Asset pricing with expectation shocks

Author’s Accepted Manuscript Asset Pricing with Expectation Shocks Christopher J. Elias

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S0165-1889(16)30013-6 http://dx.doi.org/10.1016/j.jedc.2016.02.005 DYNCON3274

To appear in: Journal of Economic Dynamics and Control Received date: 5 June 2014 Revised date: 21 January 2016 Accepted date: 16 February 2016 Cite this article as: Christopher J. Elias, Asset Pricing with Expectation Shocks, Journal of Economic Dynamics and Control, http://dx.doi.org/10.1016/j.jedc.2016.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Asset Pricing with Expectation Shocks Christopher J. Eliasa a

Department of Economics, 703 Pray-Harrold, Eastern Michigan University, Ypsilanti, MI, 48197 February 18, 2016

Abstract This paper adds persistent shocks into the adaptive learning expectation formation process in stochastic growth asset pricing production and endowment economies. These expectation shocks, designed to capture psychological elements which can arise from news, changes in sentiment, herding and bandwagon effects, etc., generate waves of optimism and pessimism in equity price forecasts. The paper estimates parameters of the expectation shock and adaptive learning process with the method of simulated moments, and compares simulation results to U.S. economic and financial market stylized facts. Numerical results for both the estimated production and endowment economies show that the expectation shock model matches several of the stylized facts better than does a model that assumes rational expectations or adaptive learning alone. Keywords: Asset Pricing, Adaptive Learning, Expectations Formation, Expectation Shocks JEL: D83, D84, G12, E37, E44 1. Introduction Early economists argued that agent expectations, and ultimately, macroeconomic phenomena, are influenced by psychology. For example, Pigou (1927) and Keynes (1936) cited psychological factors as significant drivers of business cycle fluctuations. More recently, the link between psychology and expectations formation has been re-introduced into macroeconomics in various ways. One method is to incorporate an exogenous shock into the agent’s forecasting equation. This paper adds to the literature by including these shocks, known as expectation shocks, into a standard asset pricing model to capture periods of euphoria and pessimism in asset price forecasts. Furthermore, the paper shows that the model with expectation shocks replicates a series of U.S. financial market and macroeconomic stylized facts better than a model with rational expectations or learning alone. The setup assumes a representative agent that employs the adaptive learning method of expectations formation of Evans and Honkapohja (2001). The agent utilizes a forecasting model with the same form as the rational expectations equilibrium solution (REE) and behaves like an econometrician by updating the model parameters using constant gain learning and using the model to make forecasts of the equity price. In some periods, however, the forecast is subject to an exogenous, persistent shock that causes the forecast to be either above or below the value implied by the model itself. These shocks, which can be thought of as waves of optimism and pessimism arising Email address: [email protected] (Christopher J. Elias) URL: https://sites.google.com/site/cjelias/ (Christopher J. Elias) Preprint submitted to Elsevier

February 18, 2016

from outside factors such as news, changes in market sentiment, herding and bandwagon effects, etc, introduce a new source of volatility into the equity price. The use of expectation shocks is a relatively simple way to capture the psychological component of expectation formation. Milani (2011) and Milani (2014) employ them to explain business cycle fluctuations in a DSGE model, while Evans and Honkapohja (2003) use them to study monetary policy. A related idea, known as judgemental adjustments or “add-factors”, in which forecasters make adjustments to their forecasts based on variables not necessarily incorporated in their model, are used by Bullard et al. (2008) and Bullard et al. (2010), who show in the latter work that adding persistent shocks to an asset pricing model with learning can be almost self-fulfilling, which is a similar finding here. The current paper extends this literature by examining the influence of expectation shocks in the two asset pricing model economies of Carceles-Poveda and Giannitsarou (2008), which are derived from a general equilibrium stochastic growth framework. The “production economy” allows capital to accumulate, while the “endowment economy” is analogous to the Lucas Tree model (see Lucas (1978)) and assumes capital is fixed at the steady state. The current paper’s scope is most similar to Adam et al. (2015), where the authors show that a consumption based asset pricing model with an agent that learns in a non-linear framework does well at replicating a set of stylized facts similar to those studied in the current paper. Key to those authors results is the notion that agents are learning about the growth rate of the equity price, while the current paper assumes that agents are learning in a linear framework about the level of the equity price. Another closely related work is Carceles-Poveda and Giannitsarou (2008) who show that a representative agent learning about the level of the equity price does not significantly alter the dynamics of the production and endowment economies away from those with rational expectations. This paper represents an extension of those authors’ work. After incorporating expectation shocks into the model economies, I use a two stage indirect estimation procedure to facilitate simulation. In the first stage, calibrated values consistent with the macroeconomic literature are chosen for parameters standard to the stochastic growth framework, and in the second stage I estimate parameters of the expectation shock process, as well as the adaptive learning gain, by employing the method of simulated moments. Compared to a model of rational expectations or learning alone, simulation results suggest for both the production and endowment economies that the model with expectation shocks more effectively replicates the dynamics of equity asset returns, the characteristics of the price-dividend ratio, and the ability of the price-dividend ratio to predict the future equity premium. Comparing the expectation shock model across economies, the endowment economy does better at replicating equity asset returns and the predictability of the equity premium, while the production economy generates more realistic dividend and investment variables. The model’s dynamics come from the increased volatility in the equity price generated by the expectation shock. This paper is organized as follows: Section 2 presents the stylized facts. Section 3 introduces the asset pricing model with learning and expectation shocks. Section 4 describes the calibration scheme, the indirect estimation method, and presents parameter estimates. Section 5 discusses the numerical results, section 6 concludes, and appendices cover the data methods utilized, and the derivation of the stochastic growth model. 2. Stylized Facts [Table 1 about here.] 2

Table 1 presents stylized facts of the post-war U.S. equity market and macroeconomy, based on quarterly data, that will be addressed in this paper. 1 Panel one reports the mean and standard deviation of the quarterly equity and risk free return in percentage terms. The equity and risk free rates have been about two and 0.6 percent, respectively, leading to an equity premium of about 1.5 percent (a.k.a the equity premium puzzle of Mehra and Prescott (1985)). Panel two shows statistics for the price-dividend ratio and the log growth rate of dividends. The price-dividend ratio has a mean of about 34, a standard deviation of about 16, and a first-order autocorrelation coefficient of 0.98. Log dividend growth has a standard deviation of about 29 and a first-order autocorrelation coefficient of -0.57. Panel three reports regression results of the one, two, and four year ahead equity premium on the current period price-dividend ratio divided by its standard deviation. The slope coefficients represent the effect on the cumulative excess return of stocks over the risk free rate, in natural units, of a one standard deviation change in the log-price dividend ratio, and t-statistics are calculated with Newey-West standard errors. As first pointed out by Fama and French (1988), with increasing time horizon, the slope coefficients get more negative and more statistically significant, while the adjusted R2 s get larger. Panel four displays the standard deviation and first-order autocorrelation coefficients of consumption, output, and investment growth, in log terms. Investment growth is significantly more volatile than output growth, and both are more volatile than consumption growth. All three have low autocorrelation. 3. An Asset Pricing Model with Expectation Shocks In this section an asset pricing model is developed that incorporates the psychological elements inherent in expectation formation within financial markets. The environment under consideration contains the two economies, production and endowment, studied in Carceles-Poveda and Giannitsarou (2008), which are developed from a version of the stochastic growth model modified to include a stock market so that an asset pricing equation can be derived. 2 The model is characterized by a representative discounted utility maximizing household (i.e., agent), a representative dividend maximizing firm, and productivity shocks. The difference in the two economies is how capital is treated. The production economy incorporates capital that is allowed to accumulate and depreciate, while the endowment economy, which is nested within the production economy, treats capital as fixed at the steady state (i.e., no depreciation) so that investment is absent. The reduced form equations for both economies characterize the behavior of the equity price. For the production economy, pt = a1 E t pt+1 + a2 pt−1 + b1 zt zt = ρzt−1 + εt

(1)

where pt is the equity price, zt is a productivity shock, εt is i.i.d. N(0, σ2ε ), t is a time subscript, and E t is the possibly non-rational expectations operator. 1

These facts are similar to those used in Carceles-Poveda and Giannitsarou (2008) and Adam et al. (2015). Refer to Appendix A for a detailed description of the data set used and the methods employed. 2 See Appendix B for a complete derivation of the model.

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The reduced form for the endowment economy is pt = aE t pt+1 + bdt dt = ρdt−1 + εt where dt is the equity dividend. The dividend is equal to the productivity shock so the main difference in the reduced form equations for the two economies is the presence of the lag of the equity price in the production economy. The structural parameters (a 1 , a2 , b1 , a, and b) are functions of the deep parameters of the model and are given in Appendix B. 3.1. Rational Expectations The standard assumption in the macroeconomics literature is that of rational expectations. Carceles-Poveda and Giannitsarou (2008) show that the minimum state variable rational expectations equilibrium (REE) solution for the production economy is pt = φ¯ p pt−1 + φ¯ z zt

(2)

where  1 (1 − 1 − 4a1 a2 ) 2a1 b1 φ¯ z = 1 − a1 (φ¯ p + ρ)

φ¯ p =

The condition for uniqueness and stability of this solution is |a 1 + a2 | < 1, which is the case considered in this paper.3 Additionally, the authors show that the unique and stable REE solution for the endowment economy is ¯ t pt = φd

(3)

where φ¯ =

(1 − β − γ)ρ + γ 1 − βρ

β is the household discount factor, and γ is the household risk aversion parameter. 3.2. Adaptive Learning The assumption of rational expectations is rather strong. Specifically, it assumes the agent knows 1. The value of the model’s deep parameters (household risk aversion parameter, capital depreciation rate, capital share in production, etc.). 2. The productivity shock parameters (ρ and σ ε ). 3

For the technical details of rational expectations equilibria, minimum state variable solutions, and the method of undetermined coefficients see Pesaran (1988) and McCallum (1989).

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3. The form of the equity price REE solution (equations (2) and (3)). ¯ 4. The values of the REE solution parameters (φ¯ p , φ¯ z , and φ). The thrust of the adaptive learning approach is to relax assumptions one and four with the idea that the agent doesn’t know the values of the REE parameters and behaves like an econometrician by forming estimates of the parameters through the use of econometric methods. Although several estimation methods have been studied, this paper follows the bulk of the literature and assumes the agent uses least squares regression. A crucial assumption must be made about the information available at the time of estimation. Assume the agent knows the current period value of the productivity shock (z t ), but doesn’t know (i.e., does not observe) the current period value of the equity price (p t ) when it makes forecasts of time t+1 price.4 Therefore, a timeline of events in the learning process is as follows: 1. pt−1 is realized at the end of period t − 1. 2. At the beginning of period t the agent updates its parameter estimates by adding p t−1 and zt−1 to its information set and runs a least squares regression of p t−1 on the variables in its PLM. 3. The value of zt is realized. 4. The agent uses zt , pt−1 , and the parameter estimates calculated in step two to make forecasts of pt+1 . 5. pt is realized from equation 1 and then the other endogenous variables of the model are realized from the equity price (see equation (B.3)), ending period t. 3.2.1. Production Economy For the production economy, the agent’s PLM is of the same form as the REE solution, pt = κc + κ p pt−1 + κz zt + p,t

(4)

where p is a regression error and κc , κ p , and κz are parameters to be estimated using least squares regression. The constant is included with the idea that in real-world econometric estimation constants are almost always used in least squares regression. To derive an expression for the equity price forecast, iterate the PLM one period forward: pt+1 = κc + κ p pt + κz zt+1 + p,t+1 Since pt and zt+1 are unknown at the time of forecasting, this expression must be modified by plugging in expressions for p t from the PLM and zt+1 from the productivity shock law of motion, taking expectations, and collecting terms: E t pt+1 = κc (1 + κ p ) + κ2p pt−1 + (κ p κz + κz ρ)zt 4

(5)

The assumption that the endogenous variable is unknown at the time of forecasting is not uncommon in the literature and is described in Evans and Honkapohja (2001) and employed in Berardi (2007). This makes sense considering in real-world estimation problems market participants rarely have access to current period information. For example, many macroeconomic variables are released weeks or months after the period they are measuring and these figures are often updated several times so that the finalized value can be several months old. Also, equity buyers typically submit order books dictating their demand for equity shares as a function of price that has not yet been observed.

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At the beginning of period t, the agent adds p t−1 and zt−1 to its information set and runs a regression of pt−1 on the variables in the PLM (as can be seen by iterating Equation (4) one period backward). The productivity shock is observed, and the agent makes forecasts in accordance with Equation (5). The value of the time-t equity price is then realized from equation (1), which determines the other endogenous variables in the model and allows the agent to make time-t decisions about equity holdings, etc. This concludes the events of period t. Standard practice in the adaptive learning literature is to represent the least squares updating process as a stochastic recursive algorithm (SRA) consisting of a set of parameter estimates and a precision matrix. The SRA can be written in matrix form as ⎡ ⎤ ⎡ ⎞ ⎤ ⎛⎡ ⎤ ⎢⎢⎢ κc,t ⎥⎥⎥ ⎢⎢⎢ κc,t−1 ⎥⎥⎥ ⎟⎟ ⎜⎜⎜⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎢⎢⎢κ p,t ⎥⎥⎥⎥ = ⎢⎢⎢⎢κ p,t−1 ⎥⎥⎥⎥ + gt R−1 ⎜⎜⎜⎜⎢⎢⎢⎢ pt−2 ⎥⎥⎥⎥ (pt−1 − pˆ t−1 )⎟⎟⎟⎟⎟ t ⎜⎢ ⎢⎣ ⎥⎦ ⎢⎣ ⎟⎠ ⎥⎦ ⎥⎦ ⎝⎣ κz,t κz,t−1 zt−1 ⎛⎡ ⎤ ⎞ ⎜⎜⎜⎢⎢⎢ 1 ⎥⎥⎥  ⎟⎟⎟  ⎜⎜⎢⎢⎢ ⎥⎥ ⎟ ⎜ ⎥ Rt = Rt−1 + gt ⎜⎜⎢⎢ pt−2 ⎥⎥ 1 pt−2 zt−1 − Rt−1 ⎟⎟⎟⎟ ⎝⎣ ⎦ ⎠ zt−1 where time subscripts are added to parameter estimates and the precision matrix R to emphasize their dynamic nature, the expression pt − pˆ t is the agent’s most recent prediction error of the equity price, and initial values for the precision matrix and parameter estimates are assumed to be given. The parameter g is known as the gain parameter and is set to a constant between zero and one. This convention, known as constant gain learning, implies that the agent places a weight of (1 − g)i−1 for a data point i periods in the past, which allows the estimation to be thought of as a weighted least squares regression procedure. In order to maintain stability of the system, a projection facility is used to ensure that the estimate on the lag of the equity price is never greater than one. This is done by setting all parameter estimates to their previous period values in any period when the estimate on the lag of the equity price is greater than one. 3.2.2. Endowment Economy The underlying logic of the adaptive learning mechanism in the endowment economy is the same as in the production economy. Note that d t and zt are equal and therefore interchangeable. The agent’s PLM is pt = ωc + ωd dt + e,t where e , is a regression error and ωc and ωd are parameters to be estimated using least squares regression. To derive the forecasting equation, iterate the PLM one period forward: pt+1 = ωc + ωd dt+1 + e,t+1 Plug in expressions for pt and dt+1 , take expectations, and collect terms, E t pt+1 = ωc + ωd ρdt The timing of events is similar to that of the production economy. 6

(6)

3.3. Expectation Shocks Assume that in some periods the agent’s equity price forecast is subject to an exogenous, persistent shock, implying that the agent’s forecast is either overly optimistic or pessimistic. These “waves of optimism and pessimism” can arise from outside factors such as news, changes in market sentiment, herding and bandwagon effects, etc. In the production economy, modify equation (5) to E t pt+1 = κc (1 + κ p ) + κ2p pt−1 + (κ p κz + κz ρ)zt + et and for the endowment economy, equation (6) becomes E t pt+1 = ωc + ωd ρdt + et In both economies assume et follows a first order autoregressive process, et = ρe et−1 + νt where νt ∼ i.i.d. N(0, σ2e ) and 0 < ρe < 1.5 4. Indirect Estimation A two stage indirect estimation process is utilized to obtain values for the model parameters. In the first stage, calibrated values consistent with the macroeconomic literature are chosen for the parameters typical of real business cycle models. These values are then used in the second stage in a method of simulated moments procedure to estimate the innovation standard deviation and autoregressive coefficient of the expectation shock, and the adaptive learning gain. For the production economy, the discount factor, β, is set to 0.99, coinciding with an average quarterly real interest rate of one percent. γ is set to one and represents a plausible amount of risk aversion. Ten percent of yearly capital depreciation is assumed, which coincides with δ equaling 0.025. Data from the National Income and Product Accounts suggests that the capital share in the firm’s production function, α, is 0.36. The productivity shock parameters are estimated by observing how zt behaves along with output and capital, and then employing Ordinary Least Squares (OLS). In this manner, the standard deviation of the productivity shock (σ ε ) is 0.00712, and the first-order autoregressive coefficient in the productivity shock process (ρ) is 0.95. The only difference in the endowment economy is δ is set to zero because capital is constant and does not depreciate.6 Method of Simulated Moments The estimation technique used is the method of simulated moments (MSM) proposed by Lee and Ingram (1991) and Duffie and Singleton (1993). In general, the MSM technique chooses values of selected parameters that minimizes the distance between particular empirical and simulated moments. 7 Let 5

In Evans and Honkapohja (2003) and Milani (2011) the expectation shock is iid and a first-order autoregressive process, respectively. 6 See Carceles-Poveda and Giannitsarou (2008) for a discussion and DeJong and Dave (2011) for a theoretical explanation behind this calibration scheme. 7 The notation presented here is from Franke (2009).

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Π be a n p x 1 vector of parameters that are being estimated. Assume there is empirical data covering T time periods and that there are nm descriptive statistics in the empirical data that must be matched with simulated data from the model. Additionally, let m ˆ be an n m x 1 vector of moments computed from the empirical data and uˆ (Π) be an n p x 1 vector of moments computed from the simulated data. Define the vector G(Π) as the difference between uˆ (Π) and m, ˆ or G(Π) = uˆ (Π) − m ˆ where it is assumed that n p = nm . The method of simulated moments chooses the parameter vector ˆ that minimizes the criterion function Π ˆ Q(Π) = G(Π) WG(Π) ˆ is a weighting matrix. where W Consistency of the MSM estimator requires that the simulated data be drawn from a stationary distribution. To ensure this, the MSM literature recommends that the simulated data is of length sT , with s being an integer greater than one, and that the first sT/2 data points are removed and only the second sT/2 data points are used for moment calculations. It is assumed that the random numbers used to generate shocks remain unchanged throughout the procedure to ensure that the only source of randomness comes from changes in the parameters. For both economies, Π = [σe , ρe , g] , while the moments used in the estimation are the equity return mean, and the mean and variance of the price-dividend ratio, which are key asset pricing moments of the model. In general, the moment conditions are qt (qt − qemp )2 where qt is the empirical data, qemp is the mean of the empirical data, and the time average is taken. The first moment represents the mean and the second represents the variance. Suitable ranges for parameter values were found through a grid search process. Ranges for the standard deviation of the innovation of the expectation shock vary by model economy: In the production economy 0.0001 < σ e < 0.002 and in the endowment economy 0.01 < σ e < 0.02. In both economies the expectation shock AR coefficient is constrained to be highly persistent, 0.94 < ρe < 0.99, while the adaptive learning gain is constrained at 0.01 < g < 0.5. 8 The minimization is conducted using Matlab’s constrained optimization toolbox using the interior point algorithm. The pseudo-random number seed state is set to eleven so that random number draws used to calculate shocks remain the same throughout the process, T is set to 256 to match ˆ is an the number of observations used to calculate the stylized facts in Table 1, s is set to five, W identity matrix, and median values of the moments are taken over 2000 replications. Initial values of the parameter estimates for the learning algorithms are 0.9*REE (initial values for constants are zero), which have limited impact on the results due to the removal of the first 1024 observations of each replication.9 Table 2 shows the estimation results. 8

Similar values for constant gain learning parameters are employed in Branch and Evans (2006) and Milani (2007). Sensitivity analysis was done to ensure that the simulation results were robust to changes in the initial values of the adaptive learning parameter estimates. 9

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[Table 2 about here.] The estimates show that the standard deviation of the innovation of the expectation shock and the adaptive learning gain are larger in the endowment economy. 5. Numerical Results of Several Model Variants Four variants of the model constructed in section 3 are simulated and results are compared to the stylized facts of Table 1. The first is a rational expectations model with no expectation shock, and represents the benchmark specification (RE1). The second includes rational expectations with the expectation shock (RE2), the third incorporates adaptive learning with no expectation shock (AL), while the fourth couples adaptive learning with the expectation shock (ALE). Where applicable, each model variant uses the parameter values of Table 2. This fragmentation allows the analysis to show how both adaptive learning and the expectation shock affect the results of the benchmark. To maintain consistency with the method of simulated moments procedure, a replication contains 256 periods, which is created by simulating the model for 1280 periods (i.e. 5 × 256) and then discarding the first 1024 periods, thereby removing the effects of initial conditions. 10 The statistics of Table 1 are calculated for the simulated series, the process is repeated for 2000 replications, and median values across the replications are calculated. To generate shocks, random numbers are drawn from a normal distribution with Matlab’s pseudo-random number generator seed state set to eleven. 5.1. Asset Returns [Table 3 about here.] Table 3 shows the results for asset returns. For the production economy, the RE1 model produces an equity return and premium that is too low, a risk free rate that is too large, and too little volatility. Results are similar for the endowment economy except that volatility levels are larger, but not large enough to adequately replicate the data. The results for the RE2 and AL models for both economies show that neither the expectation shock nor adaptive learning alone significantly alters the dynamics away from the benchmark. However, the ALE model for both economies generates increased levels and volatility of the equity return and equity premium, and greater volatility of the risk-free rate in the production economy. 5.2. Price-Dividend Ratio and Dividend Growth [Table 4 about here.] Results for the price-dividend ratio and dividend growth are presented in Table 4. Note that dividend growth in the endowment economy is equal to the productivity shock and thus remains constant across models. Under both RE1 economies, the price-dividend ratio and dividend growth are too stable, and the level of the price-dividend ratio is too low. However, the RE1 production 10

Carceles-Poveda and Giannitsarou (2007) show that initial conditions have a significant effect on the behavior of adaptive learning algorithms.

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economy does capture the persistence of the price-dividend ratio nicely, whereas the RE1 endowment economy does not. Neither RE1 economy is able to match the observed autocorrelation of dividend growth. Compared to the benchmark, the RE2 model generates increased volatility in the variables, most notably for dividend growth in the production economy, while both economies of the AL model generate results very similar to the benchmark. Conversely, compared to the benchmark, the ALE model for both economies generates larger levels and greater volatility of the price-dividend ratio, and more volatile dividend growth in the production economy. 5.3. Equity Premium Predictability [Table 5 about here.] Table 5 presents median regression results of the one, two, and four year ahead cumulative equity premium on the log price-dividend ratio divided by its standard deviation. The RE1 production economy generates slope coefficients and adjusted R 2 s that are nearly zero at all time horizons. Additionally, only a minority of the resulting t-statistics are statistically significant. In the RE1 endowment economy, all of the calculated slope coefficients are positive, statistically insignificant, and the resulting adjusted R2 s are decreasing as the time horizon increases. The RE2 production economy does not significantly alter the resulting slope coefficients from those of the benchmark, while in the endowment economy they become negative and slightly decrease as time horizon increases. The pattern of increasing adjusted R2 s as time horizon increases is not captured in the RE2 production economy, but is present in the endowment economy. The AL model for both economies produces results very similar to those of the benchmark. The ALE endowment economy matches the data much more closely than the ALE production economy in that as the time horizon increases, the slope coefficients are decreasing, more likely to be statistically significant, and the R2 s are increasing. 5.4. Macro Aggregates [Table 6 about here.] Statistics for consumption, output, and investment growth are displayed in Table 6. Note that the endowment economy assumes that consumption and output follow the same law of motion as the exogenous productivity process and therefore the results for these variables are identical for all models; The discussion, therefore, focuses on the production economy. The benchmark generates too little volatility in consumption and investment, and fails to capture the observed autocorrelation of output and investment growth. The RE2 model increases the volatility of consumption and investment growth, while the AL model produces results similar to the benchmark. On the other hand, compared to the benchmark, the ALE model generates large increases in consumption and investment growth, while all other results are similar. 5.5. Summary and Discussion The ALE production and endowment economies improve upon the dynamics of their respective benchmarks by more effectively replicating equity returns and premiums, and the characteristics of the price-dividend ratio. Comparing across ALE specifications, the endowment economy does better at replicating the asset return characteristics and the predictability of the equity premium, while the production economy generates more realistic dividend and investment data. It is clear 10

that the dynamics of the ALE model are coming from the expectation shock and not the adaptive learning mechanism, as evidenced by the fact that the AL model fails to alter the dynamics of the benchmark, which is consistent with the findings of Carceles-Poveda and Giannitsarou (2008). The dynamics of the ALE model are such that the expectation shock is pushing the agent’s forecast away from the correct model specification, which increases the amount of volatility in the equity price, thereby increasing the risk in holding equity and leading to higher returns, a larger equity premium, and more volatile endogenous variables. Moreover, a relatively high equity price today implies that future returns from holding equity will be low, and vice versa. This leads to a negative correlation between the current period price-dividend ratio and future excess returns. Increasing the volatility of the equity price tends to magnify this mechanism. The differences in equity price volatility of the RE1 and ALE models are shown in Figures B.1 and B.2, which display simulated trajectories of stock prices for 10,000 periods. 11 In both economies the equity price in the ALE model is more volatile than in the RE1 model, with the effect much more pronounced in the endowment economy. 12 [Figure 1 about here.] [Figure 2 about here.] It should be noted that the RE2 model is capable of generating similar results to those of the ALE model, but would require a more volatile expectation shock innovation process than what is estimated in the ALE model. Also, incorporating the expectation shock into a model with rational expectations causes problems on theoretical grounds in that it is difficult to justify an agent that has both rational expectations and is subject to waves of optimism and pessimism in expectations formation. 6. Conclusion This paper modified the adaptive learning expectation formation mechanism in stochastic growth asset pricing production and endowment economies to include expectation shocks, which were designed to capture psychological elements inherent in financial market forecasting arising from news, changes in sentiment, bandwagon and herding effects, etc. Selected expectation shock and adaptive learning parameters were estimated by employing the method of simulated moments, while other variables were calibrated using values consistent with the macroeconomic literature. Simulation results suggested for both model economies that inclusion of expectation shocks improved upon the dynamics of the rational expectations and adaptive learning variants of the economies by more effectively replicating equity returns and premiums, the characteristics 11

Figures showing the evolution of the adaptive learning coefficient estimates of the agent’s PLM for both economies are available in a supplementary appendix. 12 This is the same number of periods in simulations in Branch and Evans (2011). For all figures the (arbitrarily chosen) initial values are: Learning coefficients are 0.9 times the REE solutions, constants are zero, and precision matrices have all elements equal to 0.1. Simulations are run for 11,000 periods, and the first 1,000 periods are discarded to remove the effect of initial conditions. The same random numbers are used to generate the productivity shocks in the RE1 and ALE models for each economy. Expectation shock parameters and adaptive learning gains are the applicable estimates in Table 2.

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of the price-dividend ratio, and the ability of the price-dividend ratio to predict the future equity premium. Moreover, the model’s dynamics were shown to derive from the increased volatility of the equity price that was injected into the system from the expectation shocks. Areas for future work include incorporating expectation shocks into other financial market settings, such as models of the term structure of interest rates and exchange rates. Also, exploring alternative estimation methods of the expectation shock parameters may prove useful. Acknowledgements I am grateful to Professors William Branch, Fabio Milani, David Brownstone, Jim Saunoris, Steve Hayworth, and three anonymous reviewers for many useful comments and suggestions. I am also grateful to seminar participants at the University of California Irvine, Colby College, Miami University, Eastern Michigan University, the Federal Deposit Insurance Corporation, and the 19th International Conference on Computing in Economics and Finance for several constructive comments. All errors are my own.

12

Appendix A. Data Description/Methods The stylized facts are calculated from a quarterly dataset running from the first quarter of 1947 through the fourth quarter of 2010, producing a total of 256 observations. The observations from 1947:1 through 1998:4 are used in Campbell (1999) and Campbell (2003), are available at Campbell (2007), and have been unaltered. The techniques employed in the original data were used for the observations from 1999:1 through 2010:4. A NYSE/AMEX value weighted portfolio return index was used to calculate the equity price and dividend data. The equity price index in period t was calculated as (VWRET Xt + 1)Pt−1 , where VWRET Xt is the current period value weighted return excluding distributions and P t−1 is the value of the equity price index in the previous period. The current dividend yield, DYt , was calculated as ((1 + VWRET Dt )/(1 + VWRET Xt )) − 1 where VWRET Dt is the current period value weighted return including distributions. The dividend, D t , was calculated as Pt ∗ DYt . The price-dividend ratio was calculated as the equity price index in the current period divided by the sum of the most current four quarters of dividends (current and previous three quarters). The CPI is for all urban consumers and was converted to the same base year as the original Campbell series. The return on equity was calculated as ([(Pt + Dt )/Pt−1 ] − 1) ∗ 100 where the equity price and dividend series were deflated using the CPI. The inflation rate was calculated as (CPIt −CPIt−1 )/CPIt−1 ∗ 100. The risk free rate was calculated by subtracting the one period ahead inflation rate from the current period three-month Treasury-bill (secondary market) rate. Following Carceles-Poveda and Giannitsarou (2008), the predictability statistics are for regressions of the one, two, and four year ahead equity premium on the current log price-dividend ratio divided by its standard deviation. The slope coefficients as reported are the changes in the one, two, and four year ahead equity premiums, measured in natural units, resulting from a one standard deviation change in the current log price-dividend ratio. The one, two, and four year ahead equity premiums are found by subtracting the one, two, and four year ahead cumulative risk free returns from the one, two, and four year ahead cumulative equity returns. t-statistics are calculated with Newey-West standard errors with the truncation lag length set to floor[4(T/100) 2/9 ], where T is the number of observations in the regression data. For the macro variables, population data is total population of all ages including armed forces overseas, and real per-capita consumption was calculated by adding nominal consumption of nondurable goods and services, dividing by population, and deflating with the CPI. Output is real gross domestic product and investment is real gross private domestic investment, both three decimals. The updated output data was converted to the base year used in the original data set. All growth rates are log growth rates and are calculated as the natural logarithm of the previous period variable subtracted from the natural logarithm of the current period variable. 13 The standard deviation of the growth rates reported in Tables 1, 4, and 6 are for the real variables and are annualized by multiplying the sample standard deviation of the log growth series by 200. The autocorrelation statistics are first-order autocorrelations of the log growth series. The equity return index is available from the Center for Research in Security Prices (CRSP), while all other data are available from the Federal Reserve (FRED) (see Table 7). For data only available monthly, quarterly data was constructed using the end of period value. 13

Because quarterly output and investment data are not available before 1947 these series are missing the log growth rate for the first quarter of 1947.

13

[Table 7 about here.]

14

Appendix B. Stochastic Growth Model The model under consideration is a variant of the stochastic growth model, modified to include a stock market so that an asset pricing equation can be derived, and is along the lines of the models presented in Brock (1982), Rouwenhorst (1995), Lettau (2003), Carceles-Poveda and Giannitsarou (2007), and Carceles-Poveda and Giannitsarou (2008). The version presented here borrows heavily from the ideas and notation in Carceles-Poveda and Giannitsarou (2008). Throughout this section capital letters represent levels of variables, while lower-case letters represent deviations from a log steady-state. The economy consists of a large number of identical households and firms. Households are infinitely lived and in each period maximize a discounted utility function subject to a budget constraint: ∞  max E t β j U(Ct+ j ) Θt ,Bt

j=0

S.T. Ct + Pt Θt + Pbt Bt = (Pt + Dt )Θt−1 + Bt−1 + Wt Nt

(B.1)

where t represents the time period. The utility function is given a specific functional form depending on the household risk aversion parameter, ⎧ 1−γ C ⎪ ⎪ ⎨ 1−γ if γ > 1 U(C) = ⎪ ⎪ ⎩ln C if γ = 1 where β is the discount factor, C t is consumption, P t is the price of equity, Pbt is the price of a riskfree one period bond, Dt is equity dividends, W t is the aggregate wage rate, and Nt is the household labor supply. Additionally, Θ t and Bt represent the household’s holdings of equities and bonds, respectively. The resulting Euler equations for equity and bond prices, written in terms of gross asset returns, are Dt+1 + Pt+1 Pt 1 = b Pt

1 = E t [Mt,t+1 Rt+1 ], where Rt+1 = f f 1 = E t [Mt,t+1 Rt+1 ], where Rt+1

with stochastic discount factor



Mt,t+1

Ct+1 =β Ct

(B.2)

−γ

The representative firm produces a single good Yt using a constant returns to scale production function, α Yt = Zt Kt−1 Nt1−α

where Zt is a productivity shock following a first-order autoregressive process that is assumed to be stationary. log Zt = ρ log Zt−1 + εt 15

Kt−1 is the aggregate capital stock, Nt is the household labor input (assumed to be equal to one because leisure does not enter the utility function), ρ ∈ (0, 1), and ε t is i.i.d. N(0, σ2ε ). Gross profits are given as Xt = Yt − Wt Nt From gross profits the firm funds Investment, I t , and pays out the remainder in dividends, D t . Dt = Xt − It The model allows for capital accumulation, with the law of motion for capital following Kt = It + (1 − δ)Kt−1 with capital depreciation rate δ ∈ (0, 1). The firm’s problem is to maximize the present discounted value of dividends to its owners, ∞  max E t Mt,t+ j Dt+ j Nt ,Kt

j=0

subject to the firm’s production function and the laws of motion for the productivity shock and capital. The resulting first-order conditions are Wt = (1 − α)Yt 1−α 1 = E t {Mt,t+1 [αZt+1 Ktα−1 Nt+1 + (1 − δ)]}

Additionally, market clearing gives the following equations: Yt = Ct + Kt − (1 − δ)Kt−1 Bt = 0 Θt = 1 The substitutions of Nt = 1 and Kt = Pt complete the system of equations for the model. 14 The steady-state equations of the system are derived from the assumption that the steady-state value for the productivity shock equals one. The steady-state equations for the other variables can then be written as −1  (1−α)  1 − β + βδ K¯ = αβ P¯ = K¯ Y¯ = K¯ α P¯ b = β I¯ = δK¯ X¯ = αY¯ D¯ = X¯ − I¯ C¯ = Y¯ − δK¯ ¯ = (1 − α)Y¯ W 14

Capital is equal to the equity price because capital is the only asset in the economy not in zero net supply.

16

Production Economy The non-linear system of equations is log-linearized around the steady-states using a first-order Taylor series approximation.15 The log-linearized system is referred to by Carceles-Poveda and Giannitsarou (2008) as the production economy and is written as: zt+1 = ρzt + εt+1 yt = zt + αkt−1 1 − β(1 − δ) (1 − δ)αβ αβ yt + kt−1 − kt ct = 1 − β(1 − δ) − αβδ 1 − β(1 − δ) − αβδ 1 − β(1 − δ) − αβδ 1 − β(1 − δ) β(1 − δ) β yt + kt−1 − kt dt = 1−β 1−β 1−β pt = E t [−γ(ct+1 − ct ) + (1 − β)dt+1 + βpt+1 ] pbt = E t [−γ(ct+1 − ct )] kt + kt−1 (δ − 1) it = δ xt = yt = wt kt = pt The state variables in this system are kt and zt , and since kt = pt , the system’s reduced form is pt = a1 E t pt+1 + a2 pt−1 + b1 zt zt = ρzt−1 + εt If ψ = (1 − β + δβ)/(αβ) then the parameters of the reduced form are −γ a1 = γ(−2 + δ − αψ) + (δ − ψ)(1 + β(δ − 1 − α2 ψ)) γ(δ − 1 − αψ) a2 = γ(−2 + δ − αψ) + (δ − ψ)(1 + β(δ − 1 − α2 ψ)) ψ(γ(ρ − 1) + αβ(δ − ψ)ρ) b1 = γ(−2 + δ − αψ) + (δ − ψ)(1 + β(δ − 1 − α2 ψ)) The other remaining endogenous variables can be written in terms of the current, lagged, and expected future equity price, as well as the productivity shock. Specifically, if q is an arbitrary endogenous variable, then qt can be written as qt = γq1 pt + γq2 pt−1 + γq3 zt + γq4 E t pt+1

(B.3)

To retrieve the level of a variable from the log-linearized deviation from the steady state, multiply the steady state value by the exponential of the deviation from the steady state. That is, for an arbitrary endogenous variable qt , Qt = Q¯ exp(qt ) 15

Technical details of this process are given in McCallum (1983), Campbell (1994), and Uhlig (1999), while a less technical treatment is given in Zietz (2006).

17

Endowment Economy An alternative specification, referred to as the endowment economy by Carceles-Poveda and Giannitsarou (2008), is characterized by a constant level of capital that does not depreciate (i.e., is always at the steady state value) and is derived by setting k t = 0 and δ = 0 in the production economy system of log-linearized equations. The resulting system is zt+1 ct dt yt xt wt pt

= ρzt + εt+1 = zt = zt = zt = zt = zt = E t [−γ(dt+1 − dt ) + (1 − β)dt+1 + βpt+1 ]

pbt = E t [−γ(dt+1 − dt )] The reduced form is pt = aE t pt+1 + bdt dt = ρdt−1 + εt where the parameters a and b are a=β b = (1 − β − γ)ρ + γ

18

References Adam, K., Marcet, A., Nicolini, J. P., 2015. Stock market volatility and learning. Journal of Finance (Forthcoming). Berardi, M., October 2007. Heterogeneity and misspecifications in learning. Journal of Economic Dynamics and Control 31 (10), 3203–2337. Branch, W., Evans, G., 2006. A simple recursive forecasting model. Economics Letters 91 (2), 158–166. Branch, W., Evans, G., July 2011. Learning about risk and return: A simple model of bubbles and crashes. American Economic Journal: Macroeconomics 3 (3), 159–191. Brock, W., 1982. The Economics of Information and Uncertainty. University of Chicago Press, Ch. 1: Asset Prices in a Production Economy, pp. 1–46. Bullard, J., Evans, G., Honkapohja, S., June 2008. Monetary policy, judgement, and near-rational exuberance. The American Economic Review 98 (3), 1163–1177. Bullard, J., Evans, G., Honkapohja, S., April 2010. A model of near-rational exuberance. Macroeconomic Dynamics 14 (2), 166–188. Campbell, J., June 1994. Inspecting the mechanism: An analytical approach to the stochastic growth model. Journal of Monetary Economics 33 (3), 463–506. Campbell, J., 1999. Handbook of Macroeconomics. Vol. 1. North-Holland, Ch. 19: Asset Prices, Consumption, and the Business Cycle, pp. 1231–1303. Campbell, J., 2003. Handbook of the Economics of Finance. Elsevier, North-Holland, Amsterdam, Ch. 13: Consumption-Based Asset Pricing, pp. 803–887. Campbell, J., 2007. Replication data for: Consumption-based asset pricing. URL http://hdl.handle.net/1902.1/UQRPVVDBHI Carceles-Poveda, E., Giannitsarou, C., August 2007. Adaptive learning in practice. Journal of Economic Dynamics and Control 31 (8), 2659–2697. Carceles-Poveda, E., Giannitsarou, C., July 2008. Asset pricing with adaptive learning. Review of Economic Dynamics 11 (3), 629–651. DeJong, D., Dave, C., 2011. Structural Macroeconometrics. Princeton University Press. Duffie, D., Singleton, K., July 1993. Simulated moments estimation of markov models of asset prices. Econometrica 61 (4), 929–952. Evans, G., Honkapohja, S., 2001. Learning and Expectations in Macroeconomics. Princeton University Press, Princeton, New Jersey. Evans, G., Honkapohja, S., December 2003. Adaptive learning and monetary policy design. Journal of Money, Credit and Banking 35 (6), 1045–1072. 19

Fama, E., French, K., October 1988. Dividend yields and expected stock returns. Journal of Financial Economics 22 (1), 3–25. Franke, R., December 2009. Applying the method of simulated moments to estimate a small agentbased asset pricing model. Journal of Empirical Finance 16 (5), 804–815. Keynes, J. M., 1936. The General Theory of Employment, Interest, and Money. MacMillan and Co., London. Lee, B.-S., Ingram, B. F., February 1991. Simulation estimation of time-series models. Journal of Econometrics 47 (2-3), 197–205. Lettau, M., July 2003. Inspecting the mechanism: Closed-form solutions for asset prices in real business cycle models. The Economic Journal 113 (489), 550–575. Lucas, R., November 1978. Asset prices in an exchange economy. Econometrica 46 (6), 1429– 1445. McCallum, B., 1983. On non-uniqueness in rational expectations models: An attempt at perspective. Journal of Monetary Economics 11 (2), 139–168. McCallum, B., 1989. Monetary Economics: Theory and Policy. Macmillan. Mehra, R., Prescott, E., March 1985. The equity premium: A puzzle. Journal of Monetary Economics 15 (2), 145–161. Milani, F., 2007. Expectations, learning, and macroeconomic persistence. Journal of Monetary Economics 54, 2065–2082. Milani, F., May 2011. Expectation shocks and learning as drivers of the business cycle. The Economic Journal 121 (552), 379–401. Milani, F., September 2014. Sentiment and the u.s. business cycle, unpublished Manuscript. Pesaran, M. H., 1988. The Limits to Rational Expectations. B. Blackwell. Pigou, A., 1927. Industrial Fluctuations. MacMillan. Rouwenhorst, K. G., 1995. New Frontiers of Modern Business Cycle Research. Princeton University Press, Ch. 10: Asset Pricing Implications of Equilibrium Business Cycle Models. Uhlig, H., 1999. Computational Methods for the Study of Dynamic Economies. Oxford University Press, Oxford, Ch. A Toolkit for Analysing Nonlinear Dynamic Stochastic Models Easily. Zietz, J., December 2006. Log-linearizing around the steady state: A guide with examples, working Paper.

20

Figure B.1: Trajectory of Equity Price - Production Economy RE1

44

Equity Price

42 40 38 36 34

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

6000

7000

8000

9000

10000

Period ALE

250

Equity Price

200 150 100 50 0

0

1000

2000

3000

4000

5000

Period

21

Figure B.2: Trajectory of Equity Price - Endowment Economy RE1

290

Equity Price

280 270 260 250 240

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

6000

7000

8000

9000

10000

Period 3

×10 4

ALE

Equity Price

2.5 2 1.5 1 0.5 0

0

1000

2000

3000

4000

5000

Period

22

Table 1: U.S. Economy Stylized Facts (Quarterly Data: 1947.1-2010.4)

Panel 1 - Asset Moments Equity Return Risk Free Return Equity Premium

Mean 2.0145 0.5504 1.4641

Std Dev 8.1068 1.3442 8.2496

Panel 2 - Price-Dividend Ratio and Dividend Growth Moments Price-Dividend Ratio Dividend Growth

Mean 34.1853

Std Dev 16.0356 28.6657

Autocorrelation 0.9787 -0.5677

Panel 3 - Equity Premium Predictability Horizon 1 year 2 years 4 years

Adjusted R2 0.2060 0.3534 0.4891

Slope -0.0851 -0.1672 -0.3207

t-statistic -4.37 -5.22 -7.54

Panel 4 - Macro Aggregate Moments Std Dev Autocorrelation Consumption Growth 1.7270 0.0149 Output Growth 2.2534 0.3132 Investment Growth 10.9538 0.2231 Note: All moments except the P-D ratio are in percentage terms. See Appendix A for information on the data set used and methods employed.

23

Table 2: MSM Estimates

Parameter σe ρe g

Production Economy Endowment Economy 0.0010 0.9548 0.1607

0.0166 0.9500 0.3196

24

Table 3: Asset Return Simulation Statistics

Equity Return Mean Std Dev

Risk Free Rate Mean Std Dev

Equity Premium Mean Std Dev

2.0145

8.1068

0.5504

1.3442

1.4641

8.2496

Production Economy RE1 1.0124 0.0632 RE2 1.0213 0.0831 AL 1.0201 0.0935 ALE 1.3408 0.5817

1.0097 1.0091 1.0099 1.0739

0.0582 0.0776 0.0885 0.3502

0.0028 0.0123 0.0086 0.6208

0.0251 0.0295 0.0268 0.4822

Data

Endowment Economy RE1 1.0124 0.7270 1.0102 0.1033 0.0018 0.7166 RE2 1.0278 1.8268 1.0102 0.1033 0.0200 1.8214 AL 1.0125 0.7270 1.0102 0.1033 0.0017 0.7166 ALE 2.0659 10.3479 1.0102 0.1033 1.0584 10.3449 Median values of 2000 replications with 256 observations per replication. RE1 is for simulation under rational expectations with no expectation shocks, RE2 is for simulation under rational expectations with expectation shocks, AL is for simulation under adaptive learning with no expectation shocks, and ALE is for simulation under adaptive learning with expectation shocks. All moments are in percentage terms.

25

Table 4: Price-Dividend Ratio and Dividend Growth Simulation Statistics

Price-Dividend Ratio Dividend Growth Mean Std Dev Autocorrelation Std Dev Autocorrelation Data

34.1853 16.0356

0.9787

28.6657

-0.5677

Production Economy RE1 24.8052 1.7380 RE2 24.9696 3.6066 AL 24.9804 2.4138 ALE 28.7838 15.8277

0.9789 0.9809 0.9918 0.9889

5.8434 11.3483 5.1745 25.6550

-0.0349 -0.0312 -0.0143 -0.0018

Endowment Economy RE1 24.7501 0.1628 0.5413 1.4396 -0.0282 RE2 24.7815 1.1756 0.9294 1.4396 -0.0282 AL 24.7501 0.1628 0.5414 1.4396 -0.0282 ALE 29.8134 16.0873 0.9771 1.4396 -0.0282 Median values of 2000 replications with 256 observations per replication. RE1 is for simulation under rational expectations with no expectation shocks, RE2 is for simulation under rational expectations with expectation shocks, AL is for simulation under adaptive learning with no expectation shocks, and ALE is for simulation under adaptive learning with expectation shocks. The dividend growth standard deviation data are expressed in annualized percentage points.

26

Table 5: Equity Premium Predictability Simulation Statistics

1 year β Data

Regression Coefficients 2 years 4 years t β t β t

-0.0851

-4.37

-0.1672

-5.22

-0.3207

-7.54

Median

% Sig

Median

% Sig

Median

% Sig

0.0000 -0.0001 -0.0001 -0.0079

19.0 31.3 41.9 50.4

-0.0001 -0.0002 -0.0002 -0.0160

29.0 34.1 44.6 49.9

Production Economy RE1 0.0000 9.3 RE2 0.0000 27.5 AL 0.0000 38.7 ALE -0.0040 51.3

Adjusted R-Squared 1 year 2 years 4 years 0.2060

0.3534

0.4891

0.0067 0.0685 0.1074 0.6476

0.0174 0.0800 0.1507 0.5734

0.0264 0.0752 0.1724 0.4507

Endowment Economy RE1 0.0060 0.0 0.0061 0.0 0.0064 0.0 0.1616 0.0798 0.0380 RE2 -0.0067 49.5 -0.0183 95 -0.0371 99.6 0.0301 0.1267 0.2680 AL 0.0060 0.0 0.0061 0.0 0.0065 0.0 0.1610 0.0796 0.0382 ALE -0.0532 57.5 -0.1370 79 -0.3341 90.2 0.0556 0.1499 0.2979 Median values of 2000 replications with 256 observations per replication. RE1 is for simulation under rational expectations with no expectation shocks, RE2 is for simulation under rational expectations with expectation shocks, AL is for simulation under adaptive learning with no expectation shocks, and ALE is for simulation under adaptive learning with expectation shocks.

27

Table 6: Macro Aggregate Simulation Statistics

Data

Consumption Growth Output Growth Std Dev Autocorrelation Std Dev Autocorrelation

Investment Growth Std Dev Autocorrelation

1.7270

Production Economy RE1 0.4491 RE2 1.4451 AL 0.7014 ALE 3.8617

0.0149

2.2534

0.3132

10.9538

0.2231

0.1531 -0.0108 0.0612 -0.0031

1.4310 1.4331 1.4308 1.4791

-0.0131 -0.0090 -0.0149 0.0292

4.3544 5.8540 4.0136 10.4254

-0.0315 -0.0272 -0.0295 0.0014

Endowment Economy RE1 1.4396 -0.0282 1.4396 -0.0282 N/A N/A Median values of 2000 replications with 256 observations per replication. RE1 is for simulation under rational expectations with no expectation shocks, RE2 is for simulation under rational expectations with expectation shocks, AL is for simulation under adaptive learning with no expectation shocks, and ALE is for simulation under adaptive learning with expectation shocks. Standard deviation data are expressed in annualized percentage points. Results for all models in the endowment economy are identical so only the RE1 results are shown.

28

Table 7: Data ID and Sources

Series

Data ID

Source

Value Weighted Return Including Dividends Value Weighted Return Excluding Dividends CPI, All Urban Consumers US Govt 90 Day T-Bill Secondary Market Rate Nominal Consumption, Non-Durables Nominal Consumption, Services Real Gross Domestic Product, Three Decimals Real Gross Private Domestic Investment, Three Decimals

VWRETD VWRETX CPIAUCSL DTB3 PCND PCESV GDPC96 GDPIC96

CRSP CRSP FRED FRED FRED FRED FRED FRED

29