From Which Consumption-Based Asset Pricing Models Can Investors Profit? Evidence from Model-Based Priors

Mathias S Kruttli

doi:10.1093/jjfinec/nbaa023

From Which Consumption-Based Asset Pricing Models Can Investors Profit? Evidence from Model-Based Priors

Kruttli, Mathias S 2022-06-08 00:00:00 Abstract This article analyzes whether consumption-based asset pricing models improve the excess returns forecasts of a hypothetical investor with access to these models from 1947 onwards. The investor imposes economic constraints derived from asset pricing models as model-based priors on predictive regression parameters through a Bayesian framework. Three models are considered: habit formation, long-run risk, and prospect theory. The model-based priors generally perform better than priors that shrink the parameter estimates to the historical average model and priors that impose a positive equity premium. This analysis helps to assess the value of consumption-based asset pricing models to investors. Predicting aggregate stock returns has been of great interest to academics and investors alike. For academics, the predictability of stock returns is important for testing market efficiency. For investors, knowing whether returns are predictable is crucial for portfolio allocation decisions. A range of papers investigates which variables can predict returns.1 Welch and Goyal (2008) provide a comprehensive analysis of the major predictors and question whether an investor could have profitably used them to forecast returns. Campbell and Thompson (2008) further investigate these findings by imposing economic constraints when estimating predictive regressions. They find that sign constraints on the parameter estimates of predictive regressions and a nonnegativity restriction on the equity premium (EP) help an investor to reduce uncertainty about the regression parameters and profitably forecast aggregate stock returns. Financial economists have developed consumption-based asset pricing models that attempt to explain the empirically observed behavior of stock markets and also propose solutions for the EP puzzle through different mechanisms. A natural question to ask is whether consumption-based asset pricing models could act as a source of economically motivated parameter constraints, that is, model-based priors, and whether a hypothetical investor who had access to these models from 1947 onwards could have predicted the aggregate excess returns more accurately over the subsequent decades. This exercise helps to assess the value of consumption-based asset pricing models to finance practitioners and evaluates asset pricing models through a novel approach. In this article, an investor who forecasts aggregate excess returns with the dividend–price ratio “out-of-sample (OOS),” where OOS is in quotes to indicate qualifications discussed below, has a prior belief about the parameter estimates of the predictive regression that stems from one of the following asset pricing models: the habit formation (HF) model of Campbell and Cochrane (1999), the prospect theory (PT) model of Barberis, Huang, and Santos (2001), and the long-run risk (LRR) model of Bansal and Yaron (2004). I choose these three asset pricing models as they suggest different mechanisms to explain the EP puzzle.2 The prior belief derived from the asset pricing model helps the investor to reduce uncertainty about the parameters of the predictive regression. The investor updates her beliefs about the predictive regression parameters with empirical data and forecasts excess returns based on the posterior parameter estimates. Because the consumption-based asset pricing models considered in this article have been published around the year 2000, the presented results are not attainable for an actual investor who started forecasting excess returns in 1947. However, the analysis presented in this article is of value for finance academics and practitioners because of several reasons. First, an underperformance of the model-based priors against benchmark forecasting models that are truly OOS would strongly question the usefulness of these asset pricing models for investors. Second, a strong performance of the model-based priors emphasizes that the asset pricing models could have been of use for a hypothetical investor despite the authors of the asset pricing models not being focused on model-based priors as a model evaluation tool. Third, information on which consumption-based asset pricing model yields priors that perform well can be important for researchers who have to decide which mechanisms to incorporate in future asset pricing models. I propose a simple Bayesian econometric framework to implement the economic constraints derived from the consumption-based asset pricing models as prior distributions on the parameters of single-variable predictive regressions. Single-variable predictive regressions are generally used in the return prediction literature (see, e.g., Welch and Goyal 2008). Further, single-variable predictive regressions are used by Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004) to assess the predictability implied by their asset pricing models. My framework has two steps. First, the predictive regression is estimated based on simulated data from the asset pricing model. Second, the parameter estimates from the first step are used as prior information when estimating the predictive regression with empirical data. Further, in line with Campbell and Thompson (2008) and Pettenuzzo, Timmermann, and Valkanov (2014), the EP is restricted to be nonnegative. The main predictor is the dividend–price ratio. The dividend–price ratio is endogenous in the three asset pricing models, and thus, can be simulated. My approach is related to the macroeconometric literature, in which prior distributions from dynamic stochastic general equilibrium models are imposed on vector autoregressions to forecast macroeconomic variables (see, e.g., Ingram and Whiteman 1994; Del Negro and Schorfheide 2011). The implied return predictability differs across the asset pricing models due to the model-specific mechanisms that lead to time variation in valuation ratios. For the HF and PT models, the model-implied return predictability is higher than that found in the empirical data. However, for the LRR model, the model-implied return predictability is lower than found in the empirical data. The model-based priors’ forecast performance is compared to two OOS benchmark priors. The first prior shrinks the parameter estimates toward the historical average model. The historical average model implies that the EP will be equal to its past average. The second prior restricts the EP to be weakly positive as proposed by Pettenuzzo, Timmermann, and Valkanov (2014) in addition to shrinking the parameter estimates toward the historical average model. The model-based priors generally outperform the two benchmark priors. Over the entire forecast period from 1947 to 2018 and at an annual return frequency, the model-based priors derived from the HF model generate an OOS R2 that is 8.4%. For the LRR model-based priors the OOS R2 is 3.5%, and for the PT model-based priors the value is 9.4%. The historical average prior achieves an OOS R2 of 1.3% without the positive EP constraint and an OOS R2 of 3.5% when imposing the positive EP constraint. While these baseline results are based on the calibration of the asset pricing models as proposed by the respective authors of the models, recalibrating the asset pricing models such that the data used for the recalibration have no overlap with the forecasting period leads to comparable results. Further, the results are robust to changing the tightness of the model-based priors, using longer forecast horizons, and adding additional predictors to the model. However, the predictive power of the dividend–price ratio when forecasting the excess returns is time varying over the 1947–2018 forecast sample, and so is the performance of the model-based priors. The predictive power of the dividend–price ratio was strong in the first half of the sample, when the correlation between valuation ratios and the subsequent excess returns was strongly positive. However, the bull market of the late 1990s was a period when the dividend–price ratio had very weak predictive power as historically high prices and low dividends were followed by high stock returns for several years until the end of the dot-com boom in the early 2000s. Welch and Goyal (2008) find that the dot-com period is a large contributor to the poor OOS performance of valuation ratios, when no economic constraints are imposed on the predictive model, and that an investor would perform better by forecasting the excess returns with a historical average model that assumes valuation ratios have no predictive power. Campbell and Thompson (2008) and Pettenuzzo, Timmermann, and Valkanov (2014) show that the nonnegative EP constraint can help to mitigate the poor performance of the predictive model during this episode. The HF and PT model-based priors can improve upon the performance of the nonnegativity constraint because the return predictability that these two models imply is higher than found in the empirical data, which allows an investor to take advantage of the strong predictive power of valuation ratios in the early half of the sample. While the baseline results described above show a stronger performance of the priors derived from the HF and PT models compared to the priors derived from the LRR model, it is noteworthy that when imposing the model-based priors without a nonnegativity constraint, the HF and PT model-based priors generally underperform the LRR model-based priors. The reason is that when the nonnegativity constraint is lifted, the forecasts during the dot-com boom when imposing the HF and PT model-based priors are strongly negative, which leads to substantially larger forecast errors during this period. This article contributes to the growing literature that investigates the use of economically motivated parameter constraints for predicting the excess returns and imposes them through a type of Bayesian framework on the predictive regressions. Pastor and Stambaugh (2009) employ a prior that implies a negative correlation between expected and unexpected return shocks. Shanken and Tamayo (2012) consider prior beliefs on the risk-return tradeoff and on the extent to which mispricing drives predictability. Pettenuzzo, Timmermann, and Valkanov (2014) propose a Bayesian methodology that imposes a nonnegative EP and bounds on the conditional Sharpe ratio. Their constraints lead to return forecasts that are substantially more accurate. Wachter and Warusawitharana (2009) model skepticism of an investor over the predictability of excess returns as an informative prior over the R2 and show that a skeptical investor achieves better forecasts. Wachter and Warusawitharana (2015) analyze whether an investor who is initially skeptical about the existence of return predictability would update her prior and conclude that returns are predictable when confronted with historical data. Other Bayesian studies consider uncertainty about the predictive regression parameters through uninformative priors (see, e.g., Stambaugh 1999; Barberis 2004; Brand et al. 2005; Penasse 2016) or investigate how parameter uncertainty affects the long-run predictive variance (see, e.g., Pastor and Stambaugh 2012; Avramov, Cederburg, and Lucivjanska 2017). The remainder of this article is organized as follows. Section 1 explains the Bayesian methodology used to impose the model-based priors. Section 2 reports the main results. Section 3 discusses the robustness of the results and extensions to the analysis. Section 4 concludes the article. 1 Methodology This section describes how I impose economic constraints on the single-variable predictive regressions through priors derived from consumption-based asset pricing models and how these models are simulated to obtain the priors. 1.1 Excess Return Prediction Model The aggregate excess return at time t + 1 is denoted by rt+1 and is defined as the log rate of return on the stock market in excess of the prevailing short-term interest rate. As is common in the return prediction literature, rt+1 is regressed on a constant and a predictor, xt, which is lagged by one period: rt+1=β0+β1xt+ϵt+1, where ϵt+1∼N(0,σϵ2).(1) The OOS predictions of the excess returns are generated through recursive forecasts (see, e.g., Campbell and Thompson 2008; Welch and Goyal 2008). 1.2 Prior Distribution The prior distribution of the parameters in Equation (1)—that is β and σϵ2—is assumed to be Gamma–Normal and builds on Koop (2003) and Pettenuzzo, Timmermann, and Valkanov (2014). The prior distribution is given by β∼N(β̱,V̱), σϵ−2∼G(σϵ̱−2,v̱).(2) The mean and the variance of the Normal prior distribution are specified as β̱=[β̱0β̱1], V̱=[λσr200λσr2/σx2].(3) The parameter λ is exogenously chosen and is weakly positive. If λ is large, the prior is loose. If λ is equal to zero, the prior is dogmatic. I set λ = 1 for the baseline analysis. Section 3 reports results for different values of λ and shows that these are in line with the benchmark case. The sample moments σr2 and σx2 are scaling factors, which ensure that the results are comparable for different predictors and forecast frequencies. Such scaling factors are commonly used in Bayesian macroeconometrics and date back to Litterman (1986). The sample moments are given by σr2=1t−2∑τ=2t(rτ−r¯)2, r¯=1t−1∑τ=2trτ(4) and σx2=1t−2∑τ=1t−1(xτ−x¯)2, x¯=1t−1∑τ=1t−1xτ.(5) The sample moments can change over time as the recursive estimation uses a longer sample with every forecast iteration, that is, as t increases.3 The Gamma distribution parametrization follows Koop (2003) by specifying the distribution with mean σϵ−2 and degrees of freedom v̱ ⁠. The tightness of the prior is controlled by v̱ ⁠, which is strictly positive. A large v̱ corresponds to a tight prior, and a small v̱ corresponds to a diffuse prior. The baseline analysis sets v̱ to 10, but the results are robust to a tighter or a more diffuse prior on σϵ−2 (see Section 3). 1.3 Posterior Distribution The model-based prior distributions yield conditional posterior distributions for β and σϵ−2 ⁠. I draw from these two conditional distributions through a Gibbs sampler. The conditional posterior distribution for β is β|σϵ−2,It∼N(β¯,V¯),(6) where V¯=(V̱−1+σϵ*−2X′X)−1, β¯=V¯(V̱−1β̱+σϵ*−2X′R),(7) X is a t−1×2 matrix with rows [1 xτ] for τ=1,…,t−1 ⁠, and R is a t−1×1 vector with elements rτ for τ=2,…,t ⁠. The information set at time t is denoted by It ⁠. The conditional posterior distribution for σϵ−2 takes the form σϵ−2 |β,It∼G(s¯−2,v¯),(8) where v¯=v̱+(t−1), and s¯2=∑τ=2t(rτ−β0−β1xτ−1)2+σϵ*2v̱v¯.(9) Through the Gibbs sampling algorithm with J iterations, we obtain a series of draws for each of the parameters denoted by {βj} and {σϵ−2,j} for j=1,…,J ⁠. These simulated series can then be used to draw from the predictive return distribution p(rt+1|It)=∫β,σϵ−2p(rt+1|β,σϵ−2,It)p(β,σϵ−2|It)dβdσϵ2,(10) which yields {rt+1j} for j=1,…,J ⁠. The point forecast for the excess return in period t + 1 is given by the mean of the sampled distribution r^t+1m=1J∑j=1Jrt+1j.(11) 1.4 Priors from Asset Pricing Models I use the asset pricing models to obtain prior means for the prior distribution shown in Equation (2). The priors are derived from the three consumption-based asset pricing models, that is, HF, LRR, and PT, through a simulation procedure. All three models specify a log consumption and a log dividend growth process. These two processes drive the state variables of the models, and the state variables determine the dividend–price ratio. By simulating random shocks, time series of consumption growth and dividend growth are generated, based on which I solve the models for the log EP and the log dividend–price ratio. The dividend–price ratio is one of the most prominent predictors of the EP prediction literature and is defined as the ratio of the dividends summed over the past 12 months and the end of period price. I investigate additional predictors in Section 3.3. (A more detailed description of the models and how to solve them is provided in Online Appendix.) The simulated log excess return is denoted rM,t+1 ⁠. I can then estimate the single-variable predictive regression given in Equation (1) with simulated data, where the simulated predictor xM,t is the log dividend–price ratio: rM,t+1=βM,0+βM,1xM,t+ϵM,t+1, where ϵM,t+1∼N(0,σM,ϵ2).(12) The OLS estimates of βM=[βM,0, βM,1]′ and σM,ϵ−2 are denoted by β* and σϵ*−2 ⁠, which act as the prior means of the Gamma–Normal distribution described in Section 1.2. The same predictive regression with simulated data is also estimated by the respective authors of the consumption-based asset pricing models, that is Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004), to assess the predictability of excess returns implied by their proposed theories. I consider two calibrations when simulating data from the asset pricing models for the model-based priors. First, I use the same calibration as proposed by the authors in the respective published articles. The authors use almost identical calibration datasets, and thus, the comparison of the model-based priors’ performances should not be distorted. Second, I recalibrate the models with an empirical data sample that has no overlap with the OOS period. I assume that the investor has no uncertainty about the parameters of the asset pricing models, since the focus of this article and the literature it contributes to is the investor’s uncertainty about the parameters of the predictive regression given in Equation (1). In addition to imposing prior means derived from the asset pricing models, I also impose a nonnegativity constraint that requires a weakly positive EP. The motivation behind this constraint is two-fold. First, the three asset pricing models considered in this article have an equilibrium setting where risk-averse agents hold stocks because the EP is positive. Therefore, imposing a nonnegativity constraint on the EP is in line with this aspect of the asset pricing models.4 Second, imposing a nonnegativity constraint builds on other papers in the literature that have used such a constraint successfully (see Campbell and Thompson 2008; Pettenuzzo, Timmermann, and Valkanov 2014). I impose the nonnegativity constraint by following Pettenuzzo, Timmermann, and Valkanov (2014) and rejecting Gibbs sampler draws from the posterior in Equation (6), when the condition given by β0,β1∈At,where At=β0+β1xτ≥0(13) is not satisfied for τ=1,…,t ⁠.5 1.5 Asset Pricing Model Specifications and Simulation Procedures This section discusses the specifications and simulation procedures used to generate the simulated data from the asset pricing models to derive the priors. The description of the asset pricing models is kept short, and additional details as well as sample moments can be found in Online Appendix. 1.5.1 The HF model Campbell and Cochrane (1999) use a standard representative agent consumption-based asset pricing model but add a slow-moving habit to the basic power utility function. This slow-moving habit leads to a time-varying risk premium that is higher at business cycle troughs than at peaks. The agents are identical and maximize their utility given by E[∑t=0∞δt(Ct−Xt)(1−γ)−11−γ],(14) where Ct is the consumption level, Xt is the level of habit, δ is the time discount factor, and γ is the risk aversion. A surplus consumption ratio St≡(Ct−Xt)/Ct is defined—a small value of St indicates that the economy is in a bad state. I use the specification of the HF model that assumes perfect correlation between log consumption and log dividend growth to generate the model-based priors, which is the benchmark specification in Campbell and Cochrane (1999). There, stocks represent a claim to the consumption stream. The price–consumption ratio for a consumption claim satisfies6 PtCt(st)=Et[Mt+1Ct+1Ct[1+Pt+1Ct+1(st+1)]],(15) where st is the log of St, and the intertemporal marginal rate of substitution Mt+1 takes the form Mt+1≡δ(St+1StCt+1Ct)−γ.(16) Because the term (St+1/St)−γ correlates positively with asset returns, the HF model generates a higher EP than the standard power utility model. The simulation is at a monthly frequency, and the quarterly and annual data are constructed by time-averaging the monthly data. The same procedure is used by Campbell and Cochrane (1999). I simulate 120,000 months, estimate β* and σϵ*2 ⁠, and average the estimates over 10 iterations. 1.5.2 The PT model In the model of Barberis, Huang, and Santos (2001), the agent not only derives utility from consumption but also from financial wealth fluctuations. There are two important aspects in the way financial wealth fluctuations affect the utility of an economic agent. First, the agent is loss averse. Second, the degree of loss aversion depends on prior investment outcomes. Prior gains lead to less loss aversion, and prior losses lead to more loss aversion. The agent’s maximization problem is set up as E[∑t=0∞(ρtCt1−γ1−γ+b0C¯t−γρt+1v(Xt+1,St,zt))].(17) The second term captures the fact that the agent’s utility is affected by fluctuations in financial wealth. The variable Xt+1 denotes the change of the financial wealth between time t and t + 1 and is defined as Xt+1≡StRt+1−StRf,t.(18) The variable St measures the value of the agent’s risky assets at time t. The variable zt accounts for prior gains and losses up to time t and is defined as Zt/St ⁠, where Zt is a historical benchmark level for the value of the risky asset. If zt is smaller than one, the agent has prior gains; if zt is greater than one, the agent faces prior losses. The time discount factor is ρ, and b0C¯t−γ is a scaling term, with γ being the risk aversion. The form of the utility function over financial wealth v(.) is different conditional on prior gains or prior losses and given in Online Appendix. The price–dividend ratio is assumed to be a function of the state variable zt: Pt/Dt=f(zt).(19) Similar to the HF model, the PT model is specified by Barberis, Huang, and Santos (2001) with perfect positive correlation between the log consumption and log dividend growth processes and with imperfect positive correlation between the two processes. I only use the latter specification, as it more successfully matches the empirical data moments. The authors calibrate the model with a range of parameter values for the investor’s sensitivity to financial wealth fluctuations (b0) and the effect of prior losses on risk aversion (k). I generate priors from the parameterizations that set b0 equal to 100 and k equal 8. Of the specifications proposed by Barberis, Huang, and Santos (2001), setting b0 equal to 100 and k equal to 8 generates a log EP that is closest to the empirical data moment. Following Barberis, Huang, and Santos (2001), I simulate the model at monthly, quarterly, and annual frequencies by adjusting the model parameters accordingly. For the monthly, quarterly, and annual frequencies, I simulate 120,000, 40,000, and 10,000 periods, respectively, and average the β* and σϵ*2 estimates over 10 iterations. 1.5.3 The LRR model Bansal and Yaron (2004) propose a solution to the EP puzzle through a consumption-based asset pricing model with Epstein and Zin (1989) preferences. Their model differs from other consumption-based asset pricing models in two ways. First, they include a small persistent expected growth rate component in the consumption and dividend growth rate processes. This component causes consumption and the return on the market portfolio to covary positively, and thus, the economic agents require a higher risk premium. Second, they allow for time-varying volatility, which accounts for fluctuating economic uncertainty, in both processes. The asset pricing restriction for the real return on the market portfolio Rm,t+1 is Et[δθGc,t+1−θψRc,t+1−(1−θ)Rm,t+1]=Et[Mt+1Rm,t+1]=1,(20) where Gc,t+1 is the aggregate gross growth rate of consumption, Rc,t+1 denotes the real return on an asset that pays aggregate consumption as dividends, δ is the time discount factor, and Mt+1 is the intertemporal marginal rate of substitution. The parameter θ is defined as (1−γ)/(1−1ψ) ⁠, where γ is the risk aversion parameter, and ψ accounts for the intertemporal elasticity of substitution. The dynamics of log consumption growth, gc,t+1 ⁠, and log dividend growth, gd,t+1 ⁠, incorporate a small persistent predictable component xt and a time-varying volatility component σt, reflecting fluctuating economic uncertainty. The exact dynamics are given by xt+1=ρxt+φeσtet+1gc,t+1=μc+xt+σtηt+1gd,t+1=μd+ϕxt+φdσtut+1σt+12=σ2+v1(σt2−σ2)+σwwt+1,(21) with et+1, ut+1, ηt+1 ⁠, and wt+1 having i.i.d. standard Normal distributions. The state variables, which determine the price–consumption ratio, zt, and price–dividend ratio, zm,t ⁠, are xt and σt. The solutions for zt and zm,t are zt=A0+A1xt+A2σt2zm,t=A0,m+A1,mxt+A2,mσt2,(22) where A and Am can be derived analytically as described in Bansal and Yaron (2004). The LRR model, like the HF model, is simulated at a monthly frequency, and the quarterly and annual values are time averaged.7 Again, 120,000 months are simulated to estimate β* and σϵ*2 ⁠, and the estimates are averaged across ten iterations. Bansal and Yaron (2004) present two specifications of their model: with and without time-varying volatility of consumption growth. Because the specification that accounts for time-varying volatility of consumption growth is more successful at matching the empirical data moments and leads to a time-varying risk premium, I generate priors only from this specification. I use the benchmark calibration of Bansal and Yaron (2004) that sets the agent’s risk aversion equal to ten. 1.5.4 Simulation results Panels A and B of Table 1 show β* and σϵ*−2 estimated from simulated data of the three consumption-based asset pricing models.8 The table also reports the in-sample empirical estimates for comparison. The empirical estimates are shown for the total sample from 1926 to 2018, and the sample from 1947 to 2018 over which the excess return will be forecast in the subsequent analysis.9 Standard errors are shown for the empirical coefficient estimates. Table 1 Model-implied parameters Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A reports the in-sample coefficient estimates of the single-variable predictive regression of the log excess return given in Equation (1). The log dividend–price ratio is the predictor. The empirical in-sample estimates are for the total data sample from 1926 to 2018 and the sample from 1947 to 2018. The model-based estimates from the three asset pricing models HF, LRR, and PT are obtained through the Monte Carlo simulation procedure described in Section 1.4. Standard errors in parentheses are shown for the empirical estimates. Panel B shows the inverse of the variance of the return innovation used for the Gamma prior in Equation (2). Panel C reports the R2 (in percent) of the single-variable predictive regression. Open in new tab Table 1 Model-implied parameters Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A reports the in-sample coefficient estimates of the single-variable predictive regression of the log excess return given in Equation (1). The log dividend–price ratio is the predictor. The empirical in-sample estimates are for the total data sample from 1926 to 2018 and the sample from 1947 to 2018. The model-based estimates from the three asset pricing models HF, LRR, and PT are obtained through the Monte Carlo simulation procedure described in Section 1.4. Standard errors in parentheses are shown for the empirical estimates. Panel B shows the inverse of the variance of the return innovation used for the Gamma prior in Equation (2). Panel C reports the R2 (in percent) of the single-variable predictive regression. Open in new tab For all three asset pricing models, β1* is positive. Thus, a high dividend–price ratio predicts higher subsequent returns, which is in line with the empirical estimates. Across all return frequencies, the coefficients of the LRR model are lower than the empirical estimates. The implication is that in the LRR model, the predictive power of the dividend–price ratio is weaker. However, for the HF and PT models, the model-implied coefficient estimates are higher than the empirical estimates. These estimates indicate that the return predictability in the HF and PT models is stronger than observed empirically. The σϵ*−2 values implied by all three asset pricing models are close to the empirical values. Panel C reports the R2 for the single-variable predictive regression in Equation (12). The R2 values for the LRR model are lower than for the HF and PT models and the empirical data. The return predictability is strongest for the HF model, for which the R2 is higher than for the empirical data across all frequencies. For the PT model, the dividend–price ratio has considerably more predictive power than in the empirical data—consistent with the higher β1* in Panel A. 1.6 Benchmark priors To assess the forecast performance of the model-implied priors, a variation of the Minnesota prior, which is popular in Bayesian macroeconometrics (see Litterman 1986), is used. This prior is called empirical prior and is not based on the asset pricing models. The empirical prior simply shrinks the parameter estimates toward the historical average model. The prior distribution in Equation (2) then becomes β∼N([r¯0],V̱), σϵ−2∼G(σr−2,v̱),(23) where r¯ and σr−2 are given in Equation (4). I also apply the nonnegativity constraint described in Section 1.4 to the empirical prior. This prior is denoted positive EP prior. Pettenuzzo, Timmermann, and Valkanov (2014) show that when the dividend–price ratio is used as predictor variable, the nonnegativity constraint helps to improve the forecasting performance and outperforms priors that shrink the parameters to the historical average model. Therefore, the empirical prior with a nonnegativity constraint is a natural benchmark to assess whether the consumption-based asset pricing models could have helped an investor to forecast excess returns more accurately. 2 Results In this section, I describe the data and report the forecasting results when imposing the model-based priors derived from asset pricing models on the single-variable predictive regressions. 2.1 Data The empirical data on excess returns and the dividend–price ratio at a monthly, quarterly, and annual frequency are available on Amit Goyal’s website.10 The excess return is computed as the log return on the S&P 500 index minus the log three-month U.S. Treasury bill rate. I set the start date of the time series in 1926, as high-quality return data on the S&P 500 from the Center of Research in Security Prices became available in 1926. The time series ends in 2018. Dividends on the S&P 500 index are 12-month moving sums from 1926 to 2018. 2.2 Measuring Forecast Accuracy I assess the performances of the model-based priors via the OOS R2 (see, e.g., Campbell and Thompson 2008): ROOS2=1−∑τ=ṯT(rτ−r^τm)2∑τ=ṯT(rτ−r^τh)2,(24) where r^τm is the excess return forecast when imposing the model-based prior as given in Equation (11); r^τh is the prediction of the historical average model when no prior is imposed; and ṯ and T are the start and end dates, respectively, of the forecast period. Thus, the ROOS2 assesses the forecast performance of the model-based prior relative to the nonpredictability model, which assumes that the best excess return forecast is its historical average, that is, β1 being set equal to zero in Equation (1). 2.3 Forecasting I consider three sample periods for the predictability exercise. First, I use the full sample from 1926 to 2018 and start the recursive forecasts in 1947. This starting point guarantees that a sufficient number of data points are available to estimate the predictive regression. Next, I analyze the subsample performance by splitting the 1947–2018 forecast period in half and consider forecasts up to 1982 and forecasts starting in 1983. Table 2 shows the ROOS2 (in percent) results for all model-based priors for three return frequencies. Following the return forecasting literature, monthly, quarterly, and annual return frequencies are used. The “empirical” column reports the ROOS2 for the case in which the single-variable predictive regression in Equation (1) is estimated via the empirical prior described in Section 1.6. The “positive EP” columns reports the ROOS2 when the positive EP prior is imposed. The positive EP prior is also described in Section 1.6. The last column of the table shows the best-performing prior for the respective frequency, predictor, and time period. Whether the differences in forecast errors between the predictive regression and the historical average model are significant is tested with the test of Clark and West (2007). Table 2 Model-based prior forecast performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab Table 2 Model-based prior forecast performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab As documented in previous studies, the nonnegativity constraint of the positive EP prior improves the forecasting accuracy of the empirical prior. However, the model-based priors can help to further improve the forecast performance of the single-variable predictive regressions. For the total sample, the HF priors raise the ROOS2 substantially for all return frequencies, and the historical average model is rejected in every case. The priors of the PT model perform similarly to the HF prior, as they raise the ROOS2 relative to the positive EP prior at all return frequencies. The subsample analysis shows that the HF and PT priors are particularly strong for the first half of the sample, that is, the forecast period from 1947 to 1982. However, for the second half of the sample, that is, the forecast period from 1983 to 2018, the performance of the HF and PT model-based priors is more in line with the positive EP prior. The LRR prior performance is overall similar to the positive EP prior's performance. Generally, the forecasts are more accurate at lower return frequencies, with the highest ROOS2 being achieved at the annual frequency. The differences in performance of the three model-based priors can be explained with the differences in the model-implied parameters reported in Table 1. The HF and PT models imply values for β0 and β1 that are generally higher than the empirical estimates. Thus, imposing the HF and PT priors pushes the posterior estimates of β0 and β1 up relative to the empirical and the positive EP prior. Figure 1 shows the positive EP posterior estimates and the model-based posterior estimates for quarterly returns and the 1947–2018 forecast period. The HF and PT posterior estimates are higher than the positive EP posterior estimates. However, the LRR posterior estimates are similar to the positive EP posterior estimates. This finding is the result of the lower model-implied predictability of the LRR model reported in Table 1. Because of the low-implied predictability, the LRR prior shrinks the coefficient estimates of the predictive regression toward the historical average model, as done by the empirical and the positive EP priors. Figure 1 Open in new tabDownload slide Posterior coefficient estimates. This figure shows the posterior estimates of β0 and β1 for the forecast period from 1947 to 2018 for the positive EP, HF, LRR, and PT priors. The predictor is the log dividend–price ratio, and the predictive regression in Equation (1) is estimated recursively with quarterly data. Because of the higher posterior estimates of β0 and β1, the HF and PT priors lead to more accurate forecasts in periods during which the predictive power of valuation ratios was strong. However, the forecasts will be too volatile during periods of weak predictability. Figure 2 gives more insight into the performance differences of the model-based priors. The top panel depicts the difference between the cumulative sum of squared errors (SSEs) of the historical average model and the single-variable predictive regression estimated via positive EP prior or model-based priors. I subtract the cumulative SSE of the predictive regression from the cumulative SSE of the historical average model. A positive value implies that the predictive regression outperforms the historical average model. Figure 2 Open in new tabDownload slide Forecasts of model-based priors. The top panel shows the differences in the cumulative SSEs between the log excess return forecasts of the historical average model and the predictive regression given in Equation (1) estimated via priors. The cumulative SSE of the predictive regression is subtracted from the cumulative SSE of the historical average model. The forecast period is from 1947 to 2018 and the forecasts are at an annual frequency. The predictor is the log dividend–price ratio. The lower panel depicts the posterior point forecasts of the excess returns of the predictive regression, with the posterior point forecasts as given in Equation (11). The HF and PT prior forecasts are the most accurate, and the LRR prior forecasts are similar to the forecasts of the positive EP prior. The strong performance of the HF and PT priors is due to the strong predictive power of the log dividend–price ratio up to the 1970s. The PT and HF prior outperform the LRR and positive EP priors, and this advantage remains until the end of the sample. This finding also explains why in Table 2, the HF and PT priors achieve higher ROOS2 in the first half of the forecast period, which starts in 1947 and ends in 1982, than in the second half of the forecast period, which starts in 1983 and ends in 2018. The low predictability implied by the LRR prior is an advantage during the dot-com boom from 1994 to 1999 when the predictive power of the log dividend–price ratio collapses. During this period, the cumulative SSE of the predictive regression drops for all four estimation methods. However, the LRR model-based prior is better able to avoid forecasts of the excess returns that are too low. The lower panel of Figure 2 depicts the higher volatility of the EP forecasts when the HF and PT priors are imposed on the predictive regression. The forecasts of the HF and PT priors diverge strongly from the positive EP prior forecasts. The LRR prior forecasts are less volatile and similar to the positive EP prior forecasts. Figure 3 shows the simulated posterior densities of β0 and β1 given in Equation (6) for the year 1971. Around this time, the empirical data show a strong predictability of excess returns. The posterior densities of the positive EP prior and the three model-based priors are shown. The coefficient densities approximate Normal distributions. For both coefficients, the LRR prior results in posterior densities that are centered to the left of the HF and PT priors. Therefore, an investor who relies on priors from the HF and PT models puts more weight on the valuation ratios when predicting excess returns. Such an investor predicts excess returns more accurately when the predictive power of valuation ratios is strong. Figure 3 Open in new tabDownload slide Posterior density of coefficients. This figure shows the simulated posterior density of the coefficients β0 and β1 given in Equation (6) for 1971 for the positive EP prior and the three model-based priors: HF, LRR, and PT. The predictor is the log dividend–price ratio. Annual data from 1947 to 1970 are used to estimate the predictive regression. The densities are simulated with 20,000 draws. In-sample estimates have a forward-looking bias that allows them to optimize between periods of weak and strong predictability. Therefore, one could assume that the asset pricing model with the model-implied parameters that are closest to the empirical in-sample estimates in Table 1 would perform best, but because of the nonnegativity constraint, this is not necessarily the case as shown by the previous results. The model-implied parameters of the LRR model are below but closer to the empirical estimates than the model-implied parameters of the HF and PT models, which are above the empirical in-sample estimates. The fact that the model-based priors include the nonnegativity constraint shown in Equation (13) helps the performance of the priors from all three asset pricing models, but the nonnegativity constraint particularly helps the priors from the HF and PT models. Table 3 shows the forecast performance of the model-based priors without the nonnegativity constraint for the total sample. Of the three model-based priors, the LRR prior performs best over the full sample and the second half of the sample. The reason is that the nonnegativity constraint ensured that the strong predictability of excess returns implied by the HF and PT models does not lead to negative forecasts during the dot-com boom when high prices and low dividends were followed by positive returns. Without the nonnegativity constraint, the forecasts during the dot-com boom when imposing the HF and PT priors are strongly negative, which leads to substantially larger forecast errors during this period. Table 3 Model-based prior forecast performance without nonnegativity constraint Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The model-based priors are imposed without the nonnegativity constraint described in Equation (13). The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab Table 3 Model-based prior forecast performance without nonnegativity constraint Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The model-based priors are imposed without the nonnegativity constraint described in Equation (13). The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab For the results shown in Table 2, the consumption-based asset pricing models used to derive the priors are calibrated as proposed by the respective authors, that is, Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004). To test whether the results are robust to calibrating the asset pricing models with data from a time period that has no overlap with the forecast period, I calibrate the parameters of the asset pricing models with data from 1926 to 1967. All three asset pricing models are calibrated with annual data. Thus, using a shorter sample for the calibration makes the task of matching empirical moments too challenging for the models, as the empirical moments are likely distorted by outliers due to the Great Depression and the Second World War. I follow the calibration methodology proposed by the respective authors: some parameters are set equal to their empirical counterparts and others are chosen such that the model simulated data moments match the empirical data moments, as, for example, the mean and standard deviation of the dividend–price ratio or the excess returns. Details regarding the calibration of the models can be found in Online Appendix. Table 4 reports the results for the priors derived from the asset pricing models calibrated with data from 1926 to 1967. The OOS forecasts start in 1968, and the ROOS2 for each prior, predictor, and return frequency is shown. The model-based priors can still improve the forecast accuracy of the predictive regression in several cases. However, the empirical and the positive EP priors are generally more competitive than in Table 2, which shows that the calibration of the asset pricing models is an important component for the performance of the model-based priors. Table 4 Model-based priors calibrated with 1926–1967 data. Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Reported is the forecasting performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The consumption-based asset pricing models are calibrated with annual data from 1926 to 1967. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the greatest improvement in forecast accuracy. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab Table 4 Model-based priors calibrated with 1926–1967 data. Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Reported is the forecasting performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The consumption-based asset pricing models are calibrated with annual data from 1926 to 1967. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the greatest improvement in forecast accuracy. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab 3 Robustness and Extensions This section discusses the robustness of the results shown in the previous section and provides extensions to the baseline analysis. 3.1 Prior Parameter Sensitivity For the baseline analysis, the tightness parameters of the Gamma–Normal prior, λ and v̱ ⁠, are set equal to one and ten, respectively, as described in Section 1.2. However, the results are qualitatively the same when tightening or loosening the model-based priors. Table 5 reports the results for the total sample, that is, the 1947–2018 forecast period. Relative to the benchmark, the model-based priors are tightened and loosened by a factor of two and four, respectively. Tightening the priors further does not alter the conclusion, and loosening the priors by more than by a factor of four simply leads to forecast results of the priors that slowly converge to the OLS estimates of the single-variable predictive regression. Tightening (loosening) the prior by a factor of four results in λ=0.25 (λ = 4) and v̱=40 (⁠ v̱=2.5 ⁠). Table 5 Hyperparameter sensitivity of the forecast performance Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The prior tightness is varied, and the respective values for the hyperparameters λ and v̱ are shown in the first two columns. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab Table 5 Hyperparameter sensitivity of the forecast performance Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The prior tightness is varied, and the respective values for the hyperparameters λ and v̱ are shown in the first two columns. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab The results are fairly robust across the range of λ and v̱ values. The model-based priors can improve upon the performance of the empirical or positive EP priors in the majority of the cases and the values of the ROOS2 are similar to those reported in Table 2. The forecasts are again more accurate at longer return frequencies. 3.2 Utility Gains Based on Model-Based Prior Forecasts So far, I have analyzed how priors derived from the three consumption-based asset pricing models affect the forecast accuracy of single-variable predictive regressions. However, an investor is ultimately concerned about utility, and thus, differences in forecast accuracy need to be mapped into differences in utility. For this purpose, I follow Campbell and Thompson (2008) and Wachter and Warusawitharana (2009) and assume an investor with a mean-variance utility function and a risk aversion of three who can allocate her funds in a risky and a risk-free asset, where the risky asset pays the aggregate stock market excess return and the risk-free asset pays the risk-free rate. As explained in more detail in Online Appendix, a certainty equivalent return (CER) can be computed. The CER is defined as a constant return that equates the realized utility of a portfolio that relies on excess return forecasts from the historical average model with a portfolio that relies on excess return forecasts from the single-variable predictive regression with a prior imposed. I compute the CER for each return frequency, predictor, and forecast period. The CER values are annualized and in percent. The results are shown in Table 6, which is structured like Table 2 but with the ROOS2 values replaced by the annualized CERs. For every case, at least one model-based prior leads to a CER that is higher than the empirical and positive EP forecasts. The CER values are economically significant reaching up to 1.4%. The model-based priors perform again better for the first half of the sample, that is, for the forecast period from 1947 to 1982. For the second half of the sample, that is from 1983 to 2018. The CER values are mostly negative. Table 6 Model-based prior economic performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A, B, and C report the annualized CER in percent as discussed in Section 3.2. The CER can be interpreted as a management fee that an investor with mean-variance utility and a risk aversion of three is willing to pay each year to have access to the excess return forecasts which result from the single-variable predictive regression given in Equation (1) and estimated based on either the empirical, positive EP, or model-based priors, relative to the excess return forecasts of the historical average model. The predictor is the log dividend–price ratio. The CER values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest CER. Open in new tab Table 6 Model-based prior economic performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A, B, and C report the annualized CER in percent as discussed in Section 3.2. The CER can be interpreted as a management fee that an investor with mean-variance utility and a risk aversion of three is willing to pay each year to have access to the excess return forecasts which result from the single-variable predictive regression given in Equation (1) and estimated based on either the empirical, positive EP, or model-based priors, relative to the excess return forecasts of the historical average model. The predictor is the log dividend–price ratio. The CER values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest CER. Open in new tab Figure 4 shows the risky asset share of the portfolio for the different priors. The forecasts are at an annual frequency, and the forecast period is from 1947 to 2018. Generally, the LRR prior leads to a more stable portfolio share of the risky asset, which closely matches the risky asset share of the investor’s portfolio when forecasting with the positive EP prior. The forecasts of the HF and PT priors lead to a more volatile portfolio allocation, but the average portfolio allocation over the total forecast period is similar to the positive EP and LRR priors. Figure 4 Open in new tabDownload slide Portfolio share of risky asset. This figure shows the risky asset portfolio share for a mean-variance investor with a risk aversion of three who relies either on the positive EP, HF, LRR, and PT priors, respectively, to forecast excess returns with the log dividend–price ratio from 1947 to 2018. 3.3 Multiple Predictors The previous analysis has focused on the log dividend–price ratio as the predictor of excess returns. While the dividend–price ratio is one of the most prominent predictors in the literature, there are a couple of other predictor variables that can be generated from the asset pricing models. In this section, model-based priors are imposed on a predictive regression with two predictor variables. The first predictor being the log dividend–price ratio and the second predictor being either the log consumption growth rate or the log real risk-free rate. The consumption growth rate is time-varying in all the three asset pricing models. The real risk-free rate is constant in the HF and PT models but time-varying in the LRR model.11 Table 7 reports the results for the predictive regression with two predictors. The results are shown for the full sample with the forecast period starting 20 years into the sample. For consumption growth, monthly and quarterly data are available from 1960 and 1948, respectively. Overall, the model-based priors outperform the empirical and the positive EP prior for all the specifications. However, adding consumption growth or the real risk-free rate as predictors does not consistently improve the forecast accuracy of the model compared to a single-variable predictive regression that only includes the dividend–price ratio. The values in parentheses are the ROOS2 of the multiple predictor’s regressions with the corresponding ROOS2 of the single-variable dividend–price ratio predictive regression being subtracted. These values are often negative, implying that the single-variable predictive regression outperforms the regression with multiple predictors. Table 7 Model-based prior forecast performance for multiple predictors Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the predictive regression given in Equation (1) but adding a second predictor. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the predictive regression relative to the historical average model. The predictors are the log dividend–price ratio together with the log consumption growth or log risk free rate, respectively. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. The values in parentheses are the ROOS2 of the multiple predictors regressions with the corresponding ROOS2 of the single-variable dividend–price ratio predictive regression being subtracted. Open in new tab Table 7 Model-based prior forecast performance for multiple predictors Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the predictive regression given in Equation (1) but adding a second predictor. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the predictive regression relative to the historical average model. The predictors are the log dividend–price ratio together with the log consumption growth or log risk free rate, respectively. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. The values in parentheses are the ROOS2 of the multiple predictors regressions with the corresponding ROOS2 of the single-variable dividend–price ratio predictive regression being subtracted. Open in new tab 3.4 Alternative Asset Pricing Model Specifications Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004) suggest alternative specifications of their models. In this section, I examine the performance of model-based priors derived from these alternative specifications. The baseline model-based priors from the HF model discussed previously are based on the specification that assumes a perfect positive correlation between the log consumption and log dividend growth. However, Campbell and Cochrane (1999) also solve the HF model for an imperfect but positive correlation. For the PT model, the baseline model-based prior is derived from the model specification that generates a log EP that is closest to the empirical data moment. A second specification proposed by Barberis, Huang, and Santos (2001) sets the financial wealth fluctuations parameter b0 equal to 100 and the effect of prior losses on risk aversion k equal to 3. The simulated log EP is then lower, but the average loss aversion of the agent is 2.25, which is in line with experimental evidence. Bansal and Yaron (2004) consider two calibrations for the agent’s risk aversion to simulate the LRR model: a risk aversion of 7.5 and a risk aversion of 10. The LRR model with the risk aversion set to ten is the previously discussed baseline specification, and the specification that sets the risk aversion parameter to 7.5 is the alternative specification. The three alternative model specifications are denoted HFAlt, PTAlt, and LRRAlt and the model-implied parameters can be found in Online Appendix. The results when imposing model-based priors from the alternative specifications are presented in Table 8 and are similar to the baseline model-based priors. The model-based priors outperform the empirical and positive EP priors in most cases. Also, the model-based priors’ forecast performance leads to a rejection of the historical average model in all cases for the full sample. However, the performance is again weaker for the second half of the sample. Table 8 Model-based prior from alternative asset pricing model specifications Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HFAlt., LRRAlt., and PTAlt. The calibration of the asset pricing models is based on alternative calibrations as discussed in Section 3.4. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab Table 8 Model-based prior from alternative asset pricing model specifications Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HFAlt., LRRAlt., and PTAlt. The calibration of the asset pricing models is based on alternative calibrations as discussed in Section 3.4. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab 3.5 Long Forecast Horizons So far, the assessment of the priors was based on monthly, quarterly, and annual return forecasts. When imposing the model-based priors on the predictive regressions that forecast the excess returns at a two- and three-year horizon, respectively, the results are similar to the baseline results in Table 2. However, the number of forecast observations is small with thirty-six and twenty-four observations for the two- and three-year horizons, respectively. Table 9 shows that the model-based priors can also increase the forecast accuracy as measured by the ROOS2 for longer horizons. For example, for the default λ value of one and at a two-year horizon, the model-based priors achieve an ROOS2 of 10.9% (HF), 5.0% (LRR), and 14.3% (PT), respectively. In comparison, the positive EP prior’s ROOS2 is only 3.8%. The statistical significance is generally lower than at higher return frequencies. This is caused by the decrease in power that comes with fewer return observations due to the longer return horizons. Table 9 Model-based prior forecast performance for long horizons Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panels A and B show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on a single-variable predictive regression, given in Equation (1), which predicts excess returns at a two- and three-year horizon. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. Results are shown for different λ values, which represents the prior tightness as described in Section 1.2. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by **at the 5% level and *at the 10% level. Open in new tab Table 9 Model-based prior forecast performance for long horizons Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panels A and B show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on a single-variable predictive regression, given in Equation (1), which predicts excess returns at a two- and three-year horizon. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. Results are shown for different λ values, which represents the prior tightness as described in Section 1.2. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by **at the 5% level and *at the 10% level. Open in new tab 4 Conclusion Different theories have been proposed to resolve the EP puzzle of Mehra and Prescott (1985). Three prominent consumption-based asset pricing models that provide different explanations for the existence of the EP puzzle are the HF, the LRR, and the PT models. This article tests if these asset pricing models can profitably guide the investment decisions of a hypothetical investor who had access to these models from 1947 onwards and tries to time the aggregate stock market by forecasting returns. I propose a simple Bayesian framework in which an investor reduces the uncertainty about predictive regression parameters by imposing economic constraints derived from the three asset pricing models. The model-based priors derived from the three asset pricing models generally lead to more accurate forecasts than an empirical prior that shrinks the parameter estimates toward the historical average model and the positive EP prior that restricts the forecast of the EP to be nonnegative. However, the results are sensitive to the sample period, the return frequency, and imposing a nonnegative EP constraint. By imposing model-based priors derived from consumption-based asset pricing models on predictive regressions and showing how the forecast performances of these priors compare to each other and to benchmark priors, this article makes a novel contribution to the return prediction literature. The analysis helps to determine how useful consumption-based asset pricing models could be for finance practitioners. Supplementary Data Supplementary data are available at Journal of Financial Econometrics online. Footnotes 1 See, for example, Campbell (1987); Campbell and Shiller (1988); Fama and French (1988, 1989); Baker and Wurgler (2000); Lettau and Ludvigson (2001); Lewellen (2004); Polk, Thompson, and Vuolteenaho (2006); Cochrane (2008); Lettau and Van Nieuwerburgh (2008); Rapach, Strauss, and Zhou (2010); Li, Ng, and Swaminathan (2013); Kelly and Pruitt (2013); and Neely et al. (2014). 2 Many subsequent papers have developed modified versions of these three asset pricing models. I make the choice to focus on the versions presented in the three initial papers in order to discipline the exercise. While examining priors derived from other versions of these asset pricing models is beyond the scope of this article, such priors might lead to qualitatively different results than shown here. 3 Holding the sample moments constant by estimating them over an estimation sample that has no overlap with the forecast sample leads to qualitatively similar results. 4 A caveat is that when the equity premium of the asset pricing models is estimated based on a linear projection of excess returns on valuation ratios such as in Equation (12), the resulting equity premium can be negative. However, such negative estimates are relatively rare. For the HF and LRR models, the share of negative estimates of the equity premium is 0.1% and 0%, respectively, when simulating 10,000 years. For the PT model, the share of negative estimates is higher with 17.1%. 5 Some rare combinations of xτ, β0*, β1*, σϵ*−2 ⁠, and λ can lead to the nonnegativity constraint in Equation (13) being violated for most of the Gibbs sampler draws. When the nonnegativity constraint is violated for more than 95% of the Gibbs sampler draws, β0*, β1* ⁠, and σϵ*−2 are replaced by r¯ ⁠, 0, and σr−2 ⁠, that is, the forecast is equivalent to the forecast of the positive EP prior. The assumption is that an investor would discard the model-based priors in time periods when the model-based priors’ posterior excess return distribution forecast is predominantly negative. 6 I denote this specification the HF model. The model specification that assumes an imperfect but positive correlation between dividend and consumption growth is denoted HFAlt. and described in Online Appendix. 7 I simulate the model based on the analytical solutions as done by Bansal, Kiku, and Yaron (2010) and Beeler and Campbell (2012). 8 As discussed in Section 1.5, the coefficients are estimated on 10,000 years of simulated data and averaged across ten iterations. Online Appendix contains coefficient estimates and forecast analyses when simulating 93 years of data (corresponding to the empirical length of the sample) and averaging across 1,000 iterations. The performance of the model-based priors is qualitatively similar. 9 Adjusting the β1 empirical estimates for the bias discussed by Stambaugh (1999) leads to annual, quarterly, and monthly estimates of 0.032, 0.010, and 0.002, respectively, for the 1926–2018 sample. For the 1947–2018 sample, the estimates are 0.051, 0.009, and 0.002, respectively. 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Google Scholar Crossref Search ADS WorldCat Author notes * This article is based on a chapter of my dissertation at the University of Oxford. I am grateful for the continued support from Tarun Ramadorai and Kevin Sheppard. I also thank Nicholas Barberis, John Campbell, Ben Jensen, Lubos Pastor, Andrew Patton, Allan Timmermann (Editor), Dimitri Vayanos, Jessica Wachter, Missaka Warusawitharana, Sumudu Watugala, Ivo Welch, Mungo Wilson, two anonymous referees, and seminar participants at the Federal Reserve Board of Governors, FMA 2016, NBER Summer Institute 2016, Oxford-Man Institute of Quantitative Finance, Saïd Business School, and SEA Annual Meeting 2015 for useful comments. The analysis and conclusions set forth are those of the author and do not indicate concurrence by the Board of Governors of the Federal Reserve System or its staff. Published by Oxford University Press 2020. This work is written by a US Government employee and is in the public domain in the US. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) Published by Oxford University Press 2020. This work is written by a US Government employee and is in the public domain in the US. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Financial Econometrics Oxford University Press http://www.deepdyve.com/lp/oxford-university-press/from-which-consumption-based-asset-pricing-models-can-investors-profit-E5J72V4thn

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Publisher: Oxford University Press
Copyright: Copyright © 2022 Oxford University Press
ISSN: 1479-8409
eISSN: 1479-8417
DOI: 10.1093/jjfinec/nbaa023
Publisher site: See Article on Publisher Site

Abstract

Abstract This article analyzes whether consumption-based asset pricing models improve the excess returns forecasts of a hypothetical investor with access to these models from 1947 onwards. The investor imposes economic constraints derived from asset pricing models as model-based priors on predictive regression parameters through a Bayesian framework. Three models are considered: habit formation, long-run risk, and prospect theory. The model-based priors generally perform better than priors that shrink the parameter estimates to the historical average model and priors that impose a positive equity premium. This analysis helps to assess the value of consumption-based asset pricing models to investors. Predicting aggregate stock returns has been of great interest to academics and investors alike. For academics, the predictability of stock returns is important for testing market efficiency. For investors, knowing whether returns are predictable is crucial for portfolio allocation decisions. A range of papers investigates which variables can predict returns.1 Welch and Goyal (2008) provide a comprehensive analysis of the major predictors and question whether an investor could have profitably used them to forecast returns. Campbell and Thompson (2008) further investigate these findings by imposing economic constraints when estimating predictive regressions. They find that sign constraints on the parameter estimates of predictive regressions and a nonnegativity restriction on the equity premium (EP) help an investor to reduce uncertainty about the regression parameters and profitably forecast aggregate stock returns. Financial economists have developed consumption-based asset pricing models that attempt to explain the empirically observed behavior of stock markets and also propose solutions for the EP puzzle through different mechanisms. A natural question to ask is whether consumption-based asset pricing models could act as a source of economically motivated parameter constraints, that is, model-based priors, and whether a hypothetical investor who had access to these models from 1947 onwards could have predicted the aggregate excess returns more accurately over the subsequent decades. This exercise helps to assess the value of consumption-based asset pricing models to finance practitioners and evaluates asset pricing models through a novel approach. In this article, an investor who forecasts aggregate excess returns with the dividend–price ratio “out-of-sample (OOS),” where OOS is in quotes to indicate qualifications discussed below, has a prior belief about the parameter estimates of the predictive regression that stems from one of the following asset pricing models: the habit formation (HF) model of Campbell and Cochrane (1999), the prospect theory (PT) model of Barberis, Huang, and Santos (2001), and the long-run risk (LRR) model of Bansal and Yaron (2004). I choose these three asset pricing models as they suggest different mechanisms to explain the EP puzzle.2 The prior belief derived from the asset pricing model helps the investor to reduce uncertainty about the parameters of the predictive regression. The investor updates her beliefs about the predictive regression parameters with empirical data and forecasts excess returns based on the posterior parameter estimates. Because the consumption-based asset pricing models considered in this article have been published around the year 2000, the presented results are not attainable for an actual investor who started forecasting excess returns in 1947. However, the analysis presented in this article is of value for finance academics and practitioners because of several reasons. First, an underperformance of the model-based priors against benchmark forecasting models that are truly OOS would strongly question the usefulness of these asset pricing models for investors. Second, a strong performance of the model-based priors emphasizes that the asset pricing models could have been of use for a hypothetical investor despite the authors of the asset pricing models not being focused on model-based priors as a model evaluation tool. Third, information on which consumption-based asset pricing model yields priors that perform well can be important for researchers who have to decide which mechanisms to incorporate in future asset pricing models. I propose a simple Bayesian econometric framework to implement the economic constraints derived from the consumption-based asset pricing models as prior distributions on the parameters of single-variable predictive regressions. Single-variable predictive regressions are generally used in the return prediction literature (see, e.g., Welch and Goyal 2008). Further, single-variable predictive regressions are used by Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004) to assess the predictability implied by their asset pricing models. My framework has two steps. First, the predictive regression is estimated based on simulated data from the asset pricing model. Second, the parameter estimates from the first step are used as prior information when estimating the predictive regression with empirical data. Further, in line with Campbell and Thompson (2008) and Pettenuzzo, Timmermann, and Valkanov (2014), the EP is restricted to be nonnegative. The main predictor is the dividend–price ratio. The dividend–price ratio is endogenous in the three asset pricing models, and thus, can be simulated. My approach is related to the macroeconometric literature, in which prior distributions from dynamic stochastic general equilibrium models are imposed on vector autoregressions to forecast macroeconomic variables (see, e.g., Ingram and Whiteman 1994; Del Negro and Schorfheide 2011). The implied return predictability differs across the asset pricing models due to the model-specific mechanisms that lead to time variation in valuation ratios. For the HF and PT models, the model-implied return predictability is higher than that found in the empirical data. However, for the LRR model, the model-implied return predictability is lower than found in the empirical data. The model-based priors’ forecast performance is compared to two OOS benchmark priors. The first prior shrinks the parameter estimates toward the historical average model. The historical average model implies that the EP will be equal to its past average. The second prior restricts the EP to be weakly positive as proposed by Pettenuzzo, Timmermann, and Valkanov (2014) in addition to shrinking the parameter estimates toward the historical average model. The model-based priors generally outperform the two benchmark priors. Over the entire forecast period from 1947 to 2018 and at an annual return frequency, the model-based priors derived from the HF model generate an OOS R2 that is 8.4%. For the LRR model-based priors the OOS R2 is 3.5%, and for the PT model-based priors the value is 9.4%. The historical average prior achieves an OOS R2 of 1.3% without the positive EP constraint and an OOS R2 of 3.5% when imposing the positive EP constraint. While these baseline results are based on the calibration of the asset pricing models as proposed by the respective authors of the models, recalibrating the asset pricing models such that the data used for the recalibration have no overlap with the forecasting period leads to comparable results. Further, the results are robust to changing the tightness of the model-based priors, using longer forecast horizons, and adding additional predictors to the model. However, the predictive power of the dividend–price ratio when forecasting the excess returns is time varying over the 1947–2018 forecast sample, and so is the performance of the model-based priors. The predictive power of the dividend–price ratio was strong in the first half of the sample, when the correlation between valuation ratios and the subsequent excess returns was strongly positive. However, the bull market of the late 1990s was a period when the dividend–price ratio had very weak predictive power as historically high prices and low dividends were followed by high stock returns for several years until the end of the dot-com boom in the early 2000s. Welch and Goyal (2008) find that the dot-com period is a large contributor to the poor OOS performance of valuation ratios, when no economic constraints are imposed on the predictive model, and that an investor would perform better by forecasting the excess returns with a historical average model that assumes valuation ratios have no predictive power. Campbell and Thompson (2008) and Pettenuzzo, Timmermann, and Valkanov (2014) show that the nonnegative EP constraint can help to mitigate the poor performance of the predictive model during this episode. The HF and PT model-based priors can improve upon the performance of the nonnegativity constraint because the return predictability that these two models imply is higher than found in the empirical data, which allows an investor to take advantage of the strong predictive power of valuation ratios in the early half of the sample. While the baseline results described above show a stronger performance of the priors derived from the HF and PT models compared to the priors derived from the LRR model, it is noteworthy that when imposing the model-based priors without a nonnegativity constraint, the HF and PT model-based priors generally underperform the LRR model-based priors. The reason is that when the nonnegativity constraint is lifted, the forecasts during the dot-com boom when imposing the HF and PT model-based priors are strongly negative, which leads to substantially larger forecast errors during this period. This article contributes to the growing literature that investigates the use of economically motivated parameter constraints for predicting the excess returns and imposes them through a type of Bayesian framework on the predictive regressions. Pastor and Stambaugh (2009) employ a prior that implies a negative correlation between expected and unexpected return shocks. Shanken and Tamayo (2012) consider prior beliefs on the risk-return tradeoff and on the extent to which mispricing drives predictability. Pettenuzzo, Timmermann, and Valkanov (2014) propose a Bayesian methodology that imposes a nonnegative EP and bounds on the conditional Sharpe ratio. Their constraints lead to return forecasts that are substantially more accurate. Wachter and Warusawitharana (2009) model skepticism of an investor over the predictability of excess returns as an informative prior over the R2 and show that a skeptical investor achieves better forecasts. Wachter and Warusawitharana (2015) analyze whether an investor who is initially skeptical about the existence of return predictability would update her prior and conclude that returns are predictable when confronted with historical data. Other Bayesian studies consider uncertainty about the predictive regression parameters through uninformative priors (see, e.g., Stambaugh 1999; Barberis 2004; Brand et al. 2005; Penasse 2016) or investigate how parameter uncertainty affects the long-run predictive variance (see, e.g., Pastor and Stambaugh 2012; Avramov, Cederburg, and Lucivjanska 2017). The remainder of this article is organized as follows. Section 1 explains the Bayesian methodology used to impose the model-based priors. Section 2 reports the main results. Section 3 discusses the robustness of the results and extensions to the analysis. Section 4 concludes the article. 1 Methodology This section describes how I impose economic constraints on the single-variable predictive regressions through priors derived from consumption-based asset pricing models and how these models are simulated to obtain the priors. 1.1 Excess Return Prediction Model The aggregate excess return at time t + 1 is denoted by rt+1 and is defined as the log rate of return on the stock market in excess of the prevailing short-term interest rate. As is common in the return prediction literature, rt+1 is regressed on a constant and a predictor, xt, which is lagged by one period: rt+1=β0+β1xt+ϵt+1, where ϵt+1∼N(0,σϵ2).(1) The OOS predictions of the excess returns are generated through recursive forecasts (see, e.g., Campbell and Thompson 2008; Welch and Goyal 2008). 1.2 Prior Distribution The prior distribution of the parameters in Equation (1)—that is β and σϵ2—is assumed to be Gamma–Normal and builds on Koop (2003) and Pettenuzzo, Timmermann, and Valkanov (2014). The prior distribution is given by β∼N(β̱,V̱), σϵ−2∼G(σϵ̱−2,v̱).(2) The mean and the variance of the Normal prior distribution are specified as β̱=[β̱0β̱1], V̱=[λσr200λσr2/σx2].(3) The parameter λ is exogenously chosen and is weakly positive. If λ is large, the prior is loose. If λ is equal to zero, the prior is dogmatic. I set λ = 1 for the baseline analysis. Section 3 reports results for different values of λ and shows that these are in line with the benchmark case. The sample moments σr2 and σx2 are scaling factors, which ensure that the results are comparable for different predictors and forecast frequencies. Such scaling factors are commonly used in Bayesian macroeconometrics and date back to Litterman (1986). The sample moments are given by σr2=1t−2∑τ=2t(rτ−r¯)2, r¯=1t−1∑τ=2trτ(4) and σx2=1t−2∑τ=1t−1(xτ−x¯)2, x¯=1t−1∑τ=1t−1xτ.(5) The sample moments can change over time as the recursive estimation uses a longer sample with every forecast iteration, that is, as t increases.3 The Gamma distribution parametrization follows Koop (2003) by specifying the distribution with mean σϵ−2 and degrees of freedom v̱ ⁠. The tightness of the prior is controlled by v̱ ⁠, which is strictly positive. A large v̱ corresponds to a tight prior, and a small v̱ corresponds to a diffuse prior. The baseline analysis sets v̱ to 10, but the results are robust to a tighter or a more diffuse prior on σϵ−2 (see Section 3). 1.3 Posterior Distribution The model-based prior distributions yield conditional posterior distributions for β and σϵ−2 ⁠. I draw from these two conditional distributions through a Gibbs sampler. The conditional posterior distribution for β is β|σϵ−2,It∼N(β¯,V¯),(6) where V¯=(V̱−1+σϵ*−2X′X)−1, β¯=V¯(V̱−1β̱+σϵ*−2X′R),(7) X is a t−1×2 matrix with rows [1 xτ] for τ=1,…,t−1 ⁠, and R is a t−1×1 vector with elements rτ for τ=2,…,t ⁠. The information set at time t is denoted by It ⁠. The conditional posterior distribution for σϵ−2 takes the form σϵ−2 |β,It∼G(s¯−2,v¯),(8) where v¯=v̱+(t−1), and s¯2=∑τ=2t(rτ−β0−β1xτ−1)2+σϵ*2v̱v¯.(9) Through the Gibbs sampling algorithm with J iterations, we obtain a series of draws for each of the parameters denoted by {βj} and {σϵ−2,j} for j=1,…,J ⁠. These simulated series can then be used to draw from the predictive return distribution p(rt+1|It)=∫β,σϵ−2p(rt+1|β,σϵ−2,It)p(β,σϵ−2|It)dβdσϵ2,(10) which yields {rt+1j} for j=1,…,J ⁠. The point forecast for the excess return in period t + 1 is given by the mean of the sampled distribution r^t+1m=1J∑j=1Jrt+1j.(11) 1.4 Priors from Asset Pricing Models I use the asset pricing models to obtain prior means for the prior distribution shown in Equation (2). The priors are derived from the three consumption-based asset pricing models, that is, HF, LRR, and PT, through a simulation procedure. All three models specify a log consumption and a log dividend growth process. These two processes drive the state variables of the models, and the state variables determine the dividend–price ratio. By simulating random shocks, time series of consumption growth and dividend growth are generated, based on which I solve the models for the log EP and the log dividend–price ratio. The dividend–price ratio is one of the most prominent predictors of the EP prediction literature and is defined as the ratio of the dividends summed over the past 12 months and the end of period price. I investigate additional predictors in Section 3.3. (A more detailed description of the models and how to solve them is provided in Online Appendix.) The simulated log excess return is denoted rM,t+1 ⁠. I can then estimate the single-variable predictive regression given in Equation (1) with simulated data, where the simulated predictor xM,t is the log dividend–price ratio: rM,t+1=βM,0+βM,1xM,t+ϵM,t+1, where ϵM,t+1∼N(0,σM,ϵ2).(12) The OLS estimates of βM=[βM,0, βM,1]′ and σM,ϵ−2 are denoted by β* and σϵ*−2 ⁠, which act as the prior means of the Gamma–Normal distribution described in Section 1.2. The same predictive regression with simulated data is also estimated by the respective authors of the consumption-based asset pricing models, that is Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004), to assess the predictability of excess returns implied by their proposed theories. I consider two calibrations when simulating data from the asset pricing models for the model-based priors. First, I use the same calibration as proposed by the authors in the respective published articles. The authors use almost identical calibration datasets, and thus, the comparison of the model-based priors’ performances should not be distorted. Second, I recalibrate the models with an empirical data sample that has no overlap with the OOS period. I assume that the investor has no uncertainty about the parameters of the asset pricing models, since the focus of this article and the literature it contributes to is the investor’s uncertainty about the parameters of the predictive regression given in Equation (1). In addition to imposing prior means derived from the asset pricing models, I also impose a nonnegativity constraint that requires a weakly positive EP. The motivation behind this constraint is two-fold. First, the three asset pricing models considered in this article have an equilibrium setting where risk-averse agents hold stocks because the EP is positive. Therefore, imposing a nonnegativity constraint on the EP is in line with this aspect of the asset pricing models.4 Second, imposing a nonnegativity constraint builds on other papers in the literature that have used such a constraint successfully (see Campbell and Thompson 2008; Pettenuzzo, Timmermann, and Valkanov 2014). I impose the nonnegativity constraint by following Pettenuzzo, Timmermann, and Valkanov (2014) and rejecting Gibbs sampler draws from the posterior in Equation (6), when the condition given by β0,β1∈At,where At=β0+β1xτ≥0(13) is not satisfied for τ=1,…,t ⁠.5 1.5 Asset Pricing Model Specifications and Simulation Procedures This section discusses the specifications and simulation procedures used to generate the simulated data from the asset pricing models to derive the priors. The description of the asset pricing models is kept short, and additional details as well as sample moments can be found in Online Appendix. 1.5.1 The HF model Campbell and Cochrane (1999) use a standard representative agent consumption-based asset pricing model but add a slow-moving habit to the basic power utility function. This slow-moving habit leads to a time-varying risk premium that is higher at business cycle troughs than at peaks. The agents are identical and maximize their utility given by E[∑t=0∞δt(Ct−Xt)(1−γ)−11−γ],(14) where Ct is the consumption level, Xt is the level of habit, δ is the time discount factor, and γ is the risk aversion. A surplus consumption ratio St≡(Ct−Xt)/Ct is defined—a small value of St indicates that the economy is in a bad state. I use the specification of the HF model that assumes perfect correlation between log consumption and log dividend growth to generate the model-based priors, which is the benchmark specification in Campbell and Cochrane (1999). There, stocks represent a claim to the consumption stream. The price–consumption ratio for a consumption claim satisfies6 PtCt(st)=Et[Mt+1Ct+1Ct[1+Pt+1Ct+1(st+1)]],(15) where st is the log of St, and the intertemporal marginal rate of substitution Mt+1 takes the form Mt+1≡δ(St+1StCt+1Ct)−γ.(16) Because the term (St+1/St)−γ correlates positively with asset returns, the HF model generates a higher EP than the standard power utility model. The simulation is at a monthly frequency, and the quarterly and annual data are constructed by time-averaging the monthly data. The same procedure is used by Campbell and Cochrane (1999). I simulate 120,000 months, estimate β* and σϵ*2 ⁠, and average the estimates over 10 iterations. 1.5.2 The PT model In the model of Barberis, Huang, and Santos (2001), the agent not only derives utility from consumption but also from financial wealth fluctuations. There are two important aspects in the way financial wealth fluctuations affect the utility of an economic agent. First, the agent is loss averse. Second, the degree of loss aversion depends on prior investment outcomes. Prior gains lead to less loss aversion, and prior losses lead to more loss aversion. The agent’s maximization problem is set up as E[∑t=0∞(ρtCt1−γ1−γ+b0C¯t−γρt+1v(Xt+1,St,zt))].(17) The second term captures the fact that the agent’s utility is affected by fluctuations in financial wealth. The variable Xt+1 denotes the change of the financial wealth between time t and t + 1 and is defined as Xt+1≡StRt+1−StRf,t.(18) The variable St measures the value of the agent’s risky assets at time t. The variable zt accounts for prior gains and losses up to time t and is defined as Zt/St ⁠, where Zt is a historical benchmark level for the value of the risky asset. If zt is smaller than one, the agent has prior gains; if zt is greater than one, the agent faces prior losses. The time discount factor is ρ, and b0C¯t−γ is a scaling term, with γ being the risk aversion. The form of the utility function over financial wealth v(.) is different conditional on prior gains or prior losses and given in Online Appendix. The price–dividend ratio is assumed to be a function of the state variable zt: Pt/Dt=f(zt).(19) Similar to the HF model, the PT model is specified by Barberis, Huang, and Santos (2001) with perfect positive correlation between the log consumption and log dividend growth processes and with imperfect positive correlation between the two processes. I only use the latter specification, as it more successfully matches the empirical data moments. The authors calibrate the model with a range of parameter values for the investor’s sensitivity to financial wealth fluctuations (b0) and the effect of prior losses on risk aversion (k). I generate priors from the parameterizations that set b0 equal to 100 and k equal 8. Of the specifications proposed by Barberis, Huang, and Santos (2001), setting b0 equal to 100 and k equal to 8 generates a log EP that is closest to the empirical data moment. Following Barberis, Huang, and Santos (2001), I simulate the model at monthly, quarterly, and annual frequencies by adjusting the model parameters accordingly. For the monthly, quarterly, and annual frequencies, I simulate 120,000, 40,000, and 10,000 periods, respectively, and average the β* and σϵ*2 estimates over 10 iterations. 1.5.3 The LRR model Bansal and Yaron (2004) propose a solution to the EP puzzle through a consumption-based asset pricing model with Epstein and Zin (1989) preferences. Their model differs from other consumption-based asset pricing models in two ways. First, they include a small persistent expected growth rate component in the consumption and dividend growth rate processes. This component causes consumption and the return on the market portfolio to covary positively, and thus, the economic agents require a higher risk premium. Second, they allow for time-varying volatility, which accounts for fluctuating economic uncertainty, in both processes. The asset pricing restriction for the real return on the market portfolio Rm,t+1 is Et[δθGc,t+1−θψRc,t+1−(1−θ)Rm,t+1]=Et[Mt+1Rm,t+1]=1,(20) where Gc,t+1 is the aggregate gross growth rate of consumption, Rc,t+1 denotes the real return on an asset that pays aggregate consumption as dividends, δ is the time discount factor, and Mt+1 is the intertemporal marginal rate of substitution. The parameter θ is defined as (1−γ)/(1−1ψ) ⁠, where γ is the risk aversion parameter, and ψ accounts for the intertemporal elasticity of substitution. The dynamics of log consumption growth, gc,t+1 ⁠, and log dividend growth, gd,t+1 ⁠, incorporate a small persistent predictable component xt and a time-varying volatility component σt, reflecting fluctuating economic uncertainty. The exact dynamics are given by xt+1=ρxt+φeσtet+1gc,t+1=μc+xt+σtηt+1gd,t+1=μd+ϕxt+φdσtut+1σt+12=σ2+v1(σt2−σ2)+σwwt+1,(21) with et+1, ut+1, ηt+1 ⁠, and wt+1 having i.i.d. standard Normal distributions. The state variables, which determine the price–consumption ratio, zt, and price–dividend ratio, zm,t ⁠, are xt and σt. The solutions for zt and zm,t are zt=A0+A1xt+A2σt2zm,t=A0,m+A1,mxt+A2,mσt2,(22) where A and Am can be derived analytically as described in Bansal and Yaron (2004). The LRR model, like the HF model, is simulated at a monthly frequency, and the quarterly and annual values are time averaged.7 Again, 120,000 months are simulated to estimate β* and σϵ*2 ⁠, and the estimates are averaged across ten iterations. Bansal and Yaron (2004) present two specifications of their model: with and without time-varying volatility of consumption growth. Because the specification that accounts for time-varying volatility of consumption growth is more successful at matching the empirical data moments and leads to a time-varying risk premium, I generate priors only from this specification. I use the benchmark calibration of Bansal and Yaron (2004) that sets the agent’s risk aversion equal to ten. 1.5.4 Simulation results Panels A and B of Table 1 show β* and σϵ*−2 estimated from simulated data of the three consumption-based asset pricing models.8 The table also reports the in-sample empirical estimates for comparison. The empirical estimates are shown for the total sample from 1926 to 2018, and the sample from 1947 to 2018 over which the excess return will be forecast in the subsequent analysis.9 Standard errors are shown for the empirical coefficient estimates. Table 1 Model-implied parameters Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A reports the in-sample coefficient estimates of the single-variable predictive regression of the log excess return given in Equation (1). The log dividend–price ratio is the predictor. The empirical in-sample estimates are for the total data sample from 1926 to 2018 and the sample from 1947 to 2018. The model-based estimates from the three asset pricing models HF, LRR, and PT are obtained through the Monte Carlo simulation procedure described in Section 1.4. Standard errors in parentheses are shown for the empirical estimates. Panel B shows the inverse of the variance of the return innovation used for the Gamma prior in Equation (2). Panel C reports the R2 (in percent) of the single-variable predictive regression. Open in new tab Table 1 Model-implied parameters Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A: Coefficients (β) . . Empirical . . . . . . . . 1926–2018 . 1947–2018 . HF . LRR . PT . . β0 . β1 . β0 . β1 . β0* . β1* . β0* . β1* . β0* . β1* . Annual 0.273 0.063 0.396 0.095 0.635 0.195 0.073 0.011 0.869 0.321 (0.153) (0.044) (0.126) (0.037) Quarterly 0.082 0.020 0.092 0.022 0.166 0.051 0.016 0.002 0.469 0.220 (0.050) (0.014) (0.038) (0.011) Monthly 0.022 0.005 0.028 0.007 0.057 0.018 0.007 0.001 0.270 0.158 (0.017) (0.005) (0.012) (0.003) Panel B: Inverse variance of return innovation (⁠ σϵ−2 ⁠) Annual 26.548 39.963 50.231 36.917 18.733 Quarterly 92.050 165.583 173.088 147.964 88.708 Monthly 342.966 579.346 507.683 441.194 394.022 Panel C: Predictability (R2 in %) Annual 2.324 6.726 12.219 0.026 6.068 Quarterly 0.784 1.498 3.244 0.007 2.787 Monthly 0.189 0.476 1.100 0.001 0.925 Panel A reports the in-sample coefficient estimates of the single-variable predictive regression of the log excess return given in Equation (1). The log dividend–price ratio is the predictor. The empirical in-sample estimates are for the total data sample from 1926 to 2018 and the sample from 1947 to 2018. The model-based estimates from the three asset pricing models HF, LRR, and PT are obtained through the Monte Carlo simulation procedure described in Section 1.4. Standard errors in parentheses are shown for the empirical estimates. Panel B shows the inverse of the variance of the return innovation used for the Gamma prior in Equation (2). Panel C reports the R2 (in percent) of the single-variable predictive regression. Open in new tab For all three asset pricing models, β1* is positive. Thus, a high dividend–price ratio predicts higher subsequent returns, which is in line with the empirical estimates. Across all return frequencies, the coefficients of the LRR model are lower than the empirical estimates. The implication is that in the LRR model, the predictive power of the dividend–price ratio is weaker. However, for the HF and PT models, the model-implied coefficient estimates are higher than the empirical estimates. These estimates indicate that the return predictability in the HF and PT models is stronger than observed empirically. The σϵ*−2 values implied by all three asset pricing models are close to the empirical values. Panel C reports the R2 for the single-variable predictive regression in Equation (12). The R2 values for the LRR model are lower than for the HF and PT models and the empirical data. The return predictability is strongest for the HF model, for which the R2 is higher than for the empirical data across all frequencies. For the PT model, the dividend–price ratio has considerably more predictive power than in the empirical data—consistent with the higher β1* in Panel A. 1.6 Benchmark priors To assess the forecast performance of the model-implied priors, a variation of the Minnesota prior, which is popular in Bayesian macroeconometrics (see Litterman 1986), is used. This prior is called empirical prior and is not based on the asset pricing models. The empirical prior simply shrinks the parameter estimates toward the historical average model. The prior distribution in Equation (2) then becomes β∼N([r¯0],V̱), σϵ−2∼G(σr−2,v̱),(23) where r¯ and σr−2 are given in Equation (4). I also apply the nonnegativity constraint described in Section 1.4 to the empirical prior. This prior is denoted positive EP prior. Pettenuzzo, Timmermann, and Valkanov (2014) show that when the dividend–price ratio is used as predictor variable, the nonnegativity constraint helps to improve the forecasting performance and outperforms priors that shrink the parameters to the historical average model. Therefore, the empirical prior with a nonnegativity constraint is a natural benchmark to assess whether the consumption-based asset pricing models could have helped an investor to forecast excess returns more accurately. 2 Results In this section, I describe the data and report the forecasting results when imposing the model-based priors derived from asset pricing models on the single-variable predictive regressions. 2.1 Data The empirical data on excess returns and the dividend–price ratio at a monthly, quarterly, and annual frequency are available on Amit Goyal’s website.10 The excess return is computed as the log return on the S&P 500 index minus the log three-month U.S. Treasury bill rate. I set the start date of the time series in 1926, as high-quality return data on the S&P 500 from the Center of Research in Security Prices became available in 1926. The time series ends in 2018. Dividends on the S&P 500 index are 12-month moving sums from 1926 to 2018. 2.2 Measuring Forecast Accuracy I assess the performances of the model-based priors via the OOS R2 (see, e.g., Campbell and Thompson 2008): ROOS2=1−∑τ=ṯT(rτ−r^τm)2∑τ=ṯT(rτ−r^τh)2,(24) where r^τm is the excess return forecast when imposing the model-based prior as given in Equation (11); r^τh is the prediction of the historical average model when no prior is imposed; and ṯ and T are the start and end dates, respectively, of the forecast period. Thus, the ROOS2 assesses the forecast performance of the model-based prior relative to the nonpredictability model, which assumes that the best excess return forecast is its historical average, that is, β1 being set equal to zero in Equation (1). 2.3 Forecasting I consider three sample periods for the predictability exercise. First, I use the full sample from 1926 to 2018 and start the recursive forecasts in 1947. This starting point guarantees that a sufficient number of data points are available to estimate the predictive regression. Next, I analyze the subsample performance by splitting the 1947–2018 forecast period in half and consider forecasts up to 1982 and forecasts starting in 1983. Table 2 shows the ROOS2 (in percent) results for all model-based priors for three return frequencies. Following the return forecasting literature, monthly, quarterly, and annual return frequencies are used. The “empirical” column reports the ROOS2 for the case in which the single-variable predictive regression in Equation (1) is estimated via the empirical prior described in Section 1.6. The “positive EP” columns reports the ROOS2 when the positive EP prior is imposed. The positive EP prior is also described in Section 1.6. The last column of the table shows the best-performing prior for the respective frequency, predictor, and time period. Whether the differences in forecast errors between the predictive regression and the historical average model are significant is tested with the test of Clark and West (2007). Table 2 Model-based prior forecast performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab Table 2 Model-based prior forecast performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453*** 8.380*** 3.486*** 9.392*** PT 1926 1947–1982 4.915*** 6.135*** 14.837*** 6.026*** 15.718*** PT 1926 1983–2018 −3.219 0.497 2.254 0.003 1.528 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 1.895*** 1.955*** 2.533*** PT 1926 1947–1982 3.176*** 2.755*** 3.834*** 2.871*** 3.226*** HF 1926 1983–2018 −2.891 0.644 0.755 0.277 0.507 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.450*** 0.391*** 0.842*** PT 1926 1947–1982 1.127*** 1.017*** 1.331*** 0.819*** 0.808*** HF 1926 1983–2018 −1.025 0.275 −0.041 0.048 0.101 Positive EP Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab As documented in previous studies, the nonnegativity constraint of the positive EP prior improves the forecasting accuracy of the empirical prior. However, the model-based priors can help to further improve the forecast performance of the single-variable predictive regressions. For the total sample, the HF priors raise the ROOS2 substantially for all return frequencies, and the historical average model is rejected in every case. The priors of the PT model perform similarly to the HF prior, as they raise the ROOS2 relative to the positive EP prior at all return frequencies. The subsample analysis shows that the HF and PT priors are particularly strong for the first half of the sample, that is, the forecast period from 1947 to 1982. However, for the second half of the sample, that is, the forecast period from 1983 to 2018, the performance of the HF and PT model-based priors is more in line with the positive EP prior. The LRR prior performance is overall similar to the positive EP prior's performance. Generally, the forecasts are more accurate at lower return frequencies, with the highest ROOS2 being achieved at the annual frequency. The differences in performance of the three model-based priors can be explained with the differences in the model-implied parameters reported in Table 1. The HF and PT models imply values for β0 and β1 that are generally higher than the empirical estimates. Thus, imposing the HF and PT priors pushes the posterior estimates of β0 and β1 up relative to the empirical and the positive EP prior. Figure 1 shows the positive EP posterior estimates and the model-based posterior estimates for quarterly returns and the 1947–2018 forecast period. The HF and PT posterior estimates are higher than the positive EP posterior estimates. However, the LRR posterior estimates are similar to the positive EP posterior estimates. This finding is the result of the lower model-implied predictability of the LRR model reported in Table 1. Because of the low-implied predictability, the LRR prior shrinks the coefficient estimates of the predictive regression toward the historical average model, as done by the empirical and the positive EP priors. Figure 1 Open in new tabDownload slide Posterior coefficient estimates. This figure shows the posterior estimates of β0 and β1 for the forecast period from 1947 to 2018 for the positive EP, HF, LRR, and PT priors. The predictor is the log dividend–price ratio, and the predictive regression in Equation (1) is estimated recursively with quarterly data. Because of the higher posterior estimates of β0 and β1, the HF and PT priors lead to more accurate forecasts in periods during which the predictive power of valuation ratios was strong. However, the forecasts will be too volatile during periods of weak predictability. Figure 2 gives more insight into the performance differences of the model-based priors. The top panel depicts the difference between the cumulative sum of squared errors (SSEs) of the historical average model and the single-variable predictive regression estimated via positive EP prior or model-based priors. I subtract the cumulative SSE of the predictive regression from the cumulative SSE of the historical average model. A positive value implies that the predictive regression outperforms the historical average model. Figure 2 Open in new tabDownload slide Forecasts of model-based priors. The top panel shows the differences in the cumulative SSEs between the log excess return forecasts of the historical average model and the predictive regression given in Equation (1) estimated via priors. The cumulative SSE of the predictive regression is subtracted from the cumulative SSE of the historical average model. The forecast period is from 1947 to 2018 and the forecasts are at an annual frequency. The predictor is the log dividend–price ratio. The lower panel depicts the posterior point forecasts of the excess returns of the predictive regression, with the posterior point forecasts as given in Equation (11). The HF and PT prior forecasts are the most accurate, and the LRR prior forecasts are similar to the forecasts of the positive EP prior. The strong performance of the HF and PT priors is due to the strong predictive power of the log dividend–price ratio up to the 1970s. The PT and HF prior outperform the LRR and positive EP priors, and this advantage remains until the end of the sample. This finding also explains why in Table 2, the HF and PT priors achieve higher ROOS2 in the first half of the forecast period, which starts in 1947 and ends in 1982, than in the second half of the forecast period, which starts in 1983 and ends in 2018. The low predictability implied by the LRR prior is an advantage during the dot-com boom from 1994 to 1999 when the predictive power of the log dividend–price ratio collapses. During this period, the cumulative SSE of the predictive regression drops for all four estimation methods. However, the LRR model-based prior is better able to avoid forecasts of the excess returns that are too low. The lower panel of Figure 2 depicts the higher volatility of the EP forecasts when the HF and PT priors are imposed on the predictive regression. The forecasts of the HF and PT priors diverge strongly from the positive EP prior forecasts. The LRR prior forecasts are less volatile and similar to the positive EP prior forecasts. Figure 3 shows the simulated posterior densities of β0 and β1 given in Equation (6) for the year 1971. Around this time, the empirical data show a strong predictability of excess returns. The posterior densities of the positive EP prior and the three model-based priors are shown. The coefficient densities approximate Normal distributions. For both coefficients, the LRR prior results in posterior densities that are centered to the left of the HF and PT priors. Therefore, an investor who relies on priors from the HF and PT models puts more weight on the valuation ratios when predicting excess returns. Such an investor predicts excess returns more accurately when the predictive power of valuation ratios is strong. Figure 3 Open in new tabDownload slide Posterior density of coefficients. This figure shows the simulated posterior density of the coefficients β0 and β1 given in Equation (6) for 1971 for the positive EP prior and the three model-based priors: HF, LRR, and PT. The predictor is the log dividend–price ratio. Annual data from 1947 to 1970 are used to estimate the predictive regression. The densities are simulated with 20,000 draws. In-sample estimates have a forward-looking bias that allows them to optimize between periods of weak and strong predictability. Therefore, one could assume that the asset pricing model with the model-implied parameters that are closest to the empirical in-sample estimates in Table 1 would perform best, but because of the nonnegativity constraint, this is not necessarily the case as shown by the previous results. The model-implied parameters of the LRR model are below but closer to the empirical estimates than the model-implied parameters of the HF and PT models, which are above the empirical in-sample estimates. The fact that the model-based priors include the nonnegativity constraint shown in Equation (13) helps the performance of the priors from all three asset pricing models, but the nonnegativity constraint particularly helps the priors from the HF and PT models. Table 3 shows the forecast performance of the model-based priors without the nonnegativity constraint for the total sample. Of the three model-based priors, the LRR prior performs best over the full sample and the second half of the sample. The reason is that the nonnegativity constraint ensured that the strong predictability of excess returns implied by the HF and PT models does not lead to negative forecasts during the dot-com boom when high prices and low dividends were followed by positive returns. Without the nonnegativity constraint, the forecasts during the dot-com boom when imposing the HF and PT priors are strongly negative, which leads to substantially larger forecast errors during this period. Table 3 Model-based prior forecast performance without nonnegativity constraint Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The model-based priors are imposed without the nonnegativity constraint described in Equation (13). The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab Table 3 Model-based prior forecast performance without nonnegativity constraint Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . HF . LRR . PT . Best . 1926 1947–2018 1.304 −1.381* 1.382 −8.199* LRR 1926 1947–1982 4.915*** 14.379*** 4.969*** 14.980*** PT 1926 1983–2018 −3.219 −19.989 −2.763 −35.198 LRR Panel B: Quarterly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.157* −0.986* 0.319* −6.648* LRR 1926 1947–1982 3.176*** 3.572*** 3.009*** 0.448*** HF 1926 1983–2018 −2.891 −5.431 −2.883 −13.415 LRR Panel C: Monthly returns Priors Sample start Forecast period Empirical HF LRR PT Best 1926 1947–2018 0.010 −0.124* −0.025 −1.648* Empirical 1926 1947–1982 1.127*** 1.088*** 0.933*** 0.236*** Empirical 1926 1983–2018 −1.025 −1.284 −1.001 −3.351 LRR Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The model-based priors are imposed without the nonnegativity constraint described in Equation (13). The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level and *at the 10% level. Open in new tab For the results shown in Table 2, the consumption-based asset pricing models used to derive the priors are calibrated as proposed by the respective authors, that is, Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004). To test whether the results are robust to calibrating the asset pricing models with data from a time period that has no overlap with the forecast period, I calibrate the parameters of the asset pricing models with data from 1926 to 1967. All three asset pricing models are calibrated with annual data. Thus, using a shorter sample for the calibration makes the task of matching empirical moments too challenging for the models, as the empirical moments are likely distorted by outliers due to the Great Depression and the Second World War. I follow the calibration methodology proposed by the respective authors: some parameters are set equal to their empirical counterparts and others are chosen such that the model simulated data moments match the empirical data moments, as, for example, the mean and standard deviation of the dividend–price ratio or the excess returns. Details regarding the calibration of the models can be found in Online Appendix. Table 4 reports the results for the priors derived from the asset pricing models calibrated with data from 1926 to 1967. The OOS forecasts start in 1968, and the ROOS2 for each prior, predictor, and return frequency is shown. The model-based priors can still improve the forecast accuracy of the predictive regression in several cases. However, the empirical and the positive EP priors are generally more competitive than in Table 2, which shows that the calibration of the asset pricing models is an important component for the performance of the model-based priors. Table 4 Model-based priors calibrated with 1926–1967 data. Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Reported is the forecasting performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The consumption-based asset pricing models are calibrated with annual data from 1926 to 1967. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the greatest improvement in forecast accuracy. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab Table 4 Model-based priors calibrated with 1926–1967 data. Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1968–2018 1.451 0.936 4.146** 1.682 1.811 HF 1926 1968–1993 6.469** 2.652* 7.185** 3.657** 2.994 HF 1926 1994–2018 −2.439 −0.649 −0.836 −1.132 −0.649 Positive EP Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 0.075 0.730 0.865* 0.252 0.736 HF 1926 1968–1993 3.346*** 1.889** 1.762* 1.410* 1.979** Empirical 1926 1994–2018 −4.079 −0.793 0.044 −0.339 −0.793 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1968–2018 −0.165 0.114 0.267 −0.074 0.326* PT 1926 1968–1993 0.942** 0.563** 0.840** 0.747** 0.537* Empirical 1926 1994–2018 −1.467 −0.116 −0.235 −0.039 0.018 PT Reported is the forecasting performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The consumption-based asset pricing models are calibrated with annual data from 1926 to 1967. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the greatest improvement in forecast accuracy. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab 3 Robustness and Extensions This section discusses the robustness of the results shown in the previous section and provides extensions to the baseline analysis. 3.1 Prior Parameter Sensitivity For the baseline analysis, the tightness parameters of the Gamma–Normal prior, λ and v̱ ⁠, are set equal to one and ten, respectively, as described in Section 1.2. However, the results are qualitatively the same when tightening or loosening the model-based priors. Table 5 reports the results for the total sample, that is, the 1947–2018 forecast period. Relative to the benchmark, the model-based priors are tightened and loosened by a factor of two and four, respectively. Tightening the priors further does not alter the conclusion, and loosening the priors by more than by a factor of four simply leads to forecast results of the priors that slowly converge to the OLS estimates of the single-variable predictive regression. Tightening (loosening) the prior by a factor of four results in λ=0.25 (λ = 4) and v̱=40 (⁠ v̱=2.5 ⁠). Table 5 Hyperparameter sensitivity of the forecast performance Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The prior tightness is varied, and the respective values for the hyperparameters λ and v̱ are shown in the first two columns. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab Table 5 Hyperparameter sensitivity of the forecast performance Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panel A: Annual returns . . . Priors . λ . v̱ . Empirical . Positive EP . HF . LRR . PT . Best . . Sample start: 1926/Forecast period: 1947–2018 . 0.25 40 0.710 1.950* 9.011*** 2.153** 10.502*** PT 0.5 20 0.785 2.638** 7.438*** 2.277** 8.861*** PT 2 5 1.352 4.004*** 8.102*** 4.181*** 8.658*** PT 4 2.5 0.555 4.425*** 7.127*** 4.999*** 8.489*** PT Panel B: Quarterly returns 0.25 40 0.307 1.494*** 2.878*** 1.188*** 1.494*** HF 0.5 20 0.459 1.672*** 1.844*** 1.399*** 1.672*** HF 2 5 −0.260* 1.813*** 1.737*** 2.199*** 2.696*** PT 4 2.5 −0.316* 1.944*** 2.070*** 1.915*** 2.036*** HF Panel C: Monthly returns 0.25 40 0.049 0.452*** 0.821*** 0.558*** 0.452*** HF 0.5 20 0.065 0.550*** 0.581*** 0.728*** 0.550*** LRR 2 5 0.013* 0.738*** 0.287* 0.475*** 0.367** Positive EP 4 2.5 −0.161 0.651*** 0.741*** 0.402** 0.672*** HF Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The prior tightness is varied, and the respective values for the hyperparameters λ and v̱ are shown in the first two columns. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab The results are fairly robust across the range of λ and v̱ values. The model-based priors can improve upon the performance of the empirical or positive EP priors in the majority of the cases and the values of the ROOS2 are similar to those reported in Table 2. The forecasts are again more accurate at longer return frequencies. 3.2 Utility Gains Based on Model-Based Prior Forecasts So far, I have analyzed how priors derived from the three consumption-based asset pricing models affect the forecast accuracy of single-variable predictive regressions. However, an investor is ultimately concerned about utility, and thus, differences in forecast accuracy need to be mapped into differences in utility. For this purpose, I follow Campbell and Thompson (2008) and Wachter and Warusawitharana (2009) and assume an investor with a mean-variance utility function and a risk aversion of three who can allocate her funds in a risky and a risk-free asset, where the risky asset pays the aggregate stock market excess return and the risk-free asset pays the risk-free rate. As explained in more detail in Online Appendix, a certainty equivalent return (CER) can be computed. The CER is defined as a constant return that equates the realized utility of a portfolio that relies on excess return forecasts from the historical average model with a portfolio that relies on excess return forecasts from the single-variable predictive regression with a prior imposed. I compute the CER for each return frequency, predictor, and forecast period. The CER values are annualized and in percent. The results are shown in Table 6, which is structured like Table 2 but with the ROOS2 values replaced by the annualized CERs. For every case, at least one model-based prior leads to a CER that is higher than the empirical and positive EP forecasts. The CER values are economically significant reaching up to 1.4%. The model-based priors perform again better for the first half of the sample, that is, for the forecast period from 1947 to 1982. For the second half of the sample, that is from 1983 to 2018. The CER values are mostly negative. Table 6 Model-based prior economic performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A, B, and C report the annualized CER in percent as discussed in Section 3.2. The CER can be interpreted as a management fee that an investor with mean-variance utility and a risk aversion of three is willing to pay each year to have access to the excess return forecasts which result from the single-variable predictive regression given in Equation (1) and estimated based on either the empirical, positive EP, or model-based priors, relative to the excess return forecasts of the historical average model. The predictor is the log dividend–price ratio. The CER values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest CER. Open in new tab Table 6 Model-based prior economic performance Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 −0.331 0.145 0.695 0.360 0.631 HF 1926 1947–1982 0.099 0.460 1.339 0.681 1.217 HF 1926 1983–2018 −0.713 −0.070 −0.078 −0.068 0.049 PT Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.380 0.426 0.511 0.432 0.591 PT 1926 1947–1982 0.553 0.929 1.368 1.103 0.886 HF 1926 1983–2018 −1.240 −0.148 0.147 −0.088 −0.218 HF Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 −0.772 0.295 0.489 0.234 0.245 HF 1926 1947–1982 0.507 0.789 0.884 0.731 0.550 HF 1926 1983–2018 −2.042 −0.123 −0.378 −0.324 −0.107 PT Panel A, B, and C report the annualized CER in percent as discussed in Section 3.2. The CER can be interpreted as a management fee that an investor with mean-variance utility and a risk aversion of three is willing to pay each year to have access to the excess return forecasts which result from the single-variable predictive regression given in Equation (1) and estimated based on either the empirical, positive EP, or model-based priors, relative to the excess return forecasts of the historical average model. The predictor is the log dividend–price ratio. The CER values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest CER. Open in new tab Figure 4 shows the risky asset share of the portfolio for the different priors. The forecasts are at an annual frequency, and the forecast period is from 1947 to 2018. Generally, the LRR prior leads to a more stable portfolio share of the risky asset, which closely matches the risky asset share of the investor’s portfolio when forecasting with the positive EP prior. The forecasts of the HF and PT priors lead to a more volatile portfolio allocation, but the average portfolio allocation over the total forecast period is similar to the positive EP and LRR priors. Figure 4 Open in new tabDownload slide Portfolio share of risky asset. This figure shows the risky asset portfolio share for a mean-variance investor with a risk aversion of three who relies either on the positive EP, HF, LRR, and PT priors, respectively, to forecast excess returns with the log dividend–price ratio from 1947 to 2018. 3.3 Multiple Predictors The previous analysis has focused on the log dividend–price ratio as the predictor of excess returns. While the dividend–price ratio is one of the most prominent predictors in the literature, there are a couple of other predictor variables that can be generated from the asset pricing models. In this section, model-based priors are imposed on a predictive regression with two predictor variables. The first predictor being the log dividend–price ratio and the second predictor being either the log consumption growth rate or the log real risk-free rate. The consumption growth rate is time-varying in all the three asset pricing models. The real risk-free rate is constant in the HF and PT models but time-varying in the LRR model.11 Table 7 reports the results for the predictive regression with two predictors. The results are shown for the full sample with the forecast period starting 20 years into the sample. For consumption growth, monthly and quarterly data are available from 1960 and 1948, respectively. Overall, the model-based priors outperform the empirical and the positive EP prior for all the specifications. However, adding consumption growth or the real risk-free rate as predictors does not consistently improve the forecast accuracy of the model compared to a single-variable predictive regression that only includes the dividend–price ratio. The values in parentheses are the ROOS2 of the multiple predictor’s regressions with the corresponding ROOS2 of the single-variable dividend–price ratio predictive regression being subtracted. These values are often negative, implying that the single-variable predictive regression outperforms the regression with multiple predictors. Table 7 Model-based prior forecast performance for multiple predictors Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the predictive regression given in Equation (1) but adding a second predictor. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the predictive regression relative to the historical average model. The predictors are the log dividend–price ratio together with the log consumption growth or log risk free rate, respectively. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. The values in parentheses are the ROOS2 of the multiple predictors regressions with the corresponding ROOS2 of the single-variable dividend–price ratio predictive regression being subtracted. Open in new tab Table 7 Model-based prior forecast performance for multiple predictors Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panel A: Annual returns . . . . Priors . Sample start . Forecast period . Predictors . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 Log DP ratio and 2.161 3.757* 9.433*** 4.202** 9.703*** PT Log consumption growth (0.857) (0.304) (1.053) (0.716) (0.312) 1926 1947–2018 Log DP ratio and −2.334 3.968*** 7.175*** 3.305** 7.011*** HF Log real risk free rate (−3.639) (0.516) (−1.205) (−0.182) (−2.381) Panel B: Quarterly returns 1948 1969–2018 Log DP ratio and −0.130 0.074 0.473 0.461 0.353 HF Log consumption growth (−0.549) (−1.130) (−0.428) (−0.511) (−1.313) 1926 1947–2018 Log DP ratio and −0.178 1.622*** 1.792*** 1.254** 1.583*** HF Log real risk free rate (−0.335) (−0.426) (−0.104) (−0.702) (−0.949) Panel C: Monthly returns 1960 1981–2018 Log DP ratio and −0.533 0.008 −0.274 0.250 0.008 LRR Log consumption growth (0.003) (−0.217) (−0.605) (0.016) (−0.424) 1926 1947–2018 Log DP ratio and −0.088 0.375** 0.757*** 0.367** 0.560*** HF Log real risk free rate (−0.098) (−0.258) (0.306) (−0.024) (−0.282) Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on the predictive regression given in Equation (1) but adding a second predictor. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the predictive regression relative to the historical average model. The predictors are the log dividend–price ratio together with the log consumption growth or log risk free rate, respectively. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. The values in parentheses are the ROOS2 of the multiple predictors regressions with the corresponding ROOS2 of the single-variable dividend–price ratio predictive regression being subtracted. Open in new tab 3.4 Alternative Asset Pricing Model Specifications Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001), and Bansal and Yaron (2004) suggest alternative specifications of their models. In this section, I examine the performance of model-based priors derived from these alternative specifications. The baseline model-based priors from the HF model discussed previously are based on the specification that assumes a perfect positive correlation between the log consumption and log dividend growth. However, Campbell and Cochrane (1999) also solve the HF model for an imperfect but positive correlation. For the PT model, the baseline model-based prior is derived from the model specification that generates a log EP that is closest to the empirical data moment. A second specification proposed by Barberis, Huang, and Santos (2001) sets the financial wealth fluctuations parameter b0 equal to 100 and the effect of prior losses on risk aversion k equal to 3. The simulated log EP is then lower, but the average loss aversion of the agent is 2.25, which is in line with experimental evidence. Bansal and Yaron (2004) consider two calibrations for the agent’s risk aversion to simulate the LRR model: a risk aversion of 7.5 and a risk aversion of 10. The LRR model with the risk aversion set to ten is the previously discussed baseline specification, and the specification that sets the risk aversion parameter to 7.5 is the alternative specification. The three alternative model specifications are denoted HFAlt, PTAlt, and LRRAlt and the model-implied parameters can be found in Online Appendix. The results when imposing model-based priors from the alternative specifications are presented in Table 8 and are similar to the baseline model-based priors. The model-based priors outperform the empirical and positive EP priors in most cases. Also, the model-based priors’ forecast performance leads to a rejection of the historical average model in all cases for the full sample. However, the performance is again weaker for the second half of the sample. Table 8 Model-based prior from alternative asset pricing model specifications Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HFAlt., LRRAlt., and PTAlt. The calibration of the asset pricing models is based on alternative calibrations as discussed in Section 3.4. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab Table 8 Model-based prior from alternative asset pricing model specifications Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panel A: Annual returns . . . Priors . Sample start . Forecast period . Empirical . Positive EP . HF . LRR . PT . Best . 1926 1947–2018 1.304 3.453** 9.336*** 3.273** 8.089*** HF 1926 1947–1982 4.915*** 6.135*** 14.115*** 5.380*** 15.817*** PT 1926 1983–2018 −3.219 0.497 1.899 0.041 1.865 HF Panel B: Quarterly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.157* 2.048*** 2.468*** 1.808*** 2.495*** PT 1926 1947–1982 3.176*** 2.755*** 3.535*** 3.449*** 2.946*** HF 1926 1983–2018 −2.891 0.644 0.253 0.605 0.798 PT Panel C: Monthly returns Priors Sample start Forecast period Empirical Positive EP HF LRR PT Best 1926 1947–2018 0.010 0.633*** 0.605*** 0.505*** 0.601*** Positive EP 1926 1947–1982 1.127*** 1.017*** 1.219*** 0.871*** 0.729** HF 1926 1983–2018 −1.025 0.275 0.186 −0.115 0.600** PT Panels A, B, and C show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HFAlt., LRRAlt., and PTAlt. The calibration of the asset pricing models is based on alternative calibrations as discussed in Section 3.4. The priors are imposed on the single-variable predictive regression given in Equation (1). Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by ***at the 1% level, **at the 5% level, and *at the 10% level. Open in new tab 3.5 Long Forecast Horizons So far, the assessment of the priors was based on monthly, quarterly, and annual return forecasts. When imposing the model-based priors on the predictive regressions that forecast the excess returns at a two- and three-year horizon, respectively, the results are similar to the baseline results in Table 2. However, the number of forecast observations is small with thirty-six and twenty-four observations for the two- and three-year horizons, respectively. Table 9 shows that the model-based priors can also increase the forecast accuracy as measured by the ROOS2 for longer horizons. For example, for the default λ value of one and at a two-year horizon, the model-based priors achieve an ROOS2 of 10.9% (HF), 5.0% (LRR), and 14.3% (PT), respectively. In comparison, the positive EP prior’s ROOS2 is only 3.8%. The statistical significance is generally lower than at higher return frequencies. This is caused by the decrease in power that comes with fewer return observations due to the longer return horizons. Table 9 Model-based prior forecast performance for long horizons Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panels A and B show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on a single-variable predictive regression, given in Equation (1), which predicts excess returns at a two- and three-year horizon. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. Results are shown for different λ values, which represents the prior tightness as described in Section 1.2. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by **at the 5% level and *at the 10% level. Open in new tab Table 9 Model-based prior forecast performance for long horizons Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panel A: Two-year returns . . Priors . λ . Empirical . Positive EP . HF . LRR . PT . Best . Sample start: 1926/Forecast period: 1947–2018 0.5 −1.777 3.210 14.070** 3.101 13.968* HF 1 −2.929 3.794 10.941* 5.001 14.347* PT 2 −3.720 5.276 13.003* 6.262 11.840* HF Panel B: Three-year returns Priors λ Empirical Positive EP HF LRR PT Best Sample start: 1926/Forecast period: 1947–2018 0.5 −13.745 8.687* 11.953* 4.711 13.095* PT 1 −13.960 7.986 12.956* 6.978* 14.272* PT 2 −15.655 10.178* 15.683** 8.946* 14.700* HF Panels A and B show the forecast performance of the model-based priors derived from the three consumption-based asset pricing models: HF, LRR, and PT. The priors are imposed on a single-variable predictive regression, given in Equation (1), which predicts excess returns at a two- and three-year horizon. Reported is the ROOS2 (in percent) from Equation (24), which measures the accuracy of OOS log excess return forecasts of the single-variable predictive regression relative to the historical average model. The predictor is the log dividend–price ratio. Results are shown for different λ values, which represents the prior tightness as described in Section 1.2. The ROOS2 values of the model-based priors are compared to the values of the empirical prior, denoted “empirical,” and the empirical prior with a nonnegative forecast constraint, denoted “positive EP,” described in Section 1.6. The last column denotes which prior leads to the highest ROOS2 ⁠. The statistical significance of the difference between the forecast errors of the historical average model and the predictive regression is tested with the test of Clark and West (2007). A significant test statistic is denoted by **at the 5% level and *at the 10% level. Open in new tab 4 Conclusion Different theories have been proposed to resolve the EP puzzle of Mehra and Prescott (1985). Three prominent consumption-based asset pricing models that provide different explanations for the existence of the EP puzzle are the HF, the LRR, and the PT models. This article tests if these asset pricing models can profitably guide the investment decisions of a hypothetical investor who had access to these models from 1947 onwards and tries to time the aggregate stock market by forecasting returns. I propose a simple Bayesian framework in which an investor reduces the uncertainty about predictive regression parameters by imposing economic constraints derived from the three asset pricing models. The model-based priors derived from the three asset pricing models generally lead to more accurate forecasts than an empirical prior that shrinks the parameter estimates toward the historical average model and the positive EP prior that restricts the forecast of the EP to be nonnegative. However, the results are sensitive to the sample period, the return frequency, and imposing a nonnegative EP constraint. By imposing model-based priors derived from consumption-based asset pricing models on predictive regressions and showing how the forecast performances of these priors compare to each other and to benchmark priors, this article makes a novel contribution to the return prediction literature. The analysis helps to determine how useful consumption-based asset pricing models could be for finance practitioners. Supplementary Data Supplementary data are available at Journal of Financial Econometrics online. Footnotes 1 See, for example, Campbell (1987); Campbell and Shiller (1988); Fama and French (1988, 1989); Baker and Wurgler (2000); Lettau and Ludvigson (2001); Lewellen (2004); Polk, Thompson, and Vuolteenaho (2006); Cochrane (2008); Lettau and Van Nieuwerburgh (2008); Rapach, Strauss, and Zhou (2010); Li, Ng, and Swaminathan (2013); Kelly and Pruitt (2013); and Neely et al. (2014). 2 Many subsequent papers have developed modified versions of these three asset pricing models. I make the choice to focus on the versions presented in the three initial papers in order to discipline the exercise. While examining priors derived from other versions of these asset pricing models is beyond the scope of this article, such priors might lead to qualitatively different results than shown here. 3 Holding the sample moments constant by estimating them over an estimation sample that has no overlap with the forecast sample leads to qualitatively similar results. 4 A caveat is that when the equity premium of the asset pricing models is estimated based on a linear projection of excess returns on valuation ratios such as in Equation (12), the resulting equity premium can be negative. However, such negative estimates are relatively rare. For the HF and LRR models, the share of negative estimates of the equity premium is 0.1% and 0%, respectively, when simulating 10,000 years. For the PT model, the share of negative estimates is higher with 17.1%. 5 Some rare combinations of xτ, β0*, β1*, σϵ*−2 ⁠, and λ can lead to the nonnegativity constraint in Equation (13) being violated for most of the Gibbs sampler draws. When the nonnegativity constraint is violated for more than 95% of the Gibbs sampler draws, β0*, β1* ⁠, and σϵ*−2 are replaced by r¯ ⁠, 0, and σr−2 ⁠, that is, the forecast is equivalent to the forecast of the positive EP prior. The assumption is that an investor would discard the model-based priors in time periods when the model-based priors’ posterior excess return distribution forecast is predominantly negative. 6 I denote this specification the HF model. The model specification that assumes an imperfect but positive correlation between dividend and consumption growth is denoted HFAlt. and described in Online Appendix. 7 I simulate the model based on the analytical solutions as done by Bansal, Kiku, and Yaron (2010) and Beeler and Campbell (2012). 8 As discussed in Section 1.5, the coefficients are estimated on 10,000 years of simulated data and averaged across ten iterations. Online Appendix contains coefficient estimates and forecast analyses when simulating 93 years of data (corresponding to the empirical length of the sample) and averaging across 1,000 iterations. The performance of the model-based priors is qualitatively similar. 9 Adjusting the β1 empirical estimates for the bias discussed by Stambaugh (1999) leads to annual, quarterly, and monthly estimates of 0.032, 0.010, and 0.002, respectively, for the 1926–2018 sample. For the 1947–2018 sample, the estimates are 0.051, 0.009, and 0.002, respectively. 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Google Scholar Crossref Search ADS WorldCat Author notes * This article is based on a chapter of my dissertation at the University of Oxford. I am grateful for the continued support from Tarun Ramadorai and Kevin Sheppard. I also thank Nicholas Barberis, John Campbell, Ben Jensen, Lubos Pastor, Andrew Patton, Allan Timmermann (Editor), Dimitri Vayanos, Jessica Wachter, Missaka Warusawitharana, Sumudu Watugala, Ivo Welch, Mungo Wilson, two anonymous referees, and seminar participants at the Federal Reserve Board of Governors, FMA 2016, NBER Summer Institute 2016, Oxford-Man Institute of Quantitative Finance, Saïd Business School, and SEA Annual Meeting 2015 for useful comments. The analysis and conclusions set forth are those of the author and do not indicate concurrence by the Board of Governors of the Federal Reserve System or its staff. Published by Oxford University Press 2020. This work is written by a US Government employee and is in the public domain in the US. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) Published by Oxford University Press 2020. This work is written by a US Government employee and is in the public domain in the US.

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From Which Consumption-Based Asset Pricing Models Can Investors Profit? Evidence from Model-Based Priors

From Which Consumption-Based Asset Pricing Models Can Investors Profit? Evidence from Model-Based Priors

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From Which Consumption-Based Asset Pricing Models Can Investors Profit? Evidence from Model-Based Priors

From Which Consumption-Based Asset Pricing Models Can Investors Profit? Evidence from Model-Based Priors

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