Stock Price Booms and Expected Capital Gains?, Study notes of Dynamics

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Barcelona GSE Working Paper Series

Working Paper nº 757

Stock Price Booms and Expected Capital

Gains?

Klaus Adam

Albert Marcet

Johannes Beutel

This version: August 2016

(January 2014)

Stock Price Booms and Expected Capital Gains^1

Klaus Adam

University of Mannheim & CEPR

Albert Marcet

Institut díAn‡lisi EconÚmica (CSIC), ICREA, UAB, MOVE,

Barcelona GSE & CEPR

Johannes Beutel

University of Mannheim

August 15, 2016

(^1) This paper replaces an earlier paper titled ëBooms and Busts in Asset Pricesíby Adam and Marcet, which appeared 2010 as Bank of Japan - IMES Discussion Paper No. 2010-E-2. Thanks go to Fernando Alvarez, Chryssi Giannitsarou, Vivien Lewis, Morten Ravn, Ricardo Reis, Mike Woodford, conference participants at the Banque de France and Chicago Fed Conference on Asset Price Bubbles, ESSIM 2010 in Tarragona, MONFISPOL network, and seminar participants at Columbia University, University College London, London School of Economics, Yale University, Harvard University, Northwestern University, Stony Brook, IMF, Mitsui Conference at Michigan University, Northwestern University, University of Chicago, New York University and London Business School for helpful comments and suggestions. Research assistance from Jeanine Baumert, Oriol Carreras, Lena Gerko, Dimitry Matveev, Tongbin Zhang and especially Sebastian Merkel is greatly appreciated. Klaus Adam thanks the Bank of Japan for the hospitality o§ered during early stages of this project. Albert Marcet acknowledges support from Programa de Excelencia del Banco de EspaÒa, Plan Nacional (Ministry of Education), SGR (Generalitat de Catalunya). Klaus Adam and Albert Marcet acknowledge support from the European Research Council under the EU 7th Framework Programme (FP/2007-2013), Starting Grant Agreement No. 284262 (Adam) and Advanced Grant Agreement No. 324048 (Marcet). Klaus Adam and Albert Marcet thank Fondation Banque de France for its support on this project. All errors remain ours.

ëBull-markets are born on pessimism, grow on skepticism, mature on optimism and die on euphoriaí Sir John Templeton, Founder of Templeton Mutual Funds

1 Motivation

Following the recent boom and bust cycles in a number of asset markets around the globe, there exists renewed interest in understanding better the forces contributing to the emergence of such drastic asset price movements. This paper argues that movements in investor optimism and pessimism, as measured by the movements in investorsísubjective expectations about future capital gains, are a crucial ingredient for understanding these áuctuations. We present an asset pricing model that incorporates endogenous belief dynamics about expected capital gains. The model gives rise to sustained stock price booms and busts and is consistent with the behavior of investorsícapital gains expectations, as measured by survey data. The presented modeling approach di§ers notably from the standard approach in the consumption-based asset pricing literature, which proceeds by assuming that stock price áuctuations are fully e¢ cient. Campbell and Cochrane (1999) and Bansal and Yaron (2004), for example, present models in which stock price áuctuations reáect the interaction of investor preferences and stochastic driving forces in a setting with optimizing investors who hold rational expectations. We Örst present empirical evidence casting considerable doubt on the prevailing view that stock price áuctuations are e¢ cient. SpeciÖcally, we show that the rational ex- pectations (RE) hypothesis gives rise to an important counterfactual prediction for the behavior of investorsíreturn or capital gain expectations.^1 This counterfactual prediction is a model-independent implication of the RE hypothesis, but - as we explain below - key for understanding stock price volatility and its e¢ ciency properties. As previously noted by Fama and French (1988), the empirical behavior of asset prices implies that rational return expectations correlate negatively with the price dividend (PD) ratio. Somewhat counter-intuitively, the RE hypothesis thus predicts that investors have been particularly pessimistic about future stock returns in the early part of the year 2000, when the tech stock boom and the PD ratio of the S&P500 reached their all-time maximum. As we document, the available survey evidence implies precisely the opposite: (^1) Since most variation in returns is due to the variation in capital gains, we tend to use both terms interchangeably.

all quantitative survey measures of investorsíexpected return (or capital gain) available for the U.S. economy, unambiguously and unanimously correlate positively with the PD ratio; and perhaps not surprisingly, return expectations reached a temporary maximum rather than a minimum in the early part of the year 2000, i.e., precisely at the peak of the tech stock boom, a fact previously shown in Vissing-Jorgensen (2003). We present formal econometric tests of the null hypothesis that the survey evidence is consistent with RE and demonstrate that the hypothesis of rational return or capital gain expectations is overwhelmingly rejected by the data. Our tests correct for small sample bias, account for autocorrelations in the error structure, are immune to the presence of di§erential information on the part of agents and to the presence of measurement error in survey data. An appealing feature of the tests is that they also provide clues about why the RE hypothesis fails: the failure arises because survey expectations and RE covary di§erently with the PD ratio, a Önding that is useful for guiding the search for alternative and empirically more plausible expectations models. The positive comovement of stock prices and survey expectations suggests that price áuctuations are ampliÖed by overly optimistic beliefs at market peaks and by overly pessimistic beliefs at market troughs. Furthermore, it suggests that investorsí capital gains expectations are ináuenced - at least partly - by the capital gains observed in the past, in line with evidence presented by Malmendier and Nagel (2011). Indeed, a simple adaptive updating equation captures the time series behavior of the survey data and its correlation with the PD ratio very well. Taken together, these observations motivate the construction of an asset pricing model in which agents hold subjective beliefs about price outcomes. We do so using the frame- work of Internal Rationality (IR), developed in Adam and Marcet (2011), which allows considering maximizing investors that hold subjective price beliefs within an otherwise standard Lucas (1978) asset pricing model.^2 Within this framework, agents optimally update beliefs using Bayesílaw. With this modiÖcation, the Lucas model becomes quantitatively consistent with im- portant aspects of the data. Using conÖdence intervals based on the simulated method of moments, we Önd that the model matches key moments describing the observed volatility of stock prices and the positive correlation between the PD ratio and subjective return expectations. This is obtained even though we use the simplest version of the Lucas model with time separable preferences and standard stochastic driving processes. The (^2) As is explained in Adam and Marcet (2011), subjective price beliefs are consistent with optimizing behavior in the presence of lack of common knowledge about agentsíbeliefs and preferences.

gives rise to negative price momentum and an asset price bust. The previous arguments show how belief dynamics can temporarily delink asset prices from their fundamental value. Clearly, these price dynamics are ine¢ cient as they are not justiÖed by innovations to preferences or other fundamentals. Since we depart from RE, our model requires introducing an explicit assumption about agentsíprice beliefs. Various elements guide this modelling choice. First, we choose price beliefs such that there are no ëblack swaní like events, i.e., we insure that agents have contingency plans for all prices that they actually encounter along the equilibrium path. Second, we choose the subjective price process such that it gives rise to capital gain expectations that are consistent with the behavior of survey expectations. In particular, agents believe the average growth rate of stock prices to slowly drift over time, which is consistent with the presence of prolonged periods of price booms that are followed by price busts. Given these beliefs, equilibrium prices will indeed display prolonged periods of above average and below average growth. More generally, the present paper shows how the framework of Internal Rationality allows studying learning about market behavior in a model of intertemporal decision making. It thereby improves on shortcomings present in the learning literature, where agentsíbelief updating equations and choices are often not derived from individual max- imization and are optimal only in the limit once learning converges to the RE outcome. We thus provide explicit microfoundations for settings in which subjective beliefs about market outcomes matter for these outcomes. The bulk of the paper considers a representative agent model. This is motivated by the desire to derive results analytically and to show how a rather small deviation from the standard paradigm helps reconciling the model with the data. A range of extensions consider - amongst other things - a heterogenous agents version and more elaborate subjective belief structures. These extensions allow replicating additional data features, e.g., the equity premium. The remainder of the paper is structured as follows. The next section discusses the related literature. Section 3 then shows that the price dividend ratio (PD) ratio covaries positively with survey measures of investorsíreturn expectations and that this is incom- patible with the RE hypothesis. It also shows that the time series of survey expectations can be captured by a fairly simple belief updating equation. Section 4 introduces our asset pricing model with subjective beliefs. As a benchmark, section 5 determines the RE equilibrium. Section 6 introduces a speciÖc model of subjective price beliefs, which

relaxes agentsíprior about price behavior relative to the RE equilibrium beliefs. It also derives the Bayesian updating equations characterizing the evolution of subjective beliefs over time. After imposing market clearing in section 7, we present closed form solutions for the PD ratio in section 8 in the special case of vanishing uncertainty. Using the an- alytical solution, we explain how the interaction between belief updating dynamics and price outcomes can endogenously generate boom and bust dynamics in asset prices. Sec- tion 9 estimates the fully stochastic version of the model using a mix of calibration and simulated method of moments estimation. It shows that the model successfully replicates a number of important asset pricing moments, including the positive correlation between expected returns and the PD ratio. It also explains how the model gives rise to a high Sharpe Ratio and to a low volatility for the risk free interest rate. Section 10 shows that the estimated model can replicate the low frequency movements in the time series of the US postwar PD ratio, as well as the available time series of survey data. Section 11 presents a number of robustness checks and extensions of the basic model. A conclusion brieáy summarizes and discusses potential avenues for future research. Technical material and proofs can be found in the appendix.

2 Related Literature

Following Bob Shillerís (1981) seminal observation that stock price volatility cannot be explained by the volatility of rational dividend expectations, the asset pricing litera- ture made considerable progress in explaining stock price behavior. Bansal and Yaron (2004) and Campbell and Cochrane (1999), for example, developed consumption based RE models in which price áuctuations result from large and persistent swings in investorsí stochastic discount factor. Section 3 shows, however, that RE models fail to capture the behavior of investorsí return expectations. This strongly suggests that RE models fall short of providing a complete explanation of the sources of stock price volatility. Attributing stock price áuctuations to ësentimentí áuctuations or issues of learning has long had an intuitive appeal. A substantial part of the asset pricing literature in- troduces subjective beliefs to model investor ësentimentí. The standard approach resorts to Bayesian RE modeling, which allows for subjective beliefs about fundamentals, while keeping the assumption that investors know the equilibrium pricing function linking stock prices to fundamentals. Following early work by Timmermann (1993) and Barberis, Shleifer and Vishny (1998), a substantial literature follows this approach. It Önds that

agents entertain a distribution of prices for given fundamentals, which is non-degenerate and that does not coincide with the model at hand.^8 To show that a key element for understanding stock price volatility is investorís im- perfect knowledge about how prices are formed, we make the distinction to the Bayesian RE literature as stark as possible: we assume that agents Önd it easy to predict fun- damentals, i.e., assume agents hold RE about dividends, but Önd it di¢ cult to predict price behavior, i.e., agents do not know the equilibrium pricing function. We show that a simple asset pricing model can then replicate survey data and generate su¢ cient volatil- ity for the PD ratio, including occasional boom and bust episodes. This is achieved in a setting with standard time-separable preferences and obtained because there is a much stronger propagation of economic disturbances when agents learn about the equilibrium pricing function: belief changes then a§ect stock price behavior and stock prices feed back into belief changes; this allows movements in prices and beliefs to mutually rein- force each other during price boom and bust phases, thereby increasing price volatility. The feedback from market outcomes into beliefs is absent in a Bayesian RE setting. The literature on robust control and asset prices, e.g. Cogley and Sargent (2008), con- siders settings where investors are uncertain about the process for fundamentals. In line with Bayesian RE modeling, this literature assumes that investors know the equilibrium pricing function. The literature on adaptive learning previously considered deviations from rational price expectations using asset pricing models where investors learn about price behavior. Marcet and Sargent (1992), for example, study convergence to RE when agents estimate an incorrect model of stock prices by least squares learning. A range of papers in the adaptive learning literature argues that learning generates additional stock price volatility. Bullard and Du§y (2001) and Brock and Hommes (1998), for example, show that learning dynamics can converge to complicated attractors that increase asset return volatility, when the RE equilibrium is unstable.^9 Lansing (2010) shows how near-rational bubbles can arise in a model with learning about price behavior. Branch and Evans (2011) present a model where agents learn about risk and return and show how it gives rise to bubbles and crashes. Boswijk, Hommes and Manzan (2007) estimate a model with fundamentalist and chartist traders whose relative shares evolve according to an evolutionary performance (^8) This is related to work by Angeletos and LaíO (2013) who consider a setting in which agents are uncertain about the price at which they will be able to trade. They show how sentiment shocks can give rise to perfectly self-fulÖlling áuctuations in aggregate outcomes. Sentiment shocks in their setting result from extrinsic uncertainty; in our setting they are triggered from intrinsic sources of uncertainty. 9 Stability under learning dynamics is deÖned in Marcet and Sargent (1989).

criterion, showing that the model can generate a run-up in asset prices and subsequent mean-reversion to fundamental values. DeLong et al. (1990) show how the pricing e§ects of positive feedback trading survives or even get ampliÖed by the introduction of rational speculators. The approach used in the present paper di§ers along several dimensions from the contributions mentioned in the previous paragraph. First, we compare quantitatively the implications of our model with the data, i.e., we match a standard set of asset pricing moments capturing stock price volatility and use formal asymptotic distribution to eval- uate the goodness of Öt. Second, we compare the model to evidence obtained from survey data. Third, we present a model that derives investorsíconsumption and stockholding plans from properly speciÖed microfoundations. In particular, we consider agents that solve an inÖnite horizon decision problem and hold a consistent set of beliefs, we discuss conditions for existence and uniqueness of optimal plans, as well as conditions insuring that the optimal plan has a recursive representation. The adaptive learning literature often relies on shortcuts that amount to introducing additional behavioral elements into decision making and postulates beliefs that become well speciÖed only in the limit, if convergence to RE occurs.^10 In prior work, Adam, Marcet and Nicolini (2016) present a model in which investors learn about risk-adjusted price growth and show how such a model can quantitatively replicate a set of standard asset pricing moments describing stock price volatility. While replicating stock price volatility and postulating beliefs that are hard to reject in the light of the existing asset price data and the outcomes generated by the model, their setup falls short of explaining survey evidence. SpeciÖcally, it counterfactually implies that stock return expectations are constant over time. Adam, Marcet and Nicolini (2016) also solve for equilibrium prices under the assumption that dividend and trading income are a negligible part of total income. We solve the model without this assumption and show that it can play an important role for the model solution, for example, it gives rise to an endogenous upper bound for equilibrium prices.^11 The experimental and behavioral literature provides further evidence supporting the presence of subjective price beliefs. Hirota and Sunder (2007) and Asparouhova, Bossaerts, Roy and Zame (2013), for example, implement the Lucas asset pricing model in the (^10) See section 2 in Adam and Marcet (2011) for a detailed discussion. (^11) In line with the approach in the Bayesian RE literature, Adam Marcet and Nicolini (2016) impose an exogenous upper bound on agentsíbeliefs, a so-called ëprojection facilityí, so as to insure existence of Önite equilibrium prices.

return, as well as their expectations about the one year ahead returns on their own stock portfolio. These measures behave very similarly over the period for which they overlap, but the latter is available for a longer time period. Figure 2 reveals that there is a strong positive correlation between the PD ratio and expected returns. The correlation between the expected own portfolio returns and the PD ratio is +0.70 and even higher for expected stock market returns (+0.79). Moreover, investorsíreturn expectations were highest at the beginning of the year 2000, which is precisely the year the PD ratio reached its peak during the tech stock boom. Investors then expected annualized real returns of around 13% from stock investments, while the subsequently realized returns turned out to be particularly dismal. Conversely, investors were most pessimistic in the year 2003 when the PD ratio reached its bottom, expecting then annualized real returns of below 4%. This evidence suggests that survey data is incompatible with rational expectations and that stock prices seem to play a role in the formation of expectations about stock returns. Yet, evidence based on comparing two correlations can only be suggestive, as it is subject to several econometric shortcomings. For example, if investors possess infor- mation that is not observed by the econometrician, as might be considered likely, then the correlation between the fully rational return forecasts and the PD ratio will di§er from the correlation between realized returns and the PD ratio. The same holds true if survey expectations are measured with error, as one can reasonably expect. Further- more, results in Stambaugh (1999) imply that with the PD ratio being such a persistent process, there is considerable small sample bias in these correlations, given the relatively short time spans over which investor expectations can be tracked. Finally, a highly seri- ally correlated predictor variable (PD ratio), whose innovations are correlated with the variable that is to be predicted (future returns), gives rise to spurious regression and thus spurious correlation problems, see Ferson et al. (2003) and Campbell and Yogo (2006). There exists also no standard approach allowing to correct for these small sample issues when comparing correlations.^16 Comparisons involving correlations are thus insu¢ cient for rejecting the hypothesis that survey expectations are consistent with RE. To deal with these concerns, the next sections construct formal econometric tests that take these concerns fully into account. (^16) Any test must take into account the joint distribution of the correlation estimates in order to make statistically valid statements.

3.2 RE Test with Small Sample Adjustments

This section develops a RE test that takes into account the concerns expressed in the previous section. While the present section emphasizes the derivation of analytical results, section 3.3 provides further tests that rely entirely on Monte Carlo simulation. Let EP t denote agentsísubjective expectations operator based on information up to time t, which can di§er from the rational expectations operator Et. Let Rt;t+N denote the real cumulative stock returns between period t and t + N and let EtN = EP t Rt;t+N + Nt denote the (potentially noisy) observation of expected returns, as obtained - for example

• from survey data, where Nt is measurement error.^17 Let us linearly project the random variable E tP Rt;t+N on (^) DPtt to deÖne

E tP Rt;t+N = aN^ + cN^ DPt t

• uNt ; (1)

where E(xt uNt ) = 0; (2)

for x^0 t = (1; Pt=Dt). The operator E denotes the objective expectation for the true data generating process, whatever is the process for agentsí expectations. The projection residual uNt captures variations in agentsí actual expectations that cannot be linearly attributed to the price-dividend ratio.^18 It summarizes all other information that agents believe to be useful in predicting Rt;t+N.^19 Due to the potential presence of measurement error, one cannot directly estimate equation (1), but given the observed return expectations EtN , one can write the following regression equation EtN = aN^ + cN^ DPt t

• uNt + Nt : (3)

Assuming that the measurement error Nt is orthogonal to the current PD ratio^20 , we have the orthogonality condition

E xt(uNt + Nt )^ = 0: (4) (^17) Since the Shiller survey reports expectations about capital gains instead of returns, Rt;t+N denotes the real growth rate of stock prices between periods 18 t and t + N when using the Shiller survey. The residual uNt is likely to be correlated with current and past observables (other than the PD ratio) and thus serially correlated. 19 The projection in equation (1) and the error are well-deÖned as long as agentsíexpectations E tP Rt;t+N and 20 Pt=Dt are stationary and have bounded second moments. We allow Nt to be serially correlated and correlated with equilibrium variables other than P Dt.

bias of bcNT bcNT in the test-statistic (7) is

E(bcNT bcNT ) = cov("

P Dt+1; "Nt ) var("P Dt ) E(bT^ ^ );^ (9)

where "Nt  Rt;t+N EtRt;t+N denotes the rational prediction error and E(bT ) the small sample bias in the estimation of  for a sample of length T.

The proof of proposition 1 can be found in appendix A.3. It treats equations (3)-(6) as a seemingly unrelated regression system and uses the fact that under rational expectations one has cN^ = cN^. Part b) of the proposition follows from results in Stambaugh (1999). Part a) of proposition 1 uses minimal assumptions to obtain an asymptotically valid result. Essentially, all that is needed is stationarity of the observables and orthogonality of the measurement error. The test is asymptotically robust to serial-correlation and heteroskedasticity of the error terms. Part b) of proposition 1 deals with small sample bias in the test statistic. Since  is close to one, we have E(bT ) < 0 ; and since future stock price increases are likely to be correlated with future surprises to returns, i.e., cov("P Dt+1; "Nt ) > 0 , we tend to get E(bcNT bcNT ) < 0 in small samples, even if in fact cN^ = cN^. The fraction on the right-hand side of the bias expression in equation (9) can be estimated from observables using the calculated errors from equations (3), (5) and (8) and the fact that under the null

cov("P Dt+1; "Nt ) = cov("P Dt+1; uNt ) cov("P Dt+1; uNt + Nt ):

The bias E(bT ) in equation (9) is approximately given by E(bT ) ' 1+3T , see Marriott and Pope (1954). Since the true bias is a non-linear function of  and the analytical linear approximation less precise in the relevant range of  close to one^24 , we compute the bias E(bT ) using Monte-Carlo integration.^25 Interestingly, our RE test cN^ = cN^ is less prone to small sample bias than tests for the signiÖcance of the individual regression coe¢ cients (cN^ = 0 and cN^ = 0). This follows from the proof of proposition 1, which shows that

E(bcNT cN^ ) = E(bcNT cN^ ) + cov("

P Dt+1; "Nt ) var("P Dt ) E(bT^ ^ ): (^24) See, for instance, Ögure 1 in MacKinnon and Smith (1998). (^25) Given the estimated values of P D; ;  (^2) "P D we simulate 10.000 realizations of P D of length T , compute b for each realization, average over realizations to obtain an approximation for E(bT ) and compute the bias correction accordingly.

The previous equation implies that the small sample bias present in the individual esti- mate of cN^ , i.e., E(bcNT cN^ ), cancels when testing for cN^ = cN^. The test outcomes associated with proposition 1 are reported in table 1a.^26 The table reports the bias-corrected point estimates of cN^ and cN^ , as well as the bias-corrected p-values for the test based on proposition 1.^27 We use the UBS, CFO and Shiller surveys and consider various ways for extracting expectations from these surveys.^28 The point estimates always satisfy bcN^ > 0 and bcN^ < 0. The di§erence between the two estimates is statistically signiÖcant at the 1% level in all cases, except when using the survey median from the CFO survey, where p-values are around 3% to 5%.^29 Overall, the test results in table 1a provide strong evidence against the notion that survey expectations are compatible with rational expectations. Table 1a also shows that agents are overly optimistic when the P D ratio is high and overly pessimistic when the P D ratio is low. This suggests that current prices have an ëexcessive roleíin ináuencing current return expectations. Clearly, if the asset price and survey data were generated by a rational expectations model, say the models of Campbell and Cochrane (1999) or Bansal and Yaron (2004), the tests in table 1a would have been accepted.

3.3 Additional Small Sample Adjustments

The closed-form expressions for the small sample bias derived in the previous section are useful for understanding the nature of the bias. At same time, they fall short of completely addressing small sample issues. In particular, the result stated in part b) of proposition 1 relies on assuming that the regression residuals uNt +Nt and uNt in equations (3) and (5) are i.i.d. Yet, this is unlikely to hold in our application. Consider Örst the residual uNt. Under the null hypothesis of RE, we have uNt = uNt +"Nt , where "Nt  Rt;t+N EtRt;t+N is a prediction error from the true data-generating process. For short prediction horizons (N = 1), "Nt is indeed serially uncorrelated, but for longer horizons (N > 1 ), the residuals "Nt denote forecast errors for overlapping prediction horizons. Since we use quarterly data and prediction horizons between one and ten years, (^26) We used 4 lags for Newey West estimator and we checked that results are robust to increasing the lag length up to 12 lags. For each considered survey, we use data on actual returns (or excess returns, or price growth) for the same time period for which survey data is available when computing the p-values. (^27) The p-values are computed using the bias corrected test statistic pT bcNT bcNT bcE(cbc NT bcNT ). (^28) See appendix A.1 for information on the data sources. (^29) We conjecture that the CFO provides less signiÖcant results because the sample starts in Q3:2000, thus does not include the upswing of the tech boom period, unlike the UBS sample. As a result, the CFO sample period displays less mean reversion in prices, which accounts for the fact that the estimates of c are less negative and less signiÖcant.

As a result, the regression coe¢ cients

aN^ ; cN^

in equation (5) do not satisfy aN^ = AN^ ; cN^ = CN^ ; whenever  6 = 0. It would thus be incorrect to plug our estimates of aN^ ; cN into equation (10) for the purpose of running the Monte Carlo simulations. To estimate the parameters in equations (10) and (11) we proceed as follows. We lag equation (10) by one period, multiply by  and subtract it from equation (10). This delivers

Rt;t+N = AN^ (1 ) + Rt 1 ;t+N 1 + CN^ DPt t CN^ DPt^1 t 1

• "P Dt+N + t+N ;

which can be estimated using non-linear least squares and the observed explanatory variables (Rt 1 ;t+N 1 ; (^) DPtt ; (^) DPtt^11 ; b"P Dt+N ), because these explanatory variables are orthogonal to t+N. We thus have consistent and e¢ cient estimates for , ; ^2 ; AN^ and CN^. We plug these estimates into equations (10) and (11) to simulate Rt;t+N : To compute expected returns under the null hypothesis of RE, we compute the true expectation of returns, which are given by

Et (Rt;t+N ) = AN^ + CN^ DPt t

• ()N^ UNt :

Using these results, we can simulate all the variables involved in equations (3) and (5), compute the statistic p T

bcN^ bcN^

=bcc for each simulation and study its small sample distribution, using the sample sizes of the considered survey source. We then compute the probability that p T

bcN^ bcN^

=bcc in the Monte-Carlo simulations is smaller than the corresponding value we Önd for the data. This provides a p-value for the one-sided test of RE, when the alternative hypothesis is cN^ > cN^ , i.e., that survey returns respond more strongly to the PD ratio than actual returns. Table 1b reports the outcomes of this procedure. The second column in the table reports the estimated value for . It shows that the residuals UNt in equation (11) are indeed serially correlated. We Önd that this leads to considerable spurious regression problems, as the standard deviation of the test statistic p T bcN^ bcN^
^ =bcc is indeed around 2 to 3 times larger than its asymptotic value of 1. Table 1b also reports the bias corrected estimates bcN^ and bcN^.^31 Compared to the results in table 1a, the estimates for bcN^ are considerably less negative; the ones for the CFO and Shiller 1 year sample even become positive. Yet, the bias corrected estimates (^31) The point estimates correct for small sample bias using the mean of the estimator found in the Monte-Carlo simulations with serially correlated errors and using the fact that cN^ = CN^ + ()N^ (1 ^2 )=(1 ()^2 ).

for bcN^ in table 1b also become more positive when compared to the ones reported in table 1a. Therefore, despite the spurious regression problems, which cause an increase in the true variance of the test statistic, the RE hypothesis is soundly rejected.^32 The level of the rejection is now considerably lower than the one reported in table 1a, but still highly signiÖcant. Since the involved sample lengths are not very large, this is a remarkable result. Table 1c repeats the analysis when letting Rt;t+N denote excess stock returns rather than stock returns.^33 We construct excess return expectations following Bacchetta et al. (2009), i.e., we assume that the N period ahead risk-free interest rate is part of agentsíinformation set and subtract it from the (expected) stock return.^34 We Önd that the strength of the rejection of RE is then somewhat lower when compared to table 1b. Still, for most survey sources one obtains p-values near or below 5%. The somewhat lower p-values show that the approach based on plain stock returns, as reported in table 1b, o§ers a slightly more powerful test of the RE hypothesis. Interestingly, this is in line with the main hypothesis of this paper, namely, that agents form their expectations about stock prices by extrapolating the behavior of past price growth. Under this hypothesis, subtracting the risk-free interest rate adds noise to the independent variables on the l.h.s. of regression equations (3) and (5), which is consistent with the observed decrease in the signiÖcance levels.

3.4 How Models of Learning May Help

This section illustrates that a simple ëadaptiveíapproach to forecasting stock prices is a promising alternative to explain the joint behavior of survey expectations and stock price data. Figure 2 shows that the peaks and troughs of the PD ratio are located very closely to the peaks and troughs of investorsí return expectations. This suggests that agents become optimistic about future capital gains whenever they have observed high capital gains in the past. Such behavior can be captured by models where agentsíexpectations are ináuenced by past experience, prompting us to temporarily explore the assumption that the log of agentsí subjective conditional capital gain expectations ln Eet [Pt+1=Pt] (^32) As in table 1a, the rejection is less strong for the CFO survey, especially when using the survey median. See the discussion in footnote 29 for a discussion for the potential reasons behind this result. 33 For the Shiller survey, which reports price growth expectations, Rt;t+N now denotes excess prices growth. 34 Following Bacchetta et al. (2009), we use the constant maturity interest rates available from the FRED database at the St. Louis Federal Reserve Bank.