Asset Pricing with Distorted Beliefs: Are Equity Returns Too Good to Be True?

Asset Pricing with Distorted Beliefs: Are Equity ReturnsToo Good to Be True?

By STEPHEN G. CECCHETTI, POK-SANG LAM, AND NELSON C. MARK*

We study a Lucas asset-pricing model that is standard in all respects, except that therepresentative agent’s subjective beliefs about endowment growth are distorted.Using constant relative risk-aversion (CRRA) utility, with a CRRA coefficient below10; fluctuating beliefs that exhibit, on average, excessive pessimism over expan-sions; and excessive optimism over contractions (both ending more quickly than thedata suggest), our model is able to match the first and second moments of the equitypremium and risk-free rate, as well as the persistence and predictability of excessreturns found in the data.(JEL E44, G12)

Ever since Rajnish Mehra and Edward Pres-cott (1985) first articulated the equity premiumpuzzle, achieving an understanding of aggregateasset-price dynamics has occupied a central rolein macroeconomic research. In their originalstatement of the problem, Mehra and Prescottasked whether the rational, complete marketsasset-pricing model, where the representativeinvestor has constant relative risk-aversion(CRRA) utility with a coefficient below 10 anda discount factor between 0 and 1, could ac-count for the approximately 8-percent per an-num sample mean return on the Standard andPoor’s index and the approximately 2-percentper annum sample mean return on relativelyrisk-free short-term bonds. Their answer was aresounding no, and the inability of that model toexplain the first moment of asset-returns datahas come to be known as the equity premium

puzzle. The list of challenges taken up by ag-gregate asset-pricing theory has grown substan-tially since Mehra and Prescott’s originalinvestigation. In addition to the mean equitypremium and risk-free rate, this paper seeks tounderstand the volatility, persistence, and long-horizon predictability of asset returns as well astheir relationship to the business cycle.

Because the power of models with commonknowledge and fully rational agents to explainthese aspects of asset-return dynamics has beenpoor [see Narayana R. Kocherlakota (1996) fora survey of this research], we take an alternativeapproach that allows for small departures fromrationality in an otherwise standard Robert E.Lucas, Jr. (1978) representative-agent asset-pricing model of an endowment economy. Theagents in our model satisfy the Mehra-Prescottcriteria of having CRRA utility with a relativerisk-aversion coefficient below 10 and a dis-count factor below 1. Agents observe that theactual endowment process shifts stochasticallybetween high- and low-growth states, but theirbeliefs about the transition probabilities thatgovern this switching deviate from the trueprobabilities.

These belief distortions enter along two em-pirically plausible dimensions. First, we intro-duce systematic deviations of the subjectiveprobabilities of transitions between high- andlow-growth states. We motivate these distor-tions by showing how agents who use simplerules of thumb to estimate the transition proba-bilities governing endowment growth will form

* Department of Economics, 410 Arps Hall, Ohio StateUniversity, 1945 North High Street, Columbus, OH 43210;Cecchetti: also National Bureau of Economic Research. Wehave benefited from comments from seminar participants atthe University of Canterbury, the Chinese University ofHong Kong, Erasmus University, the Federal Reserve Bankof New York, Georgetown University, Michigan State Uni-versity, the University of Montreal, Ohio State University,Victoria University of Wellington, and the University ofWisconsin. We thank Peter Bossaerts, John Campbell, Mar-tin D. D. Evans, Peter Howitt, J. Huston McCulloch, JamesNason, and two anonymous referees for useful comments onearlier drafts. Some of this work was conducted whileCecchetti was visiting the Department of Economics at theUniversity of Melbourne, and while Mark was visiting theTinbergen Institute, Rotterdam.

787

subjective probabilities that deviate frommaximum-likelihood estimates (MLE) obtainedfrom U.S. per capita consumption growth data.Rule-of-thumb estimates that imply relativelypessimistic beliefs about the persistence of theexpansion state (with the subjective probabilityof continuation being lower than the MLE) andrelatively optimistic beliefs about the persis-tence of the contraction state (with the subjec-tive probability of continuation being lowerthan the MLE), allow us to match the meanequity premium and risk-free rate found in thedata and solve the equity premium puzzle.

Unfortunately, systematic distortions aloneare not sufficient to explain the volatility ofasset returns or the pattern of serial correlationand predictability exhibited in the data. To gobeyond an explanation of the first moments ofasset returns, we introduce a second distortionin which beliefs about the transition probabili-ties fluctuate randomly about their subjective(distorted) mean values. This randomization isour way of modeling the effects of the type ofnoisefirst suggested by Fischer Black (1986).Because the economic environment is complexand noisy, Black argued that individuals are notalways able to distinguish noise from informa-tion. For example, investors may misinterpretthe guarded statements from the chairman of theFederal Reserve System or the rise in the pop-ularity of a particular political candidate as in-formation, when it is not. These pseudosignalsbecome determinants of beliefs and of assetprices because people occasionally mistakethem for news about fundamentals. An impor-tant implication of our analysis is that random-ization of a particular type is required to matchthe data—not randomization per se. Specifi-cally, it is random fluctuations in beliefs aboutthe persistence of the low endowment growthstate that generate volatility and predictabilityin asset returns. Furthermore, it is necessary forthe subjective transition probabilities them-selves to be quite persistent.

An important ingredient in our formulation ofthe asset-pricing problem is that investors are in-capable of learning that their distorted beliefs aredistorted. With 104 years of data, Bayesian learn-ers, for example, would almost surely havelearned how to be rational by now. But, as is wellknown, the model performs poorly under rationalexpectations with CRRA utility, and so the intro-

duction of optimal learning is unlikely to aid muchin understanding asset-returns data. If there islearning, it would have to be very gradual andexplicit modeling of slow learning, although in-teresting in its own right, would introduce sub-stantial complexity, while contributing onlymodest insight into the questions at hand.

Instead, our work is part of a growing literaturethat studies the implications of local (and persis-tent) departures from full rationality. For example,John H. Cochrane (1989) demonstrates that adop-tion of rule-of-thumb behavior instead of optimaldecision rules entails only trivial economic costs,even when agents know the objective probabilitylaw governing the driving processes. Robert Bar-sky and J. Bradford DeLong (1993) find that longswings in stock prices can be explained by thepresent-value model if people believe that divi-dendgrowth, which standard statistical analysissuggests is stationary, contains a unit root. Lars P.Hansen et al. (1999) show that in their linear-quadratic specification, the combination of habitformation and distorted beliefs allows them tomimic the behavior of the market price of risk.

Introducing belief distortions contrasts sharplywith the more common approach in which fullrationality is preserved, and asset-price puzzles areresolved by introducing increasingly complexpreference structures. For example, John Y.Campbell and Cochrane (1999) are able to pro-vide a fully rational, unified explanation for thesame asset-pricing anomalies we seek to explainby using a utility function that displays a sophis-ticated form of habit persistence. Instead, we re-tain the simplicity of time-separable CRRA utilitybut allow for what we believe to be reasonabledepartures from full rationality.

The remainder of the paper is organized asfollows. The next section reports the stylizedfacts for equity and bond returns that we seek toexplain. Section II presents the fully rationalLucas asset-pricing model that serves as abenchmark for our analysis. In Section III, weintroduce distorted beliefs and discuss how themodel is solved. Section IV reports on the mod-el’s solution to the many empirical puzzles.Section V contains some concluding remarks.

I. Stylized Facts of Asset Returns

Table 1 presents the list of anomalies aroundwhich we organize our investigation. The data on

788 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2000

equity and short-term bond returns is updatedfrom Campbell and Robert J. Shiller (1988b). Theequity data are the Standard and Poor’s 500 PriceIndex and Dividends. The nominally risk-free rateis the return from six-month commercial paperbought in January and rolled over in July. Usingthe Producer Price Index, we measure expectedinflation as the fitted value from a regression ofinflation on two lags each of inflation, the nomi-nally risk-free rate, and nominal equity returns.The real risk-free rate is then constructed by sub-tracting expected inflation from the nominallyrisk-free rate.

The first two lines of panel A of Table 1 re-port our estimates of a mean level of the equitypremium somewhat above 5.5 percent, and the

mean level of theex anterisk-free rate a bit over2.5 percent. Next, we report estimates for thevolatility of the equity premium and the risk-free rate, and the correlation between these re-turns. As has been noted before, these facts posea volatility puzzle, because it is difficult tomatch simultaneously the standard deviation ofthe equity premium, which is nearly 20 percentper year, with the standard deviation of therisk-free rate, which is only 5 percent per year.1

Third, we report variance ratio statistics forthe equity premium in the fourth column ofpanel B of Table 1.2 These provide a measure ofthe serial correlation properties of the data. Thatthe variance ratio statistics are generally lessthan 1 and fall with the horizon suggests thatexcess equity returns are negatively serially cor-related, or that asset prices are mean reverting.3

Fourth, we study the phenomenon that thelog(dividend yield) predicts long-horizon real-ized excess returns. This predictability puzzle,first described in work by Campbell and Shiller(1988a) and Eugene F. Fama and Kenneth R.French (1988), can be characterized by regres-sions of thek-period realized excess return onthe log(dividend–price ratio). As can be seenfrom the second and third columns of panel B ofTable 1, the predictive regressions run on ourdata display the familiar pattern of slope coef-ficients andR2’s that increase with the returnhorizon.

Finally, we measure the cyclicality of theequity premium in two ways, both based on theNBER reference cycle chronology. In the first,we compute the mean equity premium for yearsthat both begin and end in expansions, and forthose years beginning and ending in contrac-tions. We also report the mean equity premiumcomputed from years that end in expansions andthose that end in contractions, regardless ofstate of the business cycle at the beginning ofthe year. In both cases the equity premium ishigher in expansions than that in contractions.

1 For analyses of the second moments, see George M.Constantinides (1990) and Cecchetti et al. (1993).

2 The variance ratio statistic, popularized by Cochrane(1988), is the variance of thek-year excess return divided byk times the variance of the one-year excess return.

3 Cecchetti et al. (1990) and Shmuel Kandel and RobertF. Stambaugh (1991) study the theory’s implications formean-reverting equity returns.

TABLE 1—STYLIZED FACTS OF EQUITY AND SHORT-TERM

BOND RETURNS USING ANNUAL OBSERVATIONS FROM

1871–1993

A. First and second moments as a percentage

Mean equity premium (meq) 5.75Mean risk-free rate (mf) 2.66Standard deviation

Equity premium (seq) 19.02Risk-free rate (sf) 5.13

Correlation (req,f) 20.24

B. Predictability and persistence

Horizon Regression slope R2 Variance ratio

1 0.148 0.043 1.0002 0.295 0.081 1.0383 0.370 0.096 0.9215 0.662 0.191 0.8798 0.945 0.278 0.766

C. Cyclicality of equity premium

Average of yearsbeginning and ending in Average of years ending in

Expansions Contractions Expansions Contractions

10.54 214.63 13.66 211.24

Notes:The data and methods are described in the text. Theregression slope andR2 are for regressions of thek-year(k 5 1, 2, 3, 5, 8) ahead equity premium on the current logdividend–price ratio. The variance ratio is the variance ofthe k-year equity premium divided byk times the varianceof the one-year equity premium. Cyclicality is computed asthe mean return for years that begin and end, or those thatsimply end, in either expansions or contractions. The stageof the business cycle at the beginning and end of years isdetermined by the NBER reference chronology.

789VOL. 90 NO. 4 CECCHETTI ET AL.: ASSET PRICING WITH DISTORTED BELIEFS

II. The Rational Economy

Our main purpose is to examine the implica-tions of departures from rationality. To providea benchmark for our investigation, we begin bysummarizing the properties of the fully rationalmodel—a variant of Lucas’s (1978) representa-tive agent endowment economy—that hasserved as the workhorse in aggregate asset-pricing studies. LetP be the price of the equity,which is a claim to the future stream of thenonstorable endowment called dividends (D).One perfectly divisible share of the equitytrades in a competitive market. The first-ordercondition that must hold if the agent behavesoptimally is

(1) P 5 E~@M9~P91D9!#uI !,

where primes denote next-period values,M isthe intertemporal marginal rate of substitution,and E( z uI ) is the representative individual’ssubjective expectation conditioned on currentlyavailable information (I ). Under time-separableCRRA utility defined over consumption (C),M9 5 b(C9/C)2g, whereg . 0 is the coeffi-cient of relative risk aversion and 0, b , 1 isthe subjective discount factor. Following thestandard practice of modeling per capita con-sumption as the endowment, and assuming thatit is consumed in equilibrium (D 5 C), equa-tion (1) becomes

(2) P 5 bESFSC9

C D2g

~P91C9!G U ID .

Now definev [ P/C as the price–consumptionratio and letc [ ln(C). Dividing equation (2) byCyields the stochastic difference equation inv,

(3) v 5 bE~@e~12g!Dc9~v911!#uI !,

whereD is the first difference operator.

A. The Endowment Process

We follow Cecchetti et al. (1990, 1993), whoassume thatDc evolves according to the follow-ing version of James D. Hamilton’s (1989)Markov switching process:

(4) Dc 5 a~S! 1 «,

where« is independently and identically distrib-uted (i.i.d.) asN(0, s2) andS is an underlyingtwo-point Markov state variable that assumesvalues of 0 or 1. We normalizeS 5 1 to be thegood (expansion) state of high-consumptiongrowth andS 5 0 to be the bad (contraction)state of low-consumption growth, so thata(1) . a(0). S evolves according to the fourtransition probabilities

(5) Pr~S9 5 1uS5 1! 5 p,

Pr~S9 5 0uS5 1! 5 1 2 p,

Pr~S9 5 0uS5 0! 5 q,

Pr~S9 5 1uS5 0! 5 1 2 q,

wherep is the probability of remaining in anexpansion if currently in one and 12 p is thetransition probability for moving from an ex-pansion to a contraction. Similarly,q is theprobability of remaining in a contraction if cur-rently in a contraction, whereas 12 q is thetransition probability of moving from a contrac-tion to an expansion.

Table 2 reports the maximum-likelihoodestimates of this Markov switching modelusing U.S. per capita consumption data ex-tending from 1890 to 1994.4 In our subse-

4 These data are updated from Cecchetti et al. (1990),who also perform a number of diagnostic tests on the

TABLE 2—MAXIMUM -LIKELIHOOD ESTIMATES OF THE ENDOWMENT PROCESS

p q a(1) a(0) s

Estimate 0.978 0.516 2.251 26.785 3.127(t ratio) (50.94) (1.95) (6.87) (23.60) (13.00)

Note: Per capita consumption growth rate as a percentage, 1890 to 1994.


quent analysis, we calibrate the endowmentby setting its parameter values equal to themaximum-likelihood estimates to ensure thatthe economy’s endowment growth evolvesaccording to the objectively true process.

Several features of the “truth” are worth not-ing. First, expansions are highly persistent.Given that the economy is in an expansion state,the probability that the expansion will end isless than 0.03. Moreover, the economy spendsmost of the time in the expansionary state. Theunconditional probability of being in an expan-sion is Pr(S 5 1) 5 (1 2 q)/(2 2 p 2 q) 50.96.Second, contractions are moderately per-sistent, with slightly more than an even chanceof continuing once they start. Finally, the badstate is very bad, as mean per capita consump-tion growth in the contractionary state is26.79percent.

B. Solution and Properties of the RationalEconomy

In the benchmark model, the subjective ex-pectations of individualsE[ coincide withthe mathematical expectationsE[, takenwith respect to the truth. This formulationgives rise to a particular complete marketsrational economy that is closely related to theenvironment studied by Cecchetti et al.(1990), Kandel and Stambaugh (1990), andMehra and Prescott (1985).

To solve for equilibrium returns, we first notethat because the shock« is independent ofS, wecan write the stochastic difference equation forthe price–consumption ratio,v in equation (3),in terms of the single state variableS,

(6) v~S! 5 be~12g!2~s2/2!

3 E~@1 1 v~S9!#e~12g!a~S9!uS!.

BecauseS can take on only the values 0 or 1,equation (6) is a system of two linear equationsin v(0) andv(1)

(7) F qb~0! 1 ~1 2 q!b~1!pb~1! 1 ~1 2 p!b~0! G

5 F 1 2 qb~0! 2~1 2 q!b~1!2~1 2 p!b~0! 1 2 pb~1! G

3 F v~0!v~1! G ,

where b(S) [ be(12g)2(s2/ 2)1(12g)a(S), for S5 0, 1. Solving the system (7) yields

(8) v~0! 5

qb~0! 1 ~1 2 q!b~1!2 ~ p 1 q 2 1!b~0!b~1!

D

and

(9) v~1! 5

pb~1! 1 ~1 2 p!b~0!2 ~ p 1 q 2 1!b~0!b~1!

D,

where D 5 1 2 qb(0) 2 pb(1) 1 pqb(0)b(1).5

The next step is to characterize the solutionfor one-period returns. Gross equity returnsRe

are given by

(10) Re~S9, S! 5 Fv~S9! 1 1

v~S! Gea~S9!1«9.

Since the price of a one-period risk-free assetPf

is the expected intertemporal marginal rate ofsubstitution

(11) Pf~S! 5 beg2~s2/2!E@e2ga~S9!uS#,

the implied gross risk-free rateRf is,

(12) Rf~S! 51

Pf~S!.

specification and conclude that this estimated Markov-switching model is a plausible representation of the data-generating process.

5 The information structure here corresponds to Cec-chetti et al. (1993) and Andrew B. Abel (1994), in that theinformation set, on which asset prices are based, includesSt2i and «t2i , for i 5 0, 1, 2, ... .


It is now well known that many features ofasset-returns data can be explained using thestandard model by choosing both a large valueof g (e.g., 57) and a value ofb in excess of 1.6

The challenge, as articulated by Mehra andPrescott (1985) and Kocherlakota (1996), is toproduce a model that explains the data withvalues ofb between 0 and 1, and positive valuesof g below 10. To establish the model’s bench-mark performance, we restrict our attention tovalues ofg less than 10 and associated values ofb less than 1, which predict a mean risk-freerate of 2.5 percent, subject to satisfying thetransversality condition of the model.7 Table3 displays the implied behavior of asset pricesfor selected values of the preference parameterswhen the representative agent is fully rational.The model is seen to fail badly: there is virtually

no equity premium, the volatility of equity re-turns is far below its sample value, and excessreturns have neither the persistence nor the pre-dictability found in the data.

III. A Distorted Beliefs Economy

The implied behavior of asset returns is basedon a model of preferences (M) and a model ofbeliefs (E). In the benchmark model, prefer-ences are CRRA and beliefs are rational. Inconsidering departures from this basic formula-tion, one line of asset-pricing research has beenaimed at retaining rational expectations, but en-riching the model of preferences. For example,Larry G. Epstein and Stanley E. Zin (1989,1991), Philippe Weil (1989), and Kandel andStambaugh (1991) examine recursive nonex-pected utility, whereas Abel (1990), Constantin-ides (1990), Wayne E. Ferson and Constantinides(1991), and John Heaton (1995) study versionsof habit persistence. Recently, Campbell andCochrane (1999) report success in accounting formany of the same features of the data that weconsider by retaining rational expectations but by

6 See Kocherlakota (1990), Kandel and Stambaugh(1991), and Cecchetti et al. (1993).

7 The transversality condition requires that the powerseries for the price level implied by the difference equation(2) yield finite values. Practically, this requirement can bechecked by verifying that implied values forv(S) are al-ways positive.

TABLE 3—IMPLICATIONS OF THE FULLY RATIONAL MODEL

PreferencesFirst and second

moments

Predictability and persistence

b g Horizon Slope R2 Variance ratio

0.976 0.0 meq 0.0 1 0.209 0.032 1.000mf 2.5 2 0.276 0.034 0.987seq 4.4 3 0.313 0.032 0.977sf 0.0 5 0.357 0.026 0.959req,f 0.000 8 0.421 0.023 0.933

0.985 0.5 meq 0.1 1 0.335 0.025 1.000mf 2.5 2 0.445 0.027 0.987seq 4.0 3 0.504 0.026 0.975sf 0.5 5 0.573 0.023 0.956req,f 20.007 8 0.667 0.021 0.929

1.010 2.0 meq 0.2 1 0.007 0.007 1.000mf 2.5 2 0.010 0.010 0.991seq 3.2 3 0.011 0.012 0.982sf 1.9 5 0.012 0.014 0.965req,f 0.002 8 0.013 0.016 0.939

1.026 3.0 meq 0.2 1 0.077 0.018 1.000mf 2.5 2 0.105 0.020 0.977seq 3.9 3 0.119 0.021 0.959sf 2.8 5 0.134 0.020 0.932req,f 0.034 8 0.150 0.020 0.901

Note: See Table 1 for moment definitions.


complicating preferences in a way that displays aparticular form of habit persistence.

Our line of attack corresponds to that ofCampbell and Cochrane (1999). The returnsbehavior they achieve by varyingM for fixed Ecan also be obtained for a givenM with theappropriate specification ofE—that is, by dis-torting beliefs. The interesting question, then, iswhat sorts of departures from rationality arenecessary and how large do the departures needto be?8 In the remainder of this section, weintroduce our model with distorted beliefs andcharacterize the solution for asset returns. Sec-tion IV follows with empirical results.

A. Modeling Distorted Beliefs

Because the expected endowment growthrateE(Dc9) 5 E[a(S9)] depends on the statetransition probabilities, variations in the subjec-tive transition probabilities alone will inducechanges in subjective endowment growth. Toseparate the effects of misperceived state per-sistence and misperceptions of endowmentgrowth, we explicitly model both of these as-pects of beliefs. We define all subjective beliefparameters to be nonnegative and denote themwith a tilde accent.

Agents observe the true state of the economyS, but not the true transition probabilitiesp andq or the true growth ratesa(0) anda(1). Whenthe endowment is in the expansionary state,their beliefs aboutp anda(1) are governed by

(13) p~Se! 5 m p 1 dpSe

and

(14) ae~Se! 5 me 1 teSe,

whereSe is an independent two-point Markovstate variable that assumes values 1 or21 withthe symmetric transition probabilities

(15) fe 5 Pr@S9e 5 1uSe 5 1#

5 Pr@S9e 5 21uSe 5 21#

and

(16) 12 fe 5 Pr@S9e 5 1uSe 5 21#

5 Pr@S9e 5 21uSe 5 1#,

where p(Se) is the subjective persistence of theexpansionary state and has unconditional meanmpand ae(Se) is the subjective endowment growthduring the expansion state with unconditionalmean me. Subjective persistence and subjectiveendowment growth are perfectly correlated ifdp,te . 0. We say that agents display optimism overthe expansion state whenSe 5 1 because subjec-tive persistence and subjective endowment growthexceed their mean values. Similarly, we say thatagents display pessimism over the expansion statewhen Se 5 21. Beliefs will exhibit systematicdistortion during the expansion state whenmp Þ p5 0.98 andme Þ a(1) 5 2.25.

When the economy is in the contractionarystate, agents beliefs aboutq anda(0) are gov-erned by

(17) q~Sc! 5 m q 2 dqSc

and

(18) ac~Sc! 5 mc 1 tcSc ,

8 In a linear-quadratic environment, Hansen et al. (1999)show that a model with rational expectations and recursivenonexpected utility combined with habit persistence is equiv-alent to a model with expected utility in conjunction with aparticular form of distorted beliefs. Hansen et al. interpret thesebelief distortions as the way that agents deal with Knightianuncertainty—the situation where information is too impreciseto be summarized by probabilities—as studied by Epstein andTan Wang (1994). Their representative consumer with qua-dratic expected utility can be thought of as facing a malevolentopponent, whose purpose is to minimize the welfare of theconsumer by distorting the dynamics of the economy. Fromthat perspective, they interpret the belief distortions as encod-ing the sophisticated attempts of the consumer to accommo-date specification errors. The actions of the hypotheticalmalevolent opponent generates “the worst-case scenarios” inthe Maxmin theory of Itzach Gilboa and David Schmeidler(1989).


whereSc is an independent two-point Markovstate variable that assumes values 1 or21 withthe symmetric transition probabilities

(19) fc 5 Pr@S9c 5 1uSc 5 1#

5 Pr@S9c 5 21uSc 5 21#

and

(20) 12 fc 5 Pr@S9c 5 1uSc 5 21#

5 Pr@S9c 5 21uSc 5 1#,

where mq is the mean value of the subjectivecontractionary state persistenceq(Sc) andmc isthe mean value of the subjective contractionarygrowth rateac(Sc). Beliefs exhibit contraction-ary state optimism whenSc 5 1 because sub-jective persistence of the contractionary statelies below its mean value and subjective con-tractionary state endowment growth lies aboveits mean. Beliefs exhibit contractionary statepessimism whenSc 5 21. Systematic beliefdistortions over the contraction state are presentwhen mq Þ q 5 0.52 and mc Þ a(0) 526.79.

To complete the specification, note that thestate vector is nowS# 5 (S, Se, Sc). Agents observeand condition on the true stateSwhen they formtheir expectations. Accordingly, the subjective en-dowment growth rate is given by

(21) a~S# ! 5 H ae~Se! if S5 1ac~Sc! if S5 0.

The fully rational model is nested in the distortedbeliefs model, and is obtained by settingdp 5 dq5 te 5 tc 5 0; mp 5 p 5 0.98;mq 5 q 5 0.52;me 5 2.25; andmc 5 26.79.

B. Discussion

To summarize our strategy, we assume thatinvestors believe that the underlying structureof the economy is more complex than the truth.They form overactive subjective beliefs con-cerning expansion and contraction growth andstate persistence, but are rational in every otherway. That is, given their distorted beliefs about

the endowment process, they use the correctstructure to solve for asset prices. Retaining thiselement of rationality provides a natural reduc-tion in the number of admissible models toconsider and imposes a certain discipline on ourinvestigation.

Investors are susceptible to two types of dis-tortions in their beliefs about the evolution ofthe endowment process that determines assetreturns. First, their subjective beliefs about thestate transition probabilities differ systemati-cally from the truth. A plausible explanation forthese systematic distortions is that individualsfind it too costly to acquire the skills to domaximum-likelihood estimation or to performintegration with respect to multivariate densitieswhen making decisions about everyday life. In-stead, they respond by using rules of thumb thatgive approximately the right answer. The fol-lowing simple rule-of-thumb calculations guideour modeling of the systematic distortion in thesubjective transition probabilities.

Suppose that agents realize that consumptiongrowth is governed by a two-state Markov pro-cess, but don’t know the transition probabilities.A sensible person might pursue the followingstrategy for “estimating”p and q. Begin byobtaining a copy of the NBER reference cyclechronology, and convert it to an annual fre-quency.9 Next, use this to estimateq as theproportion of years in which contractions arefollowed by contractions, and then estimatethe unconditional probability of being in anexpansion [(12 q)/(2 2 p 2 q)] as theproportion of years in expansion. An estimateof p then follows. This simple exercise yieldsan estimate forq equal to 0.273 and an esti-mate forp equal to 0.667, both well below themaximum-likelihood estimates reported inTable 2.10 We take these rule-of-thumb cal-

9 This information is available on the NBER’s homepage at http://www.nber.org. In making the conversion froma monthly to an annual frequency, years in which a majorityof the months are a contraction are assumed as being in thelow-growth state.

10 A second, and equally simple, method of generatingrule-of-thumb estimates forp and q, uses per capita con-sumption growth data and yields similar results. Begin byassigning all years in which per capita consumption con-tracted to the bad state and the remainder, those with pos-itive per capita consumption growth, to the good state. Thenuse the same simple technique based on the persistence of


culations to suggest that we should concen-trate on subjective beliefs that exhibitexcessive pessimism about expansion statesp, p and excessive optimism about contrac-tion statesq , q.

We also allow beliefs to be distorted along asecond dimension: the subjective transitionprobabilities move randomly about their subjec-tive means.That is, p and q fluctuate based onpseudosignalsSe and Sc, which are pure noise,unrelated to fundamentals, in the sense of Black(1986).11

It bears emphasizing that we are permittingdistortions only to subjective beliefs of the en-dowment and not to the actual endowment—which continues to evolve according to thetruth. This is a subtle but important point thatdistinguishes our approach from that of ThomasRietz (1988), in which the actual endowmentevolves according to a distorted process.

Finally, we note that it is also possible to inter-pret our model in the context of the “peso prob-lem” literature, in which it is the agents who arerational and the econometrician who errs in be-lieving that endowment growth follows the two-state Markov process of equations (4) and (5). Inthis interpretation, the true data-generating processfor the endowment is presumed to conform to theprobability model of equations (13) through (20),which we label as distorted. The econometrician isled to the wrong model (which we label as thetruth) by attempting estimation and inference on asample of insufficient size. Because of our limitedhistorical experience with consumption growth,realizations are absent in regions of the samplespace necessary to identify and to estimate the truemodel.

Problems of limited experience and the associ-ated difficulties associated with drawing inferencefrom small samples form the foundation for bothinterpretations. Whether the error is attributed to

the economic agents (distorted beliefs) or to theeconometrician (peso problem) is a philosophicalissue that we cannot resolve here. We simply notethat some readers may be more comfortable inter-preting our model as one of a peso problem facedby an econometrician rather than one of agentswith distorted beliefs.12

C. Solving the Distorted Beliefs Model

Returning to the analytical task, we need tocharacterize the behavior of asset prices in thedistorted beliefs model. As was the case in therational model, the solution centers on solvingfor the dynamics of the price–consumption ra-tio v, which is a function of the state vectorgoverning the endowment process. In the fullyrational model, the state vector can take on onlytwo values (S equals 0 or 1). The state vector inthe distorted beliefs model includes thepseudosignalsSe and Sc, both of which caneach take on values of21 and 1.

We define the distorted beliefs state vector asS# [ (S, Se, Sc), which can take on eightpossible values. The price–consumption ratiov(S# ) now evolves according to the stochasticdifference equation

(22) v~S# ! 5 be~12g2!~s2/2!

3 E~@11v~S# 9!#e~12g!a~S# 9!uS# !,

which is the analog to equation (6).The next step in finding the solution is to

define P, the subjective transition matrixwhoseij th element (i , j 5 1, ... , 8) isPij 5Pr[S# 9 5 S# j uS# 5 S# i ]. (A full description of Pand S# is given in the Appendix.) Then equa-tion (22) can be written as

(23) v~S# i ! 5 be~12g!2~s2/2!

3 Oj51

8

Pij @1 1 v~S# j !#e~12g!a~S# j !,

i 5 1, ... , 8.

contractions and the unconditional probability of expansions toestimate the transition probabilities. The result is an estimatefor p equal to 0.760 and an estimate forq equal to 0.269.

11 Abel (1997) models a related set of distortions. Thesystematic distortion inp and q loosely corresponds to“pessimism” in Abel’s model, which describes the situationwhen subjective distributions are first order, dominated bythe objective distribution. Randomization inp andq corre-sponds to “doubt” in Abel’s model, which is said to occurwhen the subjective distribution is a mean-preservingspread over the objective distribution.

12 In the original peso problem, the agent rationallyattaches a nonzero probability to the event in which themonetary authorities would devalue the currency, eventhough the monetary history contained no such devalua-tions. Martin D. D. Evans (1998) recently investigated pesoproblem implications for stock returns.


Rewriting equation (23) in vector notation, theanalog to equation (7) is

(24) v 5 Gv 1 f 5 ~I 2 G!21f ,

where I is an eight-dimensional identity matrix,vi 5 v(S# i), Gi j 5 be[(12g)2(s2/2)1(12g)a(S# j)]Pij ,and fi 5 be[(12g)2(s2/2)] ¥j51

8 Pij e[(12g)a(S# j)], fori, j 5 1, ... , 8. These definitions allow us to writethe implied one-period gross equity return, theanalog to equation (10), as

(25) Re~S# 9, S# ! 5 Fv~S# 9! 1 1

v~S# ! Ge@a~S# 9!1«9#.

The final task is to determine the return on aone-period risk-free bond, and the equity pre-mium. This begins with a computation of theprice of the bond, which is a function of thestate vectorS# . When the state takes on any of itseight possibilities, denoted byS# i , then the priceof the one-period risk-free bond can be writtenas

(26) Pf~S# i ! 5 be~g2s2/2! Oj51

8

Pij e2ga~S# j !.

The gross risk-free rate of return follows imme-diately as the inverse of this priceRf(S# ) 51 /Pf (S# ), and the excess return on equity isthen just Rp(S# 9, S# ) 5 Re(S# 9, S# ) 2 Rf(S# ).The moments of the equity premium and therisk-free rate are then computed with respect to theobjective probability distribution ofS# .

IV. Properties of the Distorted Beliefs Model

We assess the value added of the separatecomponents of our model by proceeding in aseries of steps. First, we investigate the role ofsystematic distortions in subjective transitionprobabilities by fixingp and q at values thatdeviate from p and q. Next, we randomizebeliefs by allowing the subjective transitionprobabilities p and q to evolve as i.i.d. pro-cesses. Third, we introduce persistence in thestochastic subjective transition probabilities byallowing the pseudosignalsfe andfc to deviate

from 0.5. Throughout these first three experi-ments, subjective endowment growth is nonsto-chastic and undistorted, witha(S# ) 5 a(S). In afourth experiment, we relax this restriction andallow for randomized subjective endowmentgrowth, in addition to randomized conditionalsubjective transition probabilities. In the fifth,and final, experiment we relax the assumptionthat the persistence of the pseudosignalsfe andfc is itself undistorted and allow investors toirrationally forecast their own irrational beliefs.

A. Systematic Distortions in SubjectiveTransition Probabilities

To what extent can the asset price puzzles setforth in Section I be resolved by the introduc-tion of perceptions that are systematically dis-torted? We begin to answer this question bysetting beliefs over (p, q) to fixed values suchthat p 5 mp Þ p 5 0.98 andq 5 mq Þ q 50.52, with dp 5 dq 5 te 5 tc 5 0. To keepconditional subjective endowment growth un-distorted, we setae 5 me 5 a(1) 5 2.25 andac 5 mc 5 a(0) 5 26.79.

Table 4 reports parameter values for prefer-ences and beliefs that solve the equity premiumpuzzle nearly exactly. These are combinationsof ( p, q, b, g) for which the mean equitypremium and mean risk-free rate predicted bythe model are 5.5 and 2.5 percent, respectively.The region of the parameter space that accom-

TABLE 4—FIXED DISTORTED BELIEFS: IMPLIED SECOND

MOMENTS FORPARAMETER VALUES THAT SOLVE THE

EQUITY PREMIUM PUZZLE

Beliefs Preferences Second moments

p q b g seq sf req,f

0.5 0.2 0.933 1.831 4.3 1.1 20.420.5 0.4 0.918 2.371 4.1 0.5 20.31

0.5 0.8 0.965 0.429 4.0 0.2 0.11

0.6 0.2 0.923 3.159 4.5 1.3 20.410.6 0.4 0.825 6.985 3.9 0.0 0.00

0.6 0.9 0.969 0.411 4.2 0.4 20.01

0.7 0.1 0.930 4.691 4.9 2.1 20.470.7 0.2 0.909 5.795 4.5 1.3 20.38

0.7 0.9 0.975 0.150 4.9 0.2 20.12

0.8 0.1 0.933 8.300 5.0 2.1 20.44


plishes this feat is quite large, as it includesvalues ofp ranging from 0.5 to 0.8, and valuesof q varying between 0.1 to 0.9. Overall, wenote that relatively mild degrees of relative riskaversion are required to match the first momentsof returns.

The systematic belief distortions presented inTable 4 fall into two categories. The first arethose in which moderately risk-averse agentsdisplay pessimism about expansion persistence( p , p), believing that expansions are shorterthan the data indicate, but display optimismabout contraction persistence (q , q), thinkingthey are shorter than suggested by the MLEtruth. For example, when (p, q) 5 (0.7, 0.2),roughly our rule-of-thumb values, (b, g) 5(0.91, 5.8) solve the equity premium puzzle.The second category of systematically distortedbeliefs that allows us to match the first momentsare those in which nearly risk-neutral individu-als are uniformly pessimistic in thinking thatboth expansions will be shorter (p , p) andcontractions longer (q . q) than the data jus-tify. For example, with (p, q) 5 (0.6, 0.9), themean value of returns can be matched with (b,g) 5 (0.97, 0.41).

The job of matching the first moments of theequity premium and the risk-free rate is accom-plished primarily through a judicious choice oftwo parameters—p andg. To see why, note firstthat lowering p generates expansionary statepessimism. This strengthens the motive for pre-cautionary saving and reducesRf. This meansthat for a given (b, g, q)–triple, we can choosep to match the mean risk-free rate. Second, notethat as we increaseg, the increase in requiredcompensation for covariance risk will force upthe equilibrium equity return and reduce theagent’s willingness to substitute intertempo-rally. In other words, a higherg increases boththe equity return and the risk-free rate. In thefully rational model, it is possible to match theequity premium only for values ofRf that aretoo high, creating a risk-free rate puzzle, buthere, we can reducep to offset the impact ofhigherg and match both of the first moments.

We now turn our attention to the impliedvolatility of the equity premium and the risk-free rate, and the correlation between them,reported in the last three columns of Table4. Although the model can account for the factthat the equity premium is substantially more

volatile than the risk-free rate, and the ratio ofthe standard deviations is not far from the 3.7 inTable 1, the levels are much too low. Asset-return volatility in the data is between two- andfourfold what we can produce using the fixedbeliefs model. Evidently, to match the volatili-ties and correlation, systematic belief distor-tions are not enough, and so we now proceed tointroduce time variation in distorted beliefs.13

B. Stochastically Independent Beliefs of StatePersistence

The next step is to assume that agents’ beliefsabout regime persistence are random but inde-pendent. By settingdp anddq to nonzero valuesand fc 5 fe 5 0.5, thesubjective transitionprobabilitiesp and q will evolve as i.i.d. pro-cesses. In this section, subjective growth ratesremain undistorted, withae 5 a(1) 5 2.25,and ac 5 a(0) 5 26.79.

Table 5 reports the mean and volatility of theequity premium and the risk-free rate, and thecorrelation between these returns, predicted bythe i.i.d. subjective transition probability model.A nontrivial region of the admissible parameterspace allows the model to match mean returnsand the volatility of the risk-free rate, in addi-tion to producing a small negative correlationbetween the equity premium and risk-free rate.But, as is clear from the column labeled “seq,”i.i.d. variations in p and q do not raise theimplied volatility of the equity premium by asufficient magnitude, as it is still only one-halfto one-third the level of the standard deviationin the sample.

To investigate why the model with i.i.d.

13 The literature suggests that introducing leverage mighthelp the fixed distorted beliefs model to match the volatilityin returns. In Simon Benninga and Aris Protopapadakis(1990) the standard model is shown to match the first twomoments of asset returns with relative risk aversion coeffi-cient near 10 and a debt to market value ratio of 0.6, butthey also require negative time discounting. Abel (1999)combines a similar degree of leverage with a “catching upwith the Joneses” preference structure and matches thevolatility of asset returns with positive time preference. Wehave examined the impact of leverage, of a magnitude foundin the actual data, and find that with CRRA utility it doesincrease volatility, but not by enough to match the data.Furthermore, we suspect that a simple model with leveragewould be incapable of matching the persistence and predict-ability of asset returns.


randomization of subjective persistence fails tomatch the second moments, we examine thebehavior of the price–consumption ratio andmean asset returns during an expansion. Specif-ically, we consider a case in whichq is fixedandp fluctuates between high and low values byan amount that allows us to match the first

moments of asset returns and the volatility ofthe risk-free rate. This exercise mimics the onein Table 5 for cases withdq 5 0. The results ofthis exercise are reported in Table 6. As isevident from the last column of the table, vari-ation in the mean equity premium induced byshocks to p fall far short of the 19-percent

TABLE 6—RESPONSE OFASSET RETURNS AND PRICE–CONSUMPTION RATIO TO CHANGES IN p

b g q p vMean equity

returnMean risk-free rate

Mean equitypremium

0.933 1.831 0.2 0.826 16.513 10.260 8.088 2.1720.174 17.155 6.137 22.963 9.099

0.918 2.371 0.4 0.756 16.338 11.399 8.319 3.0800.244 17.312 5.133 22.903 8.036

0.923 3.159 0.2 0.783 16.347 11.502 7.994 3.5080.417 17.371 4.933 22.967 7.900

0.825 6.985 0.4 0.684 16.023 13.525 8.433 5.0920.516 17.598 3.360 22.817 6.177

0.930 4.691 0.1 0.812 16.356 11.690 7.536 4.1540.588 17.458 4.635 22.906 7.542

0.909 5.795 0.2 0.794 16.209 12.533 8.009 4.5240.606 17.541 3.991 22.967 6.957

0.933 8.300 0.1 0.856 16.274 12.379 7.533 4.8460.744 17.589 3.978 22.906 6.884

Notes:The conditional expectations used to computev and the risk-free rate are based onp and q, whereas the evaluationof the mean equity return and mean equity premium are based on this value ofv, along with the true values of the transitionprobabilitiesp andq. All calculations take the current state as expansionary.

TABLE 5—MEAN, STANDARD DEVIATIONS, AND CORRELATION WITH I.I.D. SUBJECTIVE TRANSITION PROBABILITIES

Beliefs and preferences First and second moments

mp mq b g dp dq meq mf seq sf req,f

0.5 0.2 0.933 1.831 0.33 0.0 5.25 2.78 5.80 5.5 20.6310.33 0.1 5.25 2.78 5.80 5.5 20.632

0.5 0.4 0.918 2.371 0.26 0.0 5.29 2.79 5.54 5.5 20.4520.25 0.1 5.29 2.79 5.54 5.5 20.452

0.6 0.2 0.923 3.159 0.18 0.0 5.32 2.78 5.80 5.5 20.4310.18 0.1 5.32 2.78 5.80 5.5 20.433

0.6 0.4 0.825 6.985 0.08 0.0 5.43 2.79 6.13 5.5 20.0850.08 0.1 5.43 2.79 6.12 5.5 20.090

0.7 0.1 0.930 4.691 0.11 0.0 5.36 2.75 6.12 5.5 20.3900.11 0.1 5.36 2.75 6.12 5.5 20.399

0.7 0.2 0.909 5.795 0.09 0.0 5.38 2.78 6.10 5.5 20.2510.09 0.1 5.38 2.78 6.08 5.5 20.260

0.8 0.1 0.933 8.300 0.06 0.0 5.40 2.75 6.39 5.5 20.2710.05 0.1 5.40 2.74 6.30 5.5 20.307


standard deviation of the equity premium mea-sured in the data. The price–consumption ratiov is increasing in expansion state pessimism;but since a switch to pessimism is transitory, theprice–consumption ratio subsequently revertsto its mean, implying a future capital loss. Thus,low mean equity returns are associated withepisodes of expansion state pessimism. At thesame time, precautionary saving increases sothat pessimistic episodes are also associatedwith a low risk-free rate. The upshot is thatrandomized but serially independent move-ments in p generate substantial volatility ingross returns, but not in excess returns. Al-though we have focused our discussion on fluc-tuations in p with fixed q, the price–consumption ratio and mean asset returnsexhibit essentially the same behavior duringcontractions whenq switches between high andlow values with fixedp.

C. Persistently Evolving Beliefs of StatePersistence

We now relax the restriction thatfe 5 fc 50.5, andallow for persistence in the subjectivestate transition probabilities. We continue toassume that the shocks are equally persistent(fe 5 fc) and that subjective endowmentgrowth is undistorted (ae 5 a(1) 5 2.25,ac 5 a(0) 5 26.79). Wesearch for combi-nations of mp, mq, dp, dq, fe, and fc thatmatch the mean and volatility of the risk-freerate and equity premium. To narrow the scopeof the search, we limited our evaluation to val-ues ofmp 5 0.71 andmq 5 0.24, themeans ofthe subjective transition probabilities that arein the neighborhood of our rule-of-thumbcalculations.

Volatility of Asset Returns.—Table 7 reportsthree cases in which the model is able to match

all five characteristics of the data with values ofthe discount factor that are less than 1 andrisk-aversion coefficient that are less than 10. Inthe first line of the table, the representativeagent discounts the future at a rate of 17 percentand has a relative risk aversion coefficient of9.89. The subjective probability that an expan-sion will continue, given that the economy iscurrently in an expansion, fluctuates between0.73 and 0.69, whereas the subjective probabil-ity about the persistence of the next contractionfluctuates between 0.06 and 0.42. Individualsdisplay pessimism over expansions, since eventhe high value ofp lies below p. Similarly,since the high value ofq lies belowq, agentsdisplay optimism about contractions. Agents’subjective beliefs about the persistence of thestates are themselves persistent. Once agentsadopt a view, the probability that these viewswill change is 12 fc 5 0.09.

The effect of changes in subjective contrac-tion persistence is large. Consider a simple ex-ample where contraction sentiment changes,where agents display mild expansion optimism(relative to their subjective mean ofmp 5 0.71,with p 5 0.73) butbecome pessimistic aboutthe next contraction, withq shifting from 0.06to 0.42. This causes the price consumption ratiov to increase from 16.80 to 28.02, resulting in asharp jump in equity prices.14 The risk-free rate,on the other hand, is unaffected by the changeso long as the economy is in the expansion state.

Why do serially correlated subjective transi-tion probabilities generate sufficiently volatileexcess returns, without an attendant increase inthe volatility of risk-free returns? The key lies inthe fact that the risk-free rate depends only on

14 Similarly, had agents initially displayed expansionpessimism (p 5 0.69), andswitched from contractionoptimism to contraction pessimism, the price–consumptionratio rises from 18.83 to 31.84.

TABLE 7—IMPLIED FIRST AND SECOND MOMENTS WITH PERSISTENT BUTRANDOM p AND q

Beliefs and preferences First and second moments

fe 5 fc mp dp mq dq b g meq mf seq sf req,f

0.91 0.71 0.023 0.24 0.181 0.823 9.886 5.5 2.5 19.0 5.520.1070.94 0.71 0.040 0.24 0.163 0.841 9.118 5.5 2.5 19.0 5.520.1190.97 0.71 0.052 0.24 0.140 0.859 8.472 5.5 2.5 19.0 5.520.128


what happens between the current period andthe next, whereas the equity return depends onexpectations of events over the infinite future.Since the economy is normally in an expansion-ary state, changes inq reflect changes in beliefsabout a state of the world that the economyexperiences only infrequently. Changes inqwill therefore affect the risk-free rate only onthose relatively rare occasions when the econ-omy is actually in a bad state. By contrast, giventhat the economy is currently in the expansion-ary state, persistent changes inq affect equityreturns through expected marginal utility andexpected dividends in the distant, as well as inthe immediate, future.

This same mechanism, where fluctuations inbeliefs aboutq during an expansion create move-ments in equity returns but not the risk-free rate,leads fluctuations in beliefs aboutp during a con-traction state to affect excess returns; however,because the economy is only rarely in the contrac-tionary state, changes inp are relatively unimpor-tant. Indeed, the results in Table 7 are virtuallyunaffected when we fixfe 5 0.5, rather thanallowing it to switch along withfc.

Serial Correlation and Predictability of Re-turns.—We now move to an examination of theability of our model to explain the two stylizedfacts about return persistence and predictability

TABLE 8—PERSISTENCE ANDPREDICTABILITY OF EXCESS RETURNS

A. Beliefs and preferencesCase fe 5 fc mp dp mq dq b g

I 0.91 0.71 0.023 0.24 0.181 0.823 9.886II 0.94 0.71 0.040 0.24 0.163 0.841 9.188III 0.97 0.71 0.052 0.24 0.140 0.859 8.472

B. Consumption–price ratio summary statisticsData Case I Case II Case III

Mean 21.902 23.895 24.394 26.009Standard deviation 5.782 6.439 7.950 11.686r1 0.783 0.814 0.875 0.936r2 0.590 0.664 0.768 0.877r3 0.542 0.543 0.674 0.821r5 0.360 0.365 0.520 0.722

C. PredictabilityConfiguration Horizon Slope R2 Variance ratio

Data 1 0.148 0.043 1.0002 0.295 0.081 1.0383 0.370 0.096 0.9205 0.662 0.191 0.8788 0.945 0.278 0.765

Case I 1 0.192 0.077 1.0002 0.352 0.138 0.9753 0.488 0.186 0.9435 0.692 0.251 0.8748 0.881 0.297 0.778

Case II 1 0.127 0.047 1.0002 0.242 0.087 1.0093 0.346 0.120 1.0095 0.520 0.171 0.9918 0.710 0.216 0.949

Case III 1 0.070 0.028 1.0002 0.138 0.052 1.0363 0.203 0.074 1.0625 0.321 0.110 1.0948 0.472 0.152 1.116


reported in Table 1: (1) variance ratio statisticsthat decline from above 1.0 at a two-year horizonto 0.8 at an eight-year horizon, and (2) theslope coefficients andR2’s of regressions oflonghorizon excess returns on the currentlog(consumption–price ratio) that increase withhorizon.

In Table 8 we present results for three cases,labelled I, II, and III, with parameter configu-rations reported. The preference and belief pa-rameter configures are reported in panel A ofthe table (and are identical to those in Table7). The dynamics of the price–consumption ra-tio v, the model’s analog to the price–dividendratio, is key to understanding the evolution ofequity returns. For this reason, we begin bypresenting selected implied moments ofv inpanel A of Table 8, along with sample valuesfor comparison. As noted earlier, changes invare driven predominantly by changes inq. Con-sequently, the persistence ofv is governedlargely by the persistence ofq and the serialcorrelation of v for an economy withfe 5fc 5 0.91 roughly matches the persistencefound in the data.

Panel B of Table 8 presents the model-impliedvalues of the variance ratios, the regression slopecoefficients, and theR2’s, at horizons of 1, 2, 3, 5,and 8 years. To account for the small-sample biasin these statistics, we generate them using a MonteCarlo experiment. In each trial of the experiment,an artificial sequence of 123 equity and risk-freereturns and consumption–price ratio are generatedfrom the model, corresponding to the 123 annualobservations that we have in the data. A re-gression of thek-period excess return on thelog(consumption–price ratio) log(1/v), fork 5 1, 2, 3, 5, and 8, is then estimated toobtain a slope coefficient and anR2. Theslope coefficients andR2’s that we report arethe mean values from 1 million trials.

The simulated data match the sample valuesnearly exactly. The performance of the modelfollows directly from the mean reversion of theprice–consumption ratio, as excess returns in-herit their stochastic properties fromv. Whenthe process is away from its mean, the predicteddeviation from its current level is increasing inthe horizon. It is in this manner that a mean-reverting price–consumption ratio implies bothpredicted returns and predictability that increasewith the horizon. Again, the economy with

fe 5 fc 5 0.91 provides the closest match tothe data. Whenfe 5 fc exceeds 0.91, the meanreversion in the price–consumption ratio is tooslow to generate the short-horizon predictabilityobserved in the data.

The results for the variance ratio statistics arenearly identical to those for the regressions.This is not all that surprising, as both are func-tions of the same underlying autocorrelations ofthe consumption–price ratio.

Cyclicality of the Equity Premium.—Next,we compare the implied cyclical pattern of theequity premium with that in the data. Using theparameter values from the configuration labeledcase I in Table 8, and evaluating the meanequity premium under various states of endow-ment growth, the model produces a coarse cor-respondence between the predicted and actualcyclicality of the equity premium. The modelpredicts that the mean equity premium duringtransitions from expansion to expansion is 6.18percent, whereas the sample counterpart is10.54 percent (See panel C of Table 1). Thepredicted equity premium during transitionsfrom contraction to contraction is210.52 per-cent in the model, whereas the average value inthe data is214.63 percent. The model’s pre-dicted mean equity premium over periods end-ing in expansion is 6.10 percent, whereas thesample counterpart is 13.66 percent. Finally,the predicted mean equity premium over peri-ods ending in contraction is27.34 percent withsample counterpart211.24 percent.15 Thecyclicality of the equity premium emerges be-cause high endowment growth during expan-sions works to raise the return on equity,whereas the systematic distortions of expansionpessimism and contraction optimism contributeto a countercyclical risk-free rate.

Cyclicality of the Price–Dividend Ratio.—Itis common for asset-pricing models based onCRRA preferences withg . 1 to produce coun-tercyclical price–dividend ratios. Although ourmodel shares this counterfactual property, it isattenuated by the introduction of distorted beliefs.

15 To compute the sample counterparts, we used theNBER monthly reference cycle dates to classify episodes ofexpansion and contraction.


Evaluating the model-implied price–dividend ra-tios for eight possible states, using the preference–belief configuration of case I in Table 8, we findthat the implied mean price–dividend ratio is23.87 over expansions and 24.36 over contrac-tions—implying a mildly countercyclical price–dividend ratio. The extent of the cyclicality isneither quantitatively important, nor grossly atodds with the empirical values of 23.2 over ex-pansions and 19.5 over contractions.

We note that there do exist scenarios in whichthe price–dividend ratio in our model falls dur-ing a contraction. For example, if investors dis-play pessimism aboutp and optimism aboutqduring the transition from expansion to contrac-tion, the price–dividend ratio will drop by 18.4percent, from 18.83 to 15.37.

D. Distorted Subjective Endowment Growth

We now switch our focus from distortions inthe transition probabilities to the role of distor-tions in subjective endowment growth rates. Weisolate this effect by settingdp 5 dq 5 0 andfix p 5 0.71 andq 5 0.24, values that areclose to the mean values ofmp and mq ofSection IV, subsection C. We restrict the statevariables driving subjective growth during ex-pansions and during contractions to be equallypersistent (fe 5 fc), allow subjective growthrates to fluctuate about their objective meanvalues (me 5 a(1) 5 2.25, mc 5 a(0) 526.78) andsearch over values ofte, tc, fe 5fc, b, andg in an attempt to mimic the behav-ior found in the data.

Although it is possible to match the first andsecond moments of the asset-return data whenwe allow subjective growth rates to fluctuate—with fe 5 fc 5 0.98, te 5 0.53, tc 5 1.5,b 5 0.88, andg 5 7.49—distorting beliefs inthis dimension fails to resolve the predictabilitypuzzle. With this set of preference and beliefparameters, the eight-year horizon regressionslope coefficient is 0.08 withR2 5 0.08,whereas the variance ratios for implied excessreturns slightly exceed 1 for horizons of eightyears and less. The failure to match this featureof the data is a result of the fact that switches insubjective growth induce correlated movementsin equity returns and in the risk-free rate. Aswitch in subjective contractionary state growthcauses fluctuations in both equity and risk-

free returns during an expansion, whereasswitches in subjective expansionary stategrowth lead to fluctuations in both returnsduring contractions. Subjective growth ratesneed to be very persistent to generate suffi-ciently high volatility in the equity premium; atthe same time, however, these beliefs are asso-ciated with log(consumption–price ratio)s thatare so persistent that they fail to predict futureexcess returns.

E. Irrational Forecasts of Beliefs

In our distorted beliefs model, agents exhibitirrationality only to the extent that they have thewrong model in mind. Within the context oftheir subjective model, they continue to behaverationally. They are irrational in responding topseudosignals, but rational in conditioning theirbehavior on the fact that they do respond to thesignals and know their true stochastic process.In this sense, agents rationally forecast theirown irrationality. Our final task is to investigatethe consequences of relaxing this assumption.We do this by creating a distinction between theagent’s subjective transition probabilityfe 5fc and the “true” pseudosignal transition prob-abilities fe 5 fc.

We have considered three alternative valuesof the “truth,” fe 5 fc 5 0.89, 0.90, and0.91, corresponding to the parameter configura-tions in case I of Table 8. For each of thesevalues, we searched over the subjective beliefsof pseudosignal persistencefe 5 fc and anassociated preference–belief configuration thatallowed us to match the unconditional mo-ments, predictability, and mean reversion ofasset returns. We are able to find values offe 5fc that deviate fromfe 5 fc that accomplishthis task.

There are three main features of this exercisethat are worth pointing out. First, the model fitsthe data with preferences that are (arguably)more reasonable than those in Table 8. Forexample, whenfe 5 fc 5 0.91 andfe 5fc 5 0.934, themodel requiresg 5 7.22 andb 5 0.88, as compared with values of (9.886,0.823) in Table 8. Second, the model fits thedata when investors believe the pseudosignalsare more persistent than the “truth” (fe 5 fc .fe 5 fc). The model does not match thepredictability pattern when agents believe the


signals to belesspersistent than the truth (fe 5fc , fe 5 fc). Third, the beliefs about thepersistence of pseudosignals that fit the datacould be quite significantly distorted. For exam-ple, the model fits the data whenfe 5 fc 50.89 andfe 5 fc 5 0.93.This configurationimplies that investors believe the expected du-ration of a signal is 14.3 years, whereas in“reality” it is only 9.1 years.

From this exercise we conclude that allowinginvestors to rationally forecast the evolution oftheir subjective beliefs is largely innocuous. Tofit the data, the investors can be irrational inforecasting their own irrationality, so long asthe irrationality is not too large.

V. Conclusion

We have shown that a simple aggregate asset-pricing model, in which agents have distortedbeliefs about persistence of the state-transitionprobabilities of the endowment growth process,can replicate a number of features of observedasset-returns data. We are able to match themeans, standard deviations, and correlation ofthe equity premium and the risk-free rate, as wellas the persistence and predictability of long-horizon excess equity returns. In addition,the predicted cyclical patterns of the equitypremium roughly correspond to those in thedata. This is accomplished within the frameworkof a representative-agent endowment economy,

with CRRA preferences, and a two-state Markovprocess governing per capita consumption growth.The simplicity of our approach provides an attrac-tive alternative to the strategy of introducing eitherincreasingly complex preference specifications,heterogeneity, or market incompleteness, to ad-dress the failures of the standard model.

Success in our task requires that subjectivebeliefs be distorted in two ways. First, agentsmust think that both expansions and contrac-tions will be less persistent than what is impliedby careful econometric analyses of the data.Second, agents’ views of these transition prob-abilities must exhibit stochastic and persistentvariation of a very specific type. Our formula-tion can be criticized for its inability or unwill-ingness to allow agents to learn over time. Wesuspect, however, that if agents were Bayesianlearners, convergence would occur reasonablyfast and that over the time span of our data set,they would have learned by now to be rational.Under CRRA utility, this provides little help inunderstanding asset returns. On the other hand,if the true endowment process is stationary,with the passage of sufficient time, any rule-of-thumb estimates of the transition probabilitiesshould converge to the truth. Exactly how muchtime is required for convergence to occur isanother issue. It is entirely likely that slowlearning describes the real-world environmentand that 104 years of data is still too short toovercome this small sample problem.

APPENDIX

The eight possible outcomes for the distorted beliefs state vector,S# [ (S, Se, Sc) are,S#1 5 (1,1, 1), S#2 5 (1, 1, 21), S#3 5 (1, 21, 1), S#4 5 (1, 21, 21), S#5 5 (0, 1, 1),S#6 5 (0, 1, 21),S#7 5 (0, 21, 1), S#8 5 (0, 21, 21).

Let po 5 mp 1 de, pp 5 mp 2 de, qo 5 mq 2 dc, qp 5 mq 1 dc, p*o 5 1 2 po, p*p 5 1 2pp, q*o 5 1 2 qo, q*p 5 1 2 qp, f*e 5 1 2 fe, f*c 5 1 2 fc. Then the transition matrix for thestate vectorS# is

P 5 3pofefc pofef*c pof*efc pof*ef*c p*ofefc p*ofef*c p*of*efc p*of*ef*cpofef*c pofefc pof*ef*c pof*efc p*ofef*c p*ofefc p*of*ef*c p*of*efc

ppf*efc ppf*ef*c ppfefc ppfef*c p*pf*efc p*pf*ef*c p*pfefc p*pfef*cppf*ef*c ppf*efc ppfef*c ppfefc p*pf*ef*c p*pf*efc p*pfef*c p*pfefc

q*ofefc q*ofef*c q*of*efc q*of*ef*c qofefc qofef*c qof*efc qof*ef*cq*pfef*c q*pfefc q*pf*ef*c q*pf*efc qpfef*c qpfefc qpf*ef*c qpf*efc

q*of*efc q*of*ef*c q*ofefc q*ofef*c qof*efc qof*ef*c qofefc qofef*cq*pf*ef*c q*pf*efc q*pfef*c q*pfefc qpf*ef*c qpf*efc qpfef*c qpfefc

4 .


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Asset Pricing with Distorted Beliefs: Are Equity Returns Too Good to Be True?

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