stock market returns from 7.99% to 8.58%—a small increase of 0.59%—

while it pushes up the agent’s expectation about dividend growth from −0.17% to 5.33%—a much

larger increase of 5.50%. Moreover, a one-standard deviation increase in sentiment from its mean

pushes up the price-dividend ratio of the stock market from 24.73 to 36.79. Together, these results

imply that the price-dividend ratio is mainly correlated with the agent’s expectation about dividend

27Hartzmark and Solomon (2019) suggest that investors do not take the dividend yield into account when calculatingreturns. If we replace the dependent variable in (27) by Eet [dPDt /(PDt dt)], the agent’s expectation of price growth,then a 15-year simulated sample gives a regression coefficient of 1.9% with a t-statistic of 3.9. A 50-year simulatedsample gives a regression coefficient of 1.9% with a t-statistic of 5.0.

24

growth, rather than the agent’s expectation about returns.

To provide a statistical assessment of the sources of stock market movements, we follow the

procedure in Campbell and Shiller (1988) to decompose, in the context of the model, the log

price-dividend ratio of the stock market:

`n(PDt /Dt) ≈∞∑j=0

ξj(

∆d(t+j∆t)→(t+(j+1)∆t) − rD(t+j∆t)→(t+(j+1)∆t)

)− (`n(ξ) + (1− ξ)ζ)/(1− ξ),

(28)

where ζ is the in-sample average of the annual log dividend-price ratio, ξ = e−ζ/(∆t + e−ζ),

∆d(t+j∆t)→(t+(j+1)∆t) is the log dividend growth from time t+j∆t to t+(j+1)∆t, and rD(t+j∆t)→(t+(j+1)∆t)

is the log stock market return over the same period. Equation (28) says that the movement of

price-dividend ratio comes from either the movement of future dividend growth—the “cash flow

news”—or the movement of future returns—the “discount rate news.” The standard approach that

empirically addresses the relative importance of these two components is to look at future realized

dividend growth and returns, and compute

1 ≈Cov

(∑∞j=0 ξ

j∆d(t+j∆t)→(t+(j+1)∆t), `n(PDt /Dt))

Var(`n(PDt /Dt)

)︸︷︷︸CFobjective

+Cov

(−∑∞

j=0 ξjrD(t+j∆t)→(t+(j+1)∆t), `n(PDt /Dt)

)Var

(`n(PDt /Dt)

)︸︷︷︸DRobjective

.

(29)

The first term on the right hand side of (29), CFobjective, is the contribution of changes in cash flow

news to stock market movements, and the second term, DRobjective, is the contribution of changes

in discount rate news to stock market movements. Most empirical studies that have conducted a

Campbell-Shiller decomposition take this approach.

With incorrect beliefs, the model allows us to further study the relation between the agent’s

subjective expectations and stock market movements by taking the subjective expectations on both

sides of (28) and computing

1 ≈Cov

(Eet [∑∞

j=0 ξj∆d(t+j∆t)→(t+(j+1)∆t)], `n(PDt /Dt)

)Var

(`n(PDt /Dt)

)︸︷︷︸CFsubjective

+Cov

(−Eet [

∑∞j=0 ξ

jrD(t+j∆t)→(t+(j+1)∆t)], `n(PDt /Dt))

Var(`n(PDt /Dt)

)︸︷︷︸DRsubjective

.

(30)

25

The first term on the right hand side of (30), CFsubjective, is the contribution of changes in subjective

expectations about cash flow news to stock market movements, and the second term, DRsubjective,

is the contribution of changes in subjective expectations about discount rate news to stock market

movements.


Table 8 reports the four coefficients, CFobjective, DRobjective, CFsubjective, DRsubjective, as well

as their corresponding R-squared. These coefficients and R-squared are calculated using 10,000

years of monthly data simulated from the model. By using future realized dividend growth and

stock market returns, we obtain DRobjective = 0.98 with an R-squared of 0.22 and CFobjective = 0.03

with an R-squared of 2.8 × 10−4. This result replicates the empirical finding of the volatility test

literature that the variation of the price-dividend ratio comes primarily from discount rate variation

(see Cochrane (2011)).

On the other hand, by using the agent’s subjective expectations about dividend growth and

returns, we obtain DRsubjective = −0.07 with an R-squared of 0.96 and CFsubjective = 1.09 with

an R-squared of 0.98. This result unveils a very different picture that highlights the importance

of expectations data: changes in the agent’s subjective expectations about future cash flow news

explain the majority of stock market movements. Empirically, De la O and Myers (2019) find that

DRsubjective = −0.09 and CFsubjective = 1.09. These values match the theoretical values from our

model.

Importantly, the fact that prices in our model are mainly correlated with cash flow expectations

is a consequence of the Campbell-Shiller accounting identity; this statement is about correlation,

not about causality. In the model, the agent’s return expectations determine her cash flow ex-

pectations and are the cause of price movements. Together, Tables 7 and 8 suggest that our

model simultaneously explains the empirical findings of Greenwood and Shleifer (2014) on return

expectations and the empirical findings of De la O and Myers (2019) on cash flow expectations.28

We conclude Section III by discussing the role of rational arbitrageurs. The model has a

representative agent whose beliefs are biased. One natural question to ask is: if we introduce

28In the Appendix, we provide additional discussion about the relation between return expectations and cash flowexpectations.

26

rational arbitrageurs, to what extent can they counteract the mispricing caused by the behavioral

agent and therefore attenuate the significance of the model implications? Developing such a two-

agent model is beyond the scope of the paper. Nonetheless, three observations suggest that our

model implications are likely to remain intact after taking rational arbitrageurs into account.

First, in an economy with both rational and behavioral agents who have recursive preferences,

the behavioral agents may eventually dominate the market: there is a positive probability that

they take up most of the wealth in the economy in the long run. This is a key finding in Borovicka

(2019). It suggests that our model’s implications can be the limiting implications of a model with

both rational and behavioral agents in the initial period. Second, in an economy with heterogeneous

beliefs, asset prices are jointly determined by agents’ beliefs weighted by their risk tolerances. A

positive fundamental shock causes optimists to gain a larger fraction of wealth and increases their

risk tolerance relative to pessimists, which in turn gives optimists a greater weight in pushing

up asset prices. Therefore, heterogeneity in investor beliefs can be an additional source of excess

volatility; it can further amplify—rather than attenuate—our model implications.29 Lastly, as

pointed out by Barberis et al. (2015), extrapolative expectations are persistent in a dynamic model,

which means that the behavioral agents who extrapolate past returns have persistently high demand

for the stock market following high stock market returns. The persistence of this demand prevents

near-future stock market returns from becoming too low, reducing the incentive of rational agents

to counteract mispricing. In other words, the persistence of extrapolative beliefs limits the impact

of rational arbitrageurs on asset prices.

IV. Comparison with Alternative Models

In this section, we provide a quantitative comparison between our model and several alternative

models of the aggregate stock market. First, we look at a rational expectations model that is most

analogous to our behavioral model, one in which the regime-switching process characterized in

Section II represents the true data generating process. We then examine some different implications

between our model and another related model: the long-run risks model of Bansal and Yaron (2004).

Both models assume a representative agent with Epstein-Zin preferences. Our model then assumes

29See Xiong (2013) for more discussion of this amplification mechanism.

27

that the agent has extrapolative expectations about stock market returns, while the model of Bansal

and Yaron (2004) assumes that dividend growth and consumption growth share a persistent yet

stochastic component. Lastly, we compare our model with a model with fundamental extrapolation,

in which the agent holds extrapolative expectations about dividend growth of the stock market.

IV.1. A true regime-switching model

In Section II, we have assumed, in equations (6) and (7), that the agent believes that the

expected growth rate of stock market prices follows a regime-switching process. We have also

emphasized that the agent’s beliefs are incorrect : with rational expectations, the agent will realize

in the long run that the expected price growth does not follow the regime-switching process.

What if we instead assume that the regime-switching process is the true data generating process?

In this case, the agent’s beliefs about price growth become fully rational. The true evolution of the

stock market price is

dPDt /PDt = [(1− θ)gD + θµS,t]dt+ σDP (St)dω

Pt , (31)

where ωPt is a standard Brownian motion, and the latent variable µS,t follows the regime-switching

process specified in (6). As with the behavioral model, the agent in this model does not directly

observe µS,t. Instead, she uses past stock market prices to form a Bayesian estimate of µS,t:

St = E[µS,t|FPt ]. We further assume that the perceived dividend process (10) and the perceived

consumption process (14) are also rational.

By construction, this rational expectations model produces the same equilibrium prices as our

behavioral model: the solutions to the differential equations of (A.5) and (A.9) also apply to this

model. Nonetheless, the two models have statistical properties that are significantly different.

One difference, for instance, lies in the models’ implications for the predictability of stock market

returns.


Table 9 reports the regression coefficient βj and the R-squared for a regression of the log

excess return of the stock market from time t to time t+ j on the current log price-dividend ratio

`n(PDt /Dt) over various time horizons j (one to seven years), now using the true regime-switching

28

model. Table 9 shows that the model fails to produce the predictability of stock market returns

documented in Campbell and Shiller (1988) and Fama and French (1988): both the regression

coefficients and the R-squared are close to zero. In contrast, Table 4 shows that the behavioral

model produces the observed predictability of stock market returns.

With rational expectations, the agent’s beliefs about stock market returns are on average correct.

Therefore, following high past price growth, the agent properly anticipates high future price growth,

which pushes down the dividend yield in equilibrium, giving rise to flat returns in subsequent

periods. As a result, future returns do not vary significantly with the current price-dividend ratio,

leading to little return predictability in the true regime-switching model.

IV.2. The long-run risks model of Bansal and Yaron (2004)

Our model differs from the model of Bansal and Yaron (2004) in the model’s implications for

both asset prices and investor expectations. Here, we discuss two differences regarding asset prices.

First, our model produces a high equity premium and significant return volatility even when

the elasticity of intertemporal substitution is low. On the contrary, the model of Bansal and Yaron

(2004) does not generate a high equity premium with a low elasticity of intertemporal substitution.

Figure 3 shows that setting the elasticity of intertemporal substitution to 0.5, our model generates

an equity premium of 6.0% (measured as the average excess return, E[(dPDt +Dtdt)/(PDt dt)− rt]),

while the model of Bansal and Yaron (2004) produces an equity premium between 1% and 2%.


To understand this model difference, we note that the perceived dividend growth in our model

depends strongly on the state variable of sentiment, pushing up the perceived equity premium; as

a comparison, dividend growth in Bansal and Yaron (2004) depends much less on the stochastic

growth component.30 Moreover, the true equity premium in our model is significantly above the

perceived one, so it can remain high even when the elasticity of intertemporal substitution is low.

Another model difference concerns the predictability of future consumption growth and dividend

growth. On the one hand, the long-run risks models introduce persistence in consumption growth

30Bansal and Yaron (2004) set their “leverage parameter” to 3.5. In comparison, with return extrapolation andwith the default parameter values, our model effectively sets this leverage parameter to 14.5, a much higher value.

29

and dividend growth, which causes future consumption growth and dividend growth to be predicted

by current variables such as the price-dividend ratio and the interest rate.31 On the other hand, our

model does not give rise to such predictability: return extrapolation only generates perceived but

not true long-run risks, and therefore the true consumption growth and dividend growth remain

unpredictable.

IV.3. A fundamental extrapolation model

A literature in behavioral finance focuses on fundamental extrapolation, the notion that some

investors hold extrapolative expectations about asset fundamentals such as dividend growth or

GDP growth (Barberis et al. (1998); Fuster et al. (2011); Choi and Mertens (2013); Alti and

Tetlock (2014); Hirshleifer et al. (2015)). In this section, we provide a quantitative comparison

between our model and a model with fundamental extrapolation. To facilitate this comparison, we

keep the two models almost identical. The only difference is that, in the model with fundamental

extrapolation, sentiment is constructed from past dividend growth, whereas in the model with

return extrapolation, sentiment is constructed from past price growth. We provide more discussion

of this fundamental extrapolation model in the Appendix.

We now look at the model’s implications for investor expectations. Table 10 reports the re-

gression coefficient, its t-statistic, the intercept, and the R-squared for a regression of the agent’s

time-t expectation about future stock market return on the past twelve-month cumulative return,

now using the fundamental extrapolation model. In this case, we find that the t-statistics of the

regression coefficient are insignificant.


Suppose past dividend growth has been high. On the one hand, it results in high past returns.

On the other hand, fundamental extrapolation leads the agent to expect high dividend growth

moving forward, but not high returns: following high past dividend growth, the stock market

price increases to the extent that the agent’s expectation of future returns does not change sig-

nificantly. Recall from Greenwood and Shleifer (2014) that past twelve-month returns positively

31See Beeler and Campbell (2012) and Collin-Dufresne et al. (2016) for a detailed discussion of this model impli-cation.

30

predict survey expectations about future returns. Therefore, a fundamental extrapolation model

with a representative agent faces a challenge in explaining survey expectations about stock market

returns. In contrast, Table 7 shows that the return extrapolation model is able to explain these

survey expectations.32

We conclude this section by discussing a methodological contribution of the paper. In most of

asset pricing models, the state variables about agents’ beliefs can be exogenously specified without

solving the equilibrium; this greatly simplifies the models. Take the fundamental extrapolation

model described above as an example. Equation (E.1) in the Appendix characterizes the evolution of

sentiment S, the state variable that drives asset prices dynamics. And this equation can be derived

without solving the equilibrium. However, with return extrapolation, sentiment S determines—and

is endogenously determined by—asset prices. As a result, such a model requires solving beliefs and

asset prices simultaneously, and therefore imposes a greater modeling challenge. Our numerical

approach to solving a system of differential equations provides a solution to this challenge.

V. Conclusion

We build a new return extrapolation model that can be brought to the data quantitatively.

With the agent’s beliefs calibrated to fit the autocorrelations of the price-dividend ratio, the model

matches the long-run properties of stock market prices: it generates a high average equity premium,

significant excess volatility, a low average interest rate, low interest rate volatility, and a price-

dividend ratio whose average level is similar to the empirical one. The model also matches the

dynamic behavior of stock market prices: it produces the long-horizon predictability of stock market

returns, persistent asset prices, and the observed low correlation between stock market returns and

consumption growth. In addition, the model’s implications for return expectations and cash flow

expectations are both consistent with survey evidence. Lastly, we compare our model to alternative

models of the stock market, and we document their different implications for asset prices and for

investor expectations.

Our analysis leaves open several areas for future research. First, our representative-agent model

32A fundamental extrapolation model with heterogeneous agents—for instance, one with both an agent who extrap-olates past dividend growth and an agent who is fully rational—can potentially generate extrapolative expectationsof returns for the behavioral agent in the model. See the model of Ehling et al. (2018) as an example.

31

neglects an important channel that affects asset prices: the time-varying fraction of wealth held

by behavioral agents. Explicitly incorporating rational agents into our framework may lead to

additional implications. Second, our model has not allowed for time variation in the persistence

of the agent’s beliefs. Analyzing such variation may also lead to novel predictions. Finally, the

extrapolation framework is closely linked to theories of model uncertainty. A careful investigation

of this connection may produce useful insights to both literatures.

32

Appendices

A. Derivation of the Model Solution

In this section, we derive the differential equations that determine the price-dividend ratio f(St)

and the wealth-consumption ratio l(St). First, for the subjective Euler equation (4), setting Rj,t+dt,

the return on the tradeable asset, to the gross return on the stock market, the equation becomes

Eet

e−δ(1−γ)dt/(1−ψ)

(Ct+dtCt

)−ψ(1−γ)/(1−ψ)(PCt+dt + Ctdt

PCt

)(ψ−γ)/(1−ψ)PDt+dt + Dt+dtdt

PDt

= 1.

(A.1)

Using Taylor expansion, (A.1) becomes

Eet[e−δ(1−γ)dt/(1−ψ)

(Ct+dt

)−ψ(1−γ)/(1−ψ)(PCt+dt)

(ψ−γ)/(1−ψ)PDt+dt

(1 + ψ−γ

1−ψCt+dtPCt+dt

dt+Dt+dtPDt+dt

dt

)]= C

−ψ(1−γ)/(1−ψ)t (PCt )(ψ−γ)/(1−ψ)PDt .

(A.2)

Rearranging terms gives

0 = Eet

d(ΘC(ψ−γ)/(1−ψ)l(ψ−γ)/(1−ψ)Df) + ψ−γ

1−ψΘC(ψ−γ)/(1−ψ)l(2ψ−γ−1)/(1−ψ)Dfdt

+ΘC(ψ−γ)/(1−ψ)l(ψ−γ)/(1−ψ)Ddt

, (A.3)

where Θ(C, t) ≡ e−δ(1−γ)t/(1−ψ)C−ψ(1−γ)/(1−ψ). By Ito’s lemma, (A.3) leads to

0 = Eet

− δ(1−γ)1−ψ dt− γ(dC/C) + (dD/D) + (df/f) + ψ−γ

1−ψ (dl/l) + γ(γ+1)2 (dC/C)2

+12ψ−γ1−ψ

2ψ−γ−11−ψ (dl/l)2 − γ(ψ−γ)

1−ψ (dC/C)(dl/l)− γ(dC/C)(dD/D)− γ(dC/C)(df/f)

+ψ−γ1−ψ (dl/l)(dD/D) + ψ−γ

1−ψ (dl/l)(df/f) + (df/f)(dD/D) + ψ−γ1−ψ l

−1dt+ f−1dt

.

(A.4)

33

Using (7), (10), and (14) to further simplify (A.4) gives

0 =

− (1−γ)1−ψ δ − γg

eC + geD + [(f ′/f) + ψ−γ

1−ψ (l′/l)]µeS + 12 [(f ′′/f) + ψ−γ

1−ψ (l′′/l)]σ2S

+γ(γ+1)2 σ2

C + 12ψ−γ1−ψ

2ψ−γ−11−ψ (l′/l)2σ2

S −γ(ψ−γ)

1−ψ ρσCσS(l′/l)− γρσCσD − γρσCσS(f ′/f)

+ψ−γ1−ψσDσS(l′/l) + ψ−γ

1−ψσ2S(l′/l)(f ′/f) + σDσS(f ′/f) + ψ−γ

1−ψ l−1 + f−1

.

(A.5)

Similarly, setting Rj,t+dt in (4) to the gross return on the Lucas tree, the subjective Euler

equation (4) becomes

Eet

e−δ(1−γ)dt/(1−ψ)

(Ct+dtCt

)−ψ(1−γ)/(1−ψ)(PCt+dt + Ctdt

PCt

)(1−γ)/(1−ψ) = 1. (A.6)

Rearranging terms yields

0 = Eet[d(ΘC(1−γ)/(1−ψ)l(1−γ)/(1−ψ)) + 1−γ

1−ψΘC(1−γ)/(1−ψ)l(ψ−γ)/(1−ψ)dt]. (A.7)

By Ito’s lemma, (A.7) leads to

0 = Eet

− 1−γ

1−ψ δdt− (γ − 1)(dC/C) + γ(γ−1)2 (dC/C)2 + 1−γ

1−ψ (dl/l)

+12

1−γ1−ψ

ψ−γ1−ψ (dl/l)2 + (1−γ)2

1−ψ (dC/C)(dl/l) + 1−γ1−ψ l

−1dt

. (A.8)

Using (7) and (14) to further simplify (A.8) gives

0 =

− 1−γ

1−ψ δ − (γ − 1)geC + γ(γ−1)2 σ2

C + 1−γ1−ψ (l′/l)µeS + 1−γ

2(1−ψ)(l′′/l)σ2S

+12

1−γ1−ψ

ψ−γ1−ψ (l′/l)2σ2

S + (1−γ)2

1−ψ ρσCσS(l′/l) + 1−γ1−ψ l

−1

. (A.9)

Substituting µS and σS from (7), geD and σDP from (11) and (12), and geC from (15) into equa-

tions (A.5) and (A.9), we then arrive at two ordinary differential equations that jointly determine

34

f and l.33

Regarding the boundary conditions for solving these two differential equations, note that,

in (A.5) and (A.9), the second derivative terms are all multiplied by σS , and that σS goes to

zero as S approaches either µH or µL. As a result, µH and µL are both singular points, and

therefore no boundary condition is required. �

B. Steady-State Distribution for Sentiment

In this section, we derive the steady-state distribution for St as objectively measured by an

outside econometrician. To that end, we first obtain the objective evolution of sentiment by sub-

stituting the change-of-measure equation (19) into the subjective evolution of sentiment in (7)

dSt = [µeS(St) + σ−1D σS(St)(gD − geD(St))]dt+ σS(St)dω

Dt . (B.1)

Compared to the subjective evolution of sentiment, the objective evolution exhibits a larger degree

of mean reversion: the additional term σ−1D σS(St)(gD − geD(St)) in (B.1) is a negative function of

sentiment.

Denote the objective steady-state distribution for sentiment as ξ(S). Based on (B.1), we then

derive ξ(S) as the solution to the Kolmogorov forward equation (the Fokker-Planck equation)

0 = 12d2

dS2

(σ2S(S)ξ(S)

)− d

dS

([µeS(St) + σ−1

D σS(St)(gD − geD(St))]ξ(S))

= (σ′S)2ξ + σSσ′′Sξ + 2σSσ

′Sξ′ + 1

2σ2Sξ′′

− [(µeS)′ + σ−1D σ′S(gD − geD)− σ−1

D σS(geD)′]ξ − [µeS + σ−1D σS(gD − geD)]ξ′,

(B.2)

where σS and geD are from (7) and (11), respectively, and the expressions for σ′S , σ′′S , (µeS)′ and

33When θ = 0, our model reduces to a fully rational benchmark. In this case, equations (A.5) and (A.9) lead to

f =[δ + ψgC − gD − γ(ψ+1)

2σ2C + γρσCσD

]−1

, l =[δ + (ψ − 1)gC − γ(ψ−1)

2σ2C

]−1

.

35

(geD)′ are derived as follows. From the expression of σS in (7)

σ′S =θσDP (µH + µL − 2S)− θ(µH − S)(S − µL)(σDP )′

(σDP )2 ,

σ′′S =θ(µH − S)(S − µL){2[(σDP )′]

2 − σDP (σDP )′′}(σDP )

3 − 2θσDP (σDP )′(µH + µL − 2S) + (σDP )

2

(σDP )3 .

(B.3)

From the expression of µeS in (7) and the expression of geD in (11)

(µeS)′ =− (λ+ χ),

(geD)′ = θ − σD(σDP )′ − µ′S(f ′/f)− µS [f ′′/f − (f ′)2/f2]

− σSσ′S(f ′′/f)− 12σ

2S [f ′′′/f − f ′f ′′/f2],

(B.4)

where σDP is from (11), and (σDP )′ and (σDP )′′ are

(σDP )′ =θ(µH + µL − 2S)(f ′/f) + θ(µH − S)(S − µL)[f ′′/f − (f ′)2/f2]√

σ2D + 4θ(µH − S)(S − µL)(f ′/f)

,

(σDP )′′ =− 2{θ(µH + µL − 2S)(f ′/f) + θ(µH − S)(S − µL)[f ′′/f − (f ′)2/f2]}2

[σ2D + 4θ(µH − S)(S − µL)(f ′/f)]

3/2

+−2θf ′/f + 2θ(µH + µL − 2S)[f ′′/f − (f ′)2/f2]√

σ2D + 4θ(µH − S)(S − µL)(f ′/f)

+θ(µH − S)(S − µL)[f ′′′/f − 3(f ′f ′′)/f2 + 2(f ′)3/f3]√

σ2D + 4θ(µH − S)(S − µL)(f ′/f)

.

(B.5)

�

C. Numerical Procedure for Solving the Equilibrium

Equations (A.5) and (A.9) can only be solved numerically. Specifically, we apply a projection

method with Chebyshev polynomials. To start, we note that the value of the sentiment variable

S ranges from µL to µH , whereas the domain for Chebyshev polynomials is [–1, 1]. Therefore, we

36

transform S to a new state variable z

z ≡ aS + b, where a =2

µH − µL, b = −µH + µL

µH − µL, (C.1)

and we define h(z) ≡ f(S(z)) and j(z) ≡ l(S(z)). Equations (A.5) and (A.9) can be rewritten as

0 =

− (1−γ)1−ψ δ − γg

eC + geD + [(h′/h) + ψ−γ

1−ψ (j′/j)]aµeS + 12 [(h′′/h) + ψ−γ

1−ψ (j′′/j)]a2σ2S

+γ(γ+1)2 σ2

C + 12ψ−γ1−ψ

2ψ−γ−11−ψ (aj′/j)2σ2

S −γ(ψ−γ)

1−ψ ρσCσS(aj′/j)− γρσCσD − γρσCσS(ah′/h)

+ψ−γ1−ψσDσS(aj′/j) + ψ−γ

1−ψσ2Sa

2(j′/j)(h′/h) + σDσS(ah′/h) + ψ−γ1−ψ j

−1 + h−1

(C.2)

and

0 =

− 1−γ

1−ψ δ − (γ − 1)geC + γ(γ−1)2 σ2

C + 1−γ1−ψ (aj′/j)µeS + 1−γ

2(1−ψ)(a2j′′/j)σ2S

+12

1−γ1−ψ

ψ−γ1−ψ (aj′/j)2σ2

S + (1−γ)2

1−ψ ρσCσS(aj′/j) + 1−γ1−ψ j

−1

. (C.3)

We approximate h and j by

h(z) =∑n

r=0arTr(z), l(z) =

∑m

r=0brTr(z), (C.4)

where Tr(z) is the rth degree Chebyshev polynomial of the first kind.34 The projection method

chooses the coefficients {ar}nr=0 and {br}mr=0 so that the differential equations are approximately

satisfied. One criterion for a sufficient approximation is to minimize the weighted sum of squared

34See Mason and Handscomb (2003) for a detailed discussion of the properties of Chebyshev polynomials.

37

errors

∑Ni=1

1√1−z2i

− (1−γ)1−ψ δ − γg

eC + geD + [(h′/h) + ψ−γ

1−ψ (j′/j)]aµeS + 12 [(h′′/h) + ψ−γ

1−ψ (j′′/j)]a2σ2S

+γ(γ+1)2 σ2

C + 12ψ−γ1−ψ

2ψ−γ−11−ψ (aj′/j)2σ2

S −γ(ψ−γ)

1−ψ ρσCσS(aj′/j)− γρσCσD

−γρσCσS(ah′/h) + ψ−γ1−ψσDσS(aj′/j) + ψ−γ

1−ψσ2Sa

2(j′/j)(h′/h) + σDσS(ah′/h)

+ψ−γ1−ψ j

−1+ h

−1

2

z=zi

+∑N

i=11√

1−z2i

−1−γ1−ψ δ − (γ − 1)geC + γ(γ−1)

2 σ2C + 1−γ

1−ψ (aj′/j)µeS + 1−γ2(1−ψ)(a2j′′/j)σ2

S

+12

1−γ1−ψ

ψ−γ1−ψ (aj′/j)2σ2

S + (1−γ)2

1−ψ ρσCσS(aj′/j) + 1−γ1−ψ j

−1

2

z=zi

,

(C.5)

where {zi}Ni=0 are the N zeros of TN (z). By the Chebyshev interpolation theorem, if N is suffi-

ciently larger than n and m, and if the sum of weighted square in (C.5) is sufficiently small, the

approximated functions h(z) and l(z) are sufficiently close to the true solutions.

For the numerical results in the main text, we set m = 40, n = 40, N = 400. We then apply

the Levenberg-Marquardt algorithm and obtain a minimized sum of squared errors less than 10−11.

The small size of the total error indicates convergence of the numerical solution. The solution is

also insensitive to the choice of n, m, or N . Together, these findings suggest that the numerical

solutions are a sufficient approximation for the true h and j functions.

The same numerical procedure is applied to solving the Kolmogorov forward equation (B.2). �

D. Additional Discussion about Return Expectations and Cash Flow Expectations

The direct implication of return extrapolation is that the agent’s subjective expectation about

the future stock market return

Eet [PDt+dt] = 1 + Eet [rDt+dt]dt = Eet

[PDt+dtPDt

]+Dtdt

PDt(D.1)

is a positive function of the stock market’s recent past returns. Rearranging terms gives

PDtDt

=1

Eet [rDt+dt]− Eet [dPDt /(PDt dt)]. (D.2)

38

That is, the current price-dividend ratio is determined by the agent’s subjective expectation about

the future stock market return Eet [rDt+dt] and the agent’s subjective expectation about future price

growth Eet [dPDt /(PDt dt)]. Equation (D.2) does not suggest an explicit role for the agent’s expecta-

tion about dividend growth in determining the price-dividend ratio.

However, two conditions allow us to link the price-dividend ratio of the stock market to the

agent’s expectation about dividend growth. First, the law of iterated expectations must hold so

that we can iterate forward the Euler equation (4) with the stock market as the tradeable asset.

Second, the transversality condition must hold so that the economy permits no bubbles.35 These

two conditions allow us to obtain

PDtDt

= Eet

∫ ∞t

e−δ(1−γ)(s−t)/(1−ψ)

(CsCt

)−ψ(1−γ)/(1−ψ)

M(ψ−γ)/(1−ψ)t→s

(Ds

Dt

)ds

, (D.3)

where Mt→s denotes the continuously compounded gross return for holding the Lucas tree from

time t to time s (> t). Equation (D.3) says that the current price-dividend ratio of the stock market

equals the agent’s subjective expectation of the sum of discounted future dividend growths.

For an infinitely-lived agent, (D.3) further implies that the agent is aware of the fact that both

her expectation about future price growth and her expectation about future returns are linked to

her expectation about future dividend growth. The specific relationship between these expectations

are discussed in Sections II and III. �

E. Setup of the Fundamental Extrapolation Model

Here we briefly describe the key assumptions in the fundamental extrapolation model that

Section IV discussed. In this model, the agent believes that the expected growth rate of dividends—

instead of the expected growth rate of stock market prices in the case of return extrapolation—is

governed by (1−θ)gD+θµS,t, where µS,t is a latent variable that follows a regime-switching process

described in Section II. The agent does not directly observe the latent variable µS,t. Instead, she

computes its expected value given the history of past dividend growth: St ≡ E[µS,t|FDt ]. She then

35The transversality condition holds in this economy as the stock market price is bounded by a finite range.

39

applies optimal filtering theory and derives

dSt =(λµH + χµL − (λ+ χ)St)dt+ σ−1D θ(µH − St)(St − µL)dωet

≡µeS(St)dt+ σS(St)dωet ,

(E.1)

where dωet ≡ [dDt/Dt − (1− θ)gDdt− θStdt]/σD is a standard Brownian innovation term from the

agent’s perspective. That is, she perceives the evolution of dividend as

dDt/Dt = geD(St)dt+ σDdωet , (E.2)

where

geD(St) = (1− θ)gD + θSt. (E.3)

In other words, the agent’s expectation about dividend growth geD(St) is a linear combination of a

rational component gD and a sentiment component St constructed from past dividend growth. On

the other hand, the perceived evolution of the stock market price can be derived as

dPDt /PDt = µD,eP (St)dt+ σDP (St)dω

et , (E.4)

where

σDP (S) =σD + (f ′/f)σ−1D θ(µH − S)(S − µL),

µD,eP (S) =(f ′/f)µeS + 12(f ′′/f)σ2

S + (1− θ)gD + θS − σ2D + σDσ

DP (S).

(E.5)

As before, f is defined as the price-dividend ratio of the stock market.

As with the return extrapolation model, equations (A.5) and (A.9) determine f and l, the

price-dividend ratio and the wealth-consumption ratio, except that µS , σS , geD, µD,eP , and σDP are

now from (E.1), (E.3), and (E.5). �

40

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46

−0.12 −0.08 −0.04 0.00 0.04 0.08 0.12−0.80

−0.60

−0.40

−0.20

0.00

0.20

Sentiment (S)

Objective expectation about price growthSubjective expectation about price growth

Figure 1. Objective and Subjective Expectations about Price Growth. The dashed line

plots the objective (rational) expectation about price growth, Et[(dPDt )/(PDt dt)], as a function

of the sentiment variable St. The solid line plots the agent’s subjective expectation about price

growth, Eet [dPDt /(PDt dt)], as a function of the sentiment variable St. The parameter values are: gC

= 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ = 0.9, δ = 2%, θ = 0.5, χ =

0.022, µH = 12.5%, and µL = –12.5%.

47

−0.12 −0.08 −0.04 0.00 0.04 0.08 0.12−0.10

0.00

0.10

0.20

0.30

0.40

Sentiment (S)

Expectation about returnsExpectation about price growthExpectation about dividend growth

Figure 2. Agent’s Expectations about Stock Market Returns, Price Growth, and

Dividend Growth. The solid line plots the agent’s expectation about stock market returns,

Eet [(dPDt + Dtdt)/(PDt dt)], as a function of the sentiment variable St. The dashed line plots the

agent’s expectation about price growth, Eet [dPDt /(PDt dt)], as a function of the sentiment variable

St. The dotted-dashed line plots the agent’s expectation about dividend growth, Eet [dDt/(Dtdt)],

as a function of the sentiment variable St. The parameter values are: gC = 1.91%, gD = 2.45%,

σC = 3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ = 0.9, δ = 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and

µL = –12.5%.

48

6.00 8.00 10.000.01

0.02

0.03

0.04

0.05

γ

E[rD,e]

6.00 8.00 10.000.07

0.08

0.08

0.09

0.09

γ

E[RD,e]

6.00 8.00 10.00

0.28

0.30

0.32

0.34

γ

σ(rD,e)

0.50 1.00 1.50 2.00

0.04

0.06

0.08

0.10

ψ

E[rD,e]

0.50 1.00 1.50 2.00

0.06

0.08

0.10

0.12

0.14

ψ

E[RD,e]

0.50 1.00 1.50 2.000.25

0.26

0.27

0.28

ψ

σ(rD,e)

Figure 3. Comparative Statics. The upper panel plots the average log excess return E[rD,e],

the average excess return E[RD,e], and the average volatility of stock market returns σ(rD,e), each

as a function of γ, the coefficient of relative risk aversion. The lower panel plots the same quantities,

each as a function of ψ, the reciprocal of the elasticity of intertemporal substitution. The default

values for γ and ψ are 10 and 0.9, respectively. The other parameter values are: gC = 1.91%, gD

= 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2, δ = 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL =

–12.5%.

49

Table 1. Calibrating χ to Non-Price Data.

χ = 1

φ 0.58

a –0.09

b 0.14

R-squared 0.95

χ = 0.015 χ = 0.0075

λ 2.62 1.57

a –0.40 –0.13

b 0.48 0.21

R-squared 0.61 0.23

The upper panel reports the parameter φ, the intercept a, the regression coefficient b, and the

R-squared, estimated from the non-linear least squares regression


j=1wj(φ)RD(t−j∆t)→(t−(j−1)∆t) + εt,

and the lower panel reports the parameter λ, the intercept a, the regression coefficient b, and the

R-squared, estimated from


j=1wj(λ)RD(t−j∆t)→(t−(j−1)∆t) + εt.

For both regressions, Expectationt = Eet [(dPDt + Dtdt)/(PDt dt)] is the agent’s time-t subjective

expectation of the future stock market return, RD(t−j∆t)→(t−(j−1)∆t) is the past stock market return

from time t− j∆t to t− (j − 1)∆t, ∆t = 1/4 (one quarter), and n = 200. For the first regression,

wj(φ) = φj/∑n

l=1 φl. For the second regression, wj(λ) = (n − j)λλ/

∑nl=1(n − l)λ. Each reported

value is averaged over 100 trials, and each trial represents a regression using monthly data simulated

from the model over 10,000 years. As indicated in the table, the parameter value of χ is set to 1,

0.015, and 0.0075, respectively. The other parameter values are: gC = 1.91%, gD = 2.45%, σC =

3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ = 0.9, δ = 2%, θ = 0.5, µH = 12.5%, and µL = –12.5%.

50

Table 2. Parameter Values.

Parameter Variable Value

Asset parameters:

Expected consumption growth gC 1.91%

Expected dividend growth gD 2.45%

Standard deviation of consumption growth σC 3.8%

Standard deviation of dividend growth σD 11%

Correlation between dD and dC ρ 0.2

Utility parameters:

Relative risk aversion γ 10

Reciprocal of EIS ψ 0.9

Subjective discount rate δ 0.02

Belief parameters:

Degree of extrapolation θ 0.5

Perceived transition intensity between H and L χ 0.022

Upper bound of sentiment µH 12.5%

Lower bound of sentiment µL –12.5%

51

Table 3. Basic Moments.

Statistic Theoretical value Empirical value

Equity premium (E[rD,e]) 4.94% 3.90%

Return volatility (σ(rD,e)) 27.4% 18.0%

Sharpe ratio (E[rD,e]/σ(rD,e)) 0.20 0.22

Interest rate (E[r]) 2.15% 2.92%

Interest rate volatility (σ(r)) 0.19% 2.89%

Price-dividend ratio (exp(E[`n(P/D)])) 26.3 21.1

The table reports six important moments about stock market prices and returns: the long-run

average of the equity premium (the rational expectation of log excess return, E[rD,e]), the average

volatility of stock market returns (the volatility of log excess return, σ(rD,e)), the Sharpe ratio

(E[rD,e]/σ(rD,e)), the average interest rate (E[r]), interest rate volatility (σ(r)), and the average

price-dividend ratio of the stock market (exp(E[`n(P/D)])). The theoretical values for these mo-

ments are computed over the objectively measured steady-state distribution of sentiment S. The

model parameters are: gC = 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ =

0.9, δ = 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL = –12.5%. For the empirical values, five

out of six are from Campbell and Cochrane (1999); the empirical value for interest rate volatility is

not reported in Campbell and Cochrane (1999), so we report the value from Beeler and Campbell

(2012).

52

Table 4. Return Predictability Regressions.

Theoretical value Empirical value

Horizon (years)10×

coefficientR-squared

10×coefficient

R-squared

1 –2.7 0.06 –1.3 0.04

2 –4.7 0.10 –2.8 0.08

3 –6.2 0.13 –3.5 0.09

5 –8.6 0.18 –6.0 0.18

7 –10.1 0.20 –7.5 0.23

The table reports the regression coefficient βj and the R-squared for a regression of the log excess

return of stock market from time t to time t+ j on the current log price-dividend ratio `n(PDt /Dt)

rD,et→t+j = αj + βj`n(PDt /Dt) + εj,t,

where j = 1, 2, 3, 5, and 7 (years). The theoretical values are calculated using 10,000 years of

monthly data simulated from the model. The parameter values are: gC = 1.91%, gD = 2.45%, σC

= 3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ = 0.9, δ = 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL =

–12.5%. The empirical values are from Campbell and Cochrane (1999).

53

Table 5. Autocorrelations of Log Price-Dividend Ratios and Log Excess Returns.

Lag

(years)


`n(PD/D) rD,e Σji=1ρ(rD,et , rD,et−i ) `n(PD/D) rD,e Σj

i=1ρ(rD,et , rD,et−i )

1 0.77 –0.11 –0.11 0.78 0.05 0.05

2 0.60 –0.08 –0.19 0.57 –0.21 –0.16

3 0.47 –0.05 –0.24 0.50 0.08 –0.09

5 0.28 –0.04 –0.31 0.32 –0.14 –0.28

7 0.16 –0.03 –0.38 0.29 0.11 –0.15

The table reports, over various lags j, the autocorrelations of log price-dividend ratios and log

excess returns, as well as the partial sum of the autocorrelations of log excess returns. The operator

ρ(x, y) computes the sample correlation between variable x and variable y. The theoretical values

are calculated using 10,000 years of monthly data simulated from the model; for each month, we

compound the next 12 months of log excess returns to obtain an annual log excess return. The

parameter values are: gC = 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ = 0.9, δ

= 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL = –12.5%. The empirical values are from Campbell

and Cochrane (1999).

54

Table 6. Correlations between Consumption Growth and Stock Returns.

CorrelationTheoretical value Empirical value

monthly quarterly annual annual

Corr(rD,et→t+∆t, `n(Ct−∆t/Ct−2∆t)) –0.00 –0.01 –0.02 –0.05

Corr(rD,et→t+∆t, `n(Ct/Ct−∆t)) –0.01 –0.01 –0.03 –0.08

Corr(rD,et→t+∆t, `n(Ct+∆t/Ct)) 0.20 0.20 0.20 0.09

Corr(rD,et→t+∆t, `n(Ct+2∆t/Ct+∆t)) 0.00 0.00 0.01 0.49

Corr(rD,et→t+∆t, `n(Ct+3∆t/Ct+2∆t)) –0.00 0.00 0.01 0.05

The table reports correlations between log consumption growth and log excess returns of the stock

market. The log consumption growth and log excess returns are computed at either a monthly,

a quarterly, or an annual horizon. Correlations are either contemporaneous or with a lead-lag

structure; for instance, at a monthly horizon (∆t = 1/12), Corr(rD,et→t+∆t, `n(Ct+2∆t/Ct+∆t)) is

the correlation between the current monthly log excess return and the log consumption growth in

the subsequent month. The theoretical values are calculated using 10,000 years of monthly data

simulated from the model. The parameter values are: gC = 1.91%, gD = 2.45%, σC = 3.8%, σD =

11%, ρ = 0.2, γ = 10, ψ = 0.9, δ = 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL = –12.5%. The

empirical values are from Campbell and Cochrane (1999).

55

Table 7. Return Expectations.

15 yr. 50 yr.

b0.007 0.007

(3.6) (4.4)

a 0.08 0.08

R-squared 0.25 0.16

The table reports the regression coefficient and the t-statistic (in parenthesis), the intercept, and

the R-squared, estimated from

Eet [(dPDt +Dtdt)/(PDt dt)] = a+ b ·RDt−12→t + εt,

over a sample of 15 years or 50 years. Each reported value is averaged over 100 trials, and each

trial represents a regression using monthly data simulated from the model. The t-statistics are

calculated using a Newey-West estimator with twelve-month lags. The parameter values are: gC

= 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ = 0.9, δ = 2%, θ = 0.5, χ =

0.022, µH = 12.5%, and µL = –12.5%.

56

Table 8. Campbell-Shiller Decomposition.

Realized dividend growth and returns

Coefficient Value R-squared

DRobjective 0.98 0.22

CFobjective 0.03 2.8× 10−4

Subjective expectations about dividend growth and returns

Coefficient Value R-squared

DRsubjective −0.07 0.96

CFsubjective 1.09 0.98

The table reports the four coefficients, CFobjective, DRobjective, CFsubjective, and DRobjective, defined

in equations (29) and (30) of the main text, as well as their corresponding R-squared. These coeffi-

cients and R-squared are calculated using 10,000 years of monthly data simulated from the model.

At each point in time, for a given level of sentiment, subjective expectations about dividend growth

and returns are calculated as the average values of 100 trials. Each trial is 100 years of monthly

simulated data under the agent’s expectations with the given initial level of sentiment. For realized

dividend growth and returns, both∑∞

j=0 ξj∆d(t+j∆t)→(t+(j+1)∆t) and

∑∞j=0 ξ

jrD(t+j∆t)→(t+(j+1)∆t)

are approximated using 100 years of monthly simulated data. From the simulated data, ξ = 0.9970.

The other parameter values are: gC = 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2, γ =

10, ψ = 0.9, δ = 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL = –12.5%.

57

Table 9. Return Predictability Regressions in the True Regime-Switching Model.


Horizon (years)10×

coefficient

102×R-squared

10×coefficient

R-squared

1 0.2 0.08 –1.3 0.04

2 0.3 0.16 –2.8 0.08

3 0.5 0.21 –3.5 0.09

5 0.7 0.28 –6.0 0.18

7 0.8 0.26 –7.5 0.23

The table reports the regression coefficient βj and the R-squared for a regression of the log excess

return of stock market from time t to time t+ j on the current log price-dividend ratio `n(PDt /Dt)

rD,et→t+j = αj + βj`n(PDt /Dt) + εj,t,

where j = 1, 2, 3, 5, and 7 (years). The theoretical values are calculated using 10,000 years

of monthly data simulated from the true regime-switching model described in Section IV. The

parameter values are: gC = 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2, γ = 10, ψ = 0.9, δ

= 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL = –12.5%. The empirical values are from Campbell

and Cochrane (1999).

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Table 10. Return Expectations in the Fundamental Extrapolation Model.

15 yr. 50 yr.

b–0.03 –0.02

(–1.12) (–1.62)

a 0.09 0.09

R-squared 0.14 0.05

The table reports the regression coefficient and the t-statistic (in parenthesis), the intercept, and

the R-squared, estimated from

Eet [(dPDt +Dtdt)/(PDt dt)] = a+ b ·RDt−12→t + εt,

over a sample of 15 years or 50 years. Each reported value is averaged over 100 trials, and each trial

represents a regression using monthly data simulated from the fundamental extrapolation model

described in Section IV. The t-statistics are calculated using a Newey-West estimator with twelve-

month lags. The parameter values are: gC = 1.91%, gD = 2.45%, σC = 3.8%, σD = 11%, ρ = 0.2,

γ = 10, ψ = 0.9, δ = 2%, θ = 0.5, χ = 0.022, µH = 12.5%, and µL = –12.5%.

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