Does Smooth Ambiguity Matter for Asset Pricing?

Abstract

A number of asset pricing puzzles pose remarkable challenges to the standard consumption-based model with a fully rational representative agent. Among them is the equity premium puzzle first documented by Mehra and Prescott (1985), which states that the standard model requires an implausibly high level of risk aversion to explain the historical equity premium in the U.S. data. Other important stylized facts of the stock market include excess volatility of returns, countercyclical equity premium and equity volatility, and return predictability.¹ A recent strand of the literature proposes to embed ambiguity in an otherwise standard model to explain various asset pricing puzzles. An ambiguity-averse agent recognizes uncertainty about an objective law governing the state process and is averse to such uncertainty. There are two popular approaches to model ambiguity in the asset pricing literature, the multiple-priors approach and the smooth ambiguity approach.² Existing consumption-based models with ambiguity are largely confined to model calibration. However, calibration does not provide the likelihood of a model given observed macroeconomic and financial variables, and thus statistical support for the importance of ambiguity in asset pricing is still limited in the literature on estimation of structural models.

In this paper, we use Gallant and McCulloch’s (2009) general scientific models (GSM), a simulation-based Bayesian estimation method, to estimate a set of three consumption-based asset pricing models with smooth ambiguity preferences and three models with recursive utility. Due to nonlinearities inherent in structural asset pricing models, the likelihood of these models is generally not available in closed form. In addition, similar to other nonlinear applications, we face sparsity of data. As has become standard in the macrofinance empirical literature, we use Bayesian methods (in our case, GSM) coupled with prior information to overcome data sparsity. We describe this estimation method in Section 2.2. We are interested in estimates of preferences parameters and parameters governing state dynamics jointly obtained from fitting structural models to data. We consider models with an ambiguity-averse representative agent who is uncertain about the conditional mean growth rate of aggregate consumption. The agent’s preferences are represented by generalized recursive smooth ambiguity utility advanced by Hayashi and Miao (2011) and Ju and Miao (2012). This class of preferences builds on the seminal work of Klibanoff, Marinacci, and Mukerji (2005, 2009) and allows for the separation among risk aversion, ambiguity aversion, and the elasticity of intertemporal substitution (EIS). Our estimation suggests a clear distinction of the EIS from risk aversion for all models and statistical evidence supporting a preference for early resolution of uncertainty.

We examine three models featuring smooth ambiguity. The first is the original model of Ju and Miao (2012). The growth rate of consumption follows a Markov-switching process in which the mean growth rate depends on a hidden state. The hidden state evolves according to a two-state Markov chain. The agent cannot observe the state but can learn about the state in a Bayesian fashion by observing realized growth rates of consumption. Ambiguity arises since the mean growth rate is unobservable. Because the hidden state evolves dynamically over time, learning cannot resolve the agent’s ambiguity in the long run. The agent is ambiguity-averse in that he dislikes a mean-preserving spread in the continuation value led by the agent’s belief about the hidden state. As a result, compared with a solely risk-averse agent, the ambiguity-averse agent effectively assigns more probability weight to “bad” states that are associated with lower levels of the continuation values.

The second model is an extension of the first and incorporates time-varying conditional volatility. We postulate that conditional volatility of consumption growth follows another two-state Markov chain that is independent of the chain for the mean growth state, as in McConnell and Perez-Quiros (2000) and Lettau, Ludvigson, and Wachter (2008). A number of studies have examined the role of time-varying volatility and found that volatility risk is significantly priced in the stock market, see Bollerslev, Tauchen, and Zhou (2009), Drechsler (2013), and Bansal et al. (2014), among others. By estimating a model with ambiguity, learning, and time-varying volatility, we aim to investigate whether (i) inclusion of time-varying volatility affects the estimated impact of ambiguity on asset prices, and (ii) the model with time-varying volatility represents an improvement over the original model. In addition, our estimation results allow us to disentangle the contribution of time-varying volatility and ambiguity aversion to improvements in fitting the data.

The third model is built on the long-run risk model of Bansal and Yaron (2004) and the smooth ambiguity model of Collard et al. (2018). The motivation for examining this model is to study the impact of ambiguity in a long-run risk setting. This model features consumption growth dynamics similar to the model studied by Bansal, Kiku, and Yaron (2012). It features a long-run risk component and stochastic volatility in conditional mean and volatility of the consumption growth process, respectively. The persistent long-run risk component in the conditional mean of consumption growth is empirically difficult to detect. Thus, it is reasonable to postulate that the agent also faces the same difficulty as an econometrician does.³ Similar to the model setup of Ju and Miao (2012), the agent cannot observe the long-run risk component governing mean consumption growth but can learn about it in a Bayesian fashion by observing realizations of consumption and dividend growth rates. In addition, we incorporate stochastic volatility as an exogenous process as in Bansal and Yaron (2004).⁴ By estimating this model, we want to investigate to what extent the estimated level of ambiguity aversion depends on specifications of state processes and the information structure.

In all three models, the agent’s ambiguity aversion endogenously generates pessimistic beliefs about the distribution of consumption growth. In contrast to an ambiguity-neutral investor, the ambiguity-averse agent always slants his belief toward states with low levels of conditional mean growth of consumption. This pessimism is manifested by a sharp increase in the stochastic discount factor (SDF) when the economy experiences a negative shock after staying at the “normal” growth rate for several periods. The pessimistic distortion to the SDF raises its volatility and thus implies a high market price of risk and high equity premium.

In addition to models with smooth ambiguity, we also estimate three baseline models with Epstein and Zin’s (1989) recursive utility for model comparison. The first baseline model is the long-run risk model studied by Bansal, Kiku, and Yaron (2012), which is an improved formulation of the original model of Bansal and Yaron (2004). The long-run risk model allows for heteroscedasticity consumption growth and generates time-varying volatility in returns. The second baseline model is a special case of Ju and Miao’s model that suppresses ambiguity aversion. In this model, the agent without endogenous pessimism uses Bayesian inference to evaluate mean consumption growth. The third model is an extension of the second and incorporates time-varying conditional volatility. By estimating a series of structural models with and without smooth ambiguity, we address two important questions: (i) Does a structural estimation with macrofinance data lend statistical support to the class of smooth ambiguity preferences that have a sound decision-theoretic basis? (ii) Based on a standard Bayesian model comparison between those featuring smooth ambiguity and Epstein-Zin’s preferences, which estimated model is preferred?

We find a significant distinction between risk aversion and ambiguity aversion in the estimated models. This distinction is robust to different specifications of consumption dynamics. Incorporating smooth ambiguity aversion in consumption-based asset pricing models greatly reduces the burden on risk aversion, and yields posterior distributions of the risk-aversion parameter that are centered on values between 0.8 and 5. These values are noticeably smaller than median and mean values of the posterior distributions of the ambiguity aversion parameter, which range between 6.9 and 23.5. We find that incorporating smooth ambiguity in models lowers the estimated values of the EIS parameter. However, our estimation indicates that the EIS parameter is greater than 1 and thus lends support to the preference for early resolution of uncertainty. A model comparison exercise based on posterior likelihoods and the Bayesian information criteria (BIC) shows that the two models featuring smooth ambiguity and time-varying volatility are preferred to the long-run risk model and Epstein-Zin’s recursive utility model with regime-switching mean consumption growth, learning, and time-varying volatility. Prior to our study, Bansal, Gallant, and Tauchen (2007) and Aldrich and Gallant (2011) concluded that the long-run risk model is a preferred model.⁵ In addition, we find that the estimated smooth ambiguity models can match moments of asset returns better than the Epstein-Zin models do. The estimated Ju and Miao model, while receiving less statistical support than the long-run risk model, can match the equity premium and variance risk premium in the data well.

We use the projection method to solve all models examined in this paper. The widely used log-linear approximation method in the long-run risk literature is not applicable to smooth ambiguity models. This is because learning is an important ingredient in smooth ambiguity models and induces nonlinearities in the dynamics of the agent’s beliefs. Additionally, the smooth ambiguity utility function is highly nonlinear. To keep our quantitative analysis consistent, we also use the projection method to solve the long-run risk model. In a recent work, Pohl, Schmedders, and Wilms (2018) assess numerical accuracy of the log-linear approximation method and find that applying log-linearization to solve long-run risk models can yield biased results due to neglecting higher-order effects. The bias becomes more pronounced when the long-run risk and stochastic volatility components are highly persistent. Using the log-linear approximation and a mixed data frequency approach, Schorfheide, Song, and Yaron (2018) perform Bayesian estimation of long-run risk models with several specifications of stochastic volatility and find that the long-run risk component and stochastic volatilities are highly persistent. While our estimation is based on annual data and Bayesian indirect inference, we also find persistent long-run risk and stochastic volatility components as well as a high EIS.

This paper contributes to a growing body of literature on ambiguity, learning, and macrofinance. We discuss closely related papers here. Epstein and Schneider (2007) develop a model with learning under ambiguity. They use the multiple-priors approach to model ambiguity and assume a set of priors and a set of likelihoods for signals. Beliefs are updated by Bayes’ rule in an appropriate way. Epstein and Schneider (2008) apply this model to study information quality and asset prices. Leippold, Trojani, and Vanini (2008) adopt the continuous-time multiple-priors framework of Z. Chen and Epstein (2002) to analyze asset pricing implications of learning under ambiguity. Cogley and Sargent (2008) examine the impacts of pessimistic beliefs on the market price of risk and equity premium. Hansen and Sargent (2010) consider robustness concerns in learning and study time-varying model uncertainty premia. Collard et al. (2018) assume that a representative agent with smooth ambiguity preferences faces both model uncertainty and state uncertainty and analyze the dynamics of risk premia conditioning on the historical data. Johannes, Lochstoer, and Mou (2016) and Collin-Dufresne, Johannes, and Lochstoer (2016) study parameter learning and asset prices in the consumption-based framework with recursive utility. Jeong, Kim, and Park (2015) estimate an asset pricing model in which the agent has multiple-priors utility. Their estimation results suggest that ambiguity on the true probability law governing fundamentals carries a sizable premium. Ai and Bansal (2018) show that a wide class of non-expected utility models, including the smooth ambiguity model, implies demands for risk compensation for news affecting continuation values and generates a premium related to macroeconomic announcements.

Jahan-Parvar and Liu (2014), Backus, Ferriere, and Zin (2015), and Altug et al. (2017) examine both business cycle and asset pricing implications in dynamic stochastic general equilibrium (DSGE) models with smooth ambiguity. Thimme and Völkert (2015) use generalized method of moments (GMM) of Hansen (1982) to estimate a reduced-form, locally linearized version of the smooth ambiguity model without characterizing dynamics of beliefs. Their estimation crucially relies on an approximation of conditional expectation of future utility, a proxy for the wealth-consumption ratio, and omission of any specification for consumption dynamics. Ilut and Schneider (2014) estimate a DSGE model with multiple-priors utility. Their estimation suggests that time-varying confidence in future total factor productivity explains a significant fraction of the business cycle fluctuations. Bianchi, Ilut, and Schneider (2018) estimate another DSGE model to explain joint dynamics of asset prices and real economic activity in the postwar data. They show that time-varying ambiguity about corporate profits leads to a high equity premium and excess volatility. They further show that the recursive multiple-priors utility model provides a tractable way to analyze DSGE models with time-varying uncertainty and facilitates estimation by means of likelihood methods.

1. Asset Pricing Models

1.1 Models Featuring Smooth Ambiguity

We examine three consumption-based asset pricing models in which a representative agent is endowed with smooth ambiguity preferences. These models include (i) Ju and Miao’s (2012) model in which the mean of consumption growth follows a hidden Markov chain with two states, abbreviated as AAMS, (ii) an extended version of Ju and Miao’s model with time-varying conditional volatility, abbreviated as AAMSTV, and (iii) a long-run risk model featuring ambiguity in which the long-run risk component is assumed to be unobservable, abbreviated as AALRRSV. The latter model shares many features with the models introduced by Collard et al. (2018). In all these models, the agent cannot observe the state determining mean consumption growth but learns about the state in a Bayesian fashion. The unobservable mean growth state implies that the agent is ambiguous about the data-generating process of fundamentals. Smooth ambiguity utility captures the agent’s aversion toward this ambiguity.

1.1.1 The AAMS model

Ambiguity aversion is characterized by the parametric restriction |$\eta>\gamma$|⁠, where |$\eta$| is the ambiguity aversion parameter. By setting |$\eta=\gamma$|⁠, we obtain Epstein-Zin’s recursive utility under ambiguity neutrality.⁶ In the certainty equivalent in Equation (2), the expectation operator |$\mathbb{E}_{s_{t+1},t}\left[ \cdot \right] $| is taken with respect to the conditional distribution of consumption growth in state |$s_{t+1}$| and other information at time |$t$|⁠. The expectation operator |$\mathbb{E}_{\pi _{t}}$| is taken with respect to the posterior belief about the unobservable state.

As long as |$\eta>\gamma$|⁠, distorted beliefs are not equivalent to Bayesian beliefs. The distortion driven by ambiguity aversion is an equilibrium outcome and implies pessimistic beliefs; see Section 3.

1.1.2 The AAMSTV model

To ease the analysis, we assume that the mean state |$s^{\mu}_{t}$| is unobservable while the volatility state |$s^{\sigma}_{t}$| is observable (thus, no ambiguity about the volatility state). We make this simplifying assumption for three reasons. First, empirical studies such as Bryzgalova and Julliard (2015) have established that estimation and characterization of mean consumption growth is more difficult than consumption volatility. Second, according to the existing literature, the volatility state is very persistent, leading to filtered probabilities of the volatility state close to 1. These results suggest that ambiguity has limited room with respect to the consumption volatility.⁷ Third, while Epstein and Ji (2013) and Branger, Schlag, and Thimme (2016) argue that ambiguity about volatility may have certain asset pricing implications, Veronesi (1999), Ju and Miao (2012), and Miao, Wei, and Zhou (Forthcoming) show that learning about the conditional mean of fundamentals is sufficient to characterize salient features of macrofinancial variables. We confirm their findings based on structural estimation.

1.1.3 The AALRRSV model

In Bansal and Yaron’s calibration, |$x_{t}$| is a highly persistent component, and |$\sigma _{t}$| is the highly persistent stochastic volatility component representing time-varying economic uncertainty. The long-run risks literature assumes that |$x_{t}$| is fully observable and thus appears as a state variable in the wealth-consumption ratio and price-dividend ratio. However, this component is difficult to identify using empirically observed economic variables, as documented by Bansal, Gallant, and Tauchen (2007), Ma (2013), and Johannes, Lochstoer, and Mou (2016), among others. The difficulty in estimating |$x_{t}$| gives rise to the agent’s ambiguity about mean consumption growth. As a result, we adopt a more plausible information structure by assuming that |$x_{t}$| is unobservable. Collard et al. (2018) provide ample theoretical support for this assumption.

In particular, we maintain that the agent observes the realizations of |$\Delta c_{t+1}$| and |$\Delta d_{t+1}$| contemporaneously but never observes the realization of |$x_{t}$| or |$\left( \epsilon _{c,t},\epsilon _{d,t},\epsilon _{x,t}\right)$|⁠. This feature of the model characterizes ambiguity, that is, the agent’s lack of confidence in estimating the conditional mean of consumption growth. Instead, the agent uses consumption and dividend growth realizations to filter the unobserved long-run risk component |$x_{t}$|⁠. To make the model tractable and comparable to the long-run risks model, we assume that the conditional volatility of consumption growth, |$\sigma _{t}$|⁠, is observable. We also assume that values of the parameter vector |$\left( \mu _{c},\mu _{d},\phi_{c},\phi_{d},\phi_{x},\rho_{x},\lambda, \mu_{s}, \rho_{s}, \sigma_{w} \right)$| are known to the agent.

The certainty equivalent |$\mathcal{R}_{t}\left( V_{t+1}\right)$| reflects the agent’s aversion toward ambiguity in estimating the long-run risk component |$x_{t}$|⁠. The agent lacks confidence in the Gaussian posterior of |$x_{t}$| and thus applies pessimistic distortion to the posterior. This distortion is visible in Figure 1. In what follows, we describe the mechanism of how ambiguity aversion leads to distortion in the posterior.

Figure 1

Model AALRRSV: Bayesian and distorted densities of |$x$|

This figure plots Bayesian density and distorted density of the long-run risk component |$x$| for the AALRRSV model. The Bayesian density is |$x_{t}\sim N\left( \hat{x}_{t},\nu_{t} \right)$|⁠, and the distorted density is |$\tilde{f}\left( x_{t}|\hat{x}_{t},\nu _{t},t\right)$|⁠. The distorted density is generated from solving the model. The state vector is assumed to take the value (⁠|$\hat{x}_{t}=0$|⁠, |$\nu _{t}=\bar{\nu}$| [steady-state] and |$\sigma_{t}=\mu_{s}$|⁠). Model parameters are set at posterior mean estimates presented in Table 5.

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1.2 Alternative Models Featuring Ambiguity-Neutral Preferences

By setting |$\eta=\gamma$| in the generalized recursive smooth ambiguity utility function, we suppress ambiguity aversion and obtain Epstein-Zin’s recursive utility model as a special case. We impose this parametric restriction to obtain the EZMS model as the ambiguity-neutral version of the AAMS model.

The second alternative model, which we call EZMSTV, is the ambiguity-neutral version of AAMSTV where we suppress ambiguity aversion by setting |$\eta=\gamma$|⁠, as in the derivation of EZMS. EZMSTV has the same consumption growth dynamics as the model studied by Lettau, Ludvigson, and Wachter (2008) and features Epstein-Zin preferences.

We present structural parameters to be estimated for each model in Table 2. In estimating AALRRSV and EZLRRSV models, we impose that |$\mu_c=\mu_d$|⁠. We solve all the models examined in this paper using the collocation projection method with Chebyshev polynomials. Pohl, Schmedders, and Wilms (2018) show that this is a reliable solution method for nonlinear asset pricing models. The details of the implementation and numerical accuracy assessment are available in the Online Appendix.

2. Data and the Estimation Method

2.1 Data

Throughout this paper, lower case denotes the natural logarithm of an upper case variable; for example, |$c_{t}=\ln (C_{t}),$| where |$C_{t}$| is the observed consumption in period |$t,$| and |$d_{t}=\ln (D_{t}),$| where |$D_{t}$| is dividends paid in period |$t$|⁠. Similarly, we use logarithmic risk-free interest rate (⁠|$r^f_t$|⁠) and aggregate equity market return inclusive of dividends (⁠|$r_{t}=\ln{(P_{t}+D_{t})}-\ln P_{t-1}$|⁠) in the analysis, where |$P_{t}$| is the stock price in period |$t$|⁠.

We use real annual data from 1941 to 2015. The sample period 1941–1949 provides initial lags for the recursive parts of our estimation, and the sample period 1950–2015 yields estimation results and diagnostics. Our measure for the risk-free rate is the one-year U.S. Treasury bill rate. To construct the real risk-free rate, we regress the ex post real one-year Treasury bill yield on the nominal rate and past annual inflation, available from the Wharton Research Data Services (WRDS) Treasury and Inflation database. The fitted values from this regression are the proxy for the ex ante real interest rate. Using other estimates of expected inflation to construct the real rate does not lead to significant changes in our results. Our proxy for risky assets is the value-weighted returns (including dividends) on the aggregate stock market portfolio of the NYSE/AMEX/NASDAQ, which is obtained from the Center for Research in Security Prices (CRSP) and deflated using the consumer price index (CPI) data. We use the sum of real nondurable and services consumption, items 16 and 17 in national income and product accounts (NIPA) Table 7.1 “Selected Per Capita Product and Income Series in Current and Chained Dollars,” published by the Bureau of Economic Analysis (BEA) as our measure of real consumption. These values are reported in chained 2009 U.S. dollars and constructed using mid-year population data. We construct the dividend growth rate series by first computing the gross dividend level from the value-weighted returns including and excluding dividends and lagged index levels. We then obtain the real dividend growth rate by deflating the nominal growth rate.

Table 1 presents the summary statistics of the data used in estimation. The |$p$|-values of the Jarque and Bera (1980) test of normality imply that the assumption of normality is not rejected for the consumption growth series, but it is rejected for other variables. Real equity returns, interest rates, and dividend growth rates all exhibit negative skewness. In addition, both real interest rates and dividend growth rates show significant excess kurtosis. Figure 2 plots the data.

Figure 2

Time series of variables

The figure shows CRSP value-weighted index returns, one-year Treasury bill rates, excess returns, per-capita log consumption growth, and log dividend growth rates for the 1941–2015 period. All series plotted are at an annual frequency and in real terms. Shaded areas represent NBER recessions.

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2.2 GSM: Estimation of the structural model

We use the Bayesian method proposed by Gallant and McCulloch (2009), which they termed general scientific models (GSM) to estimate the asset pricing models. The GSM methodology was refined in Aldrich and Gallant (2011), abbreviated AG hereafter.⁸ The discussion here incorporates those refinements and is to a considerable extent a paraphrase of AG. GSM is a Bayesian simulation estimator. It is useful when a computationally tractable likelihood function is not available, data are sparse, but the structural model can be solved and simulated. It shares certain similarities with the classical “indirect inference” and “efficient method of moments” (hereafter, EMM) methods introduced by Gouriéroux, Monfort, and Renault (1993) and Gallant and Tauchen (1996, 1998, 2010). These are simulation-based inference methods that rely on an auxiliary model for implementation. The GSM method relies on the theoretical results of Gallant and Long (1997) in its construction of a likelihood. In particular, Gallant and McCulloch synthesize a likelihood by means of an auxiliary model and simulations from the structural model. A comparison of AG with Bansal, Gallant, and Tauchen (2007) displays the advantages of a Bayesian simulation approach relative to a frequentist EMM approach, particularly for the purpose of model comparison. GSM is an appropriate estimation methodology in the context of this study since the estimated equilibrium model is highly nonlinear and does not admit analytically tractable solutions, thereby severely inhibiting accurate numerical construction of a likelihood by means other than GSM.

GSM uses a sieve specially tailored to macroeconomic and financial time-series applications as the auxiliary model. When a suitable sieve is used as the auxiliary model, as in this study, the GSM method synthesizes the exact likelihood implied by the model.⁹ In this instance, the synthesized likelihood model departs significantly from a normal-errors likelihood, which suggests that alternative econometric methods based on normal approximations will give biased results. In particular, in addition to the generalized autoregressive conditional heteroscedasticity (GARCH) effect, the four-dimensional error distribution implied by the smooth ambiguity model is skewed in all four components and has fat tails for consumption growth, dividend growth, and stock returns, and thin tails for bond returns. Implementing GSM requires fitting the data with an over-parameterized auxiliary model (not rooted in theory) and then recovering parameter estimates from the structural model (founded on theory) by computing the mapping linking the parameter spaces of these two models.

The difficulty in implementing GM’s proposal is to compute the implied map |$g$| accurately enough that the accept/reject decision in an MCMC chain (step 5 in the algorithm for the |$\theta$| chain) is correct when |$f$| is a nonlinear model. The algorithm proposed by AG to address this difficulty is described next.

We use |$N=1,000$| in the estimation of all the six models. Results (posterior means, posterior standard deviations, etc.) are not sensitive to |$N$|⁠; doubling |$N$| makes no difference other than doubling computational time. It is essential that the same seed of the random number generator be used to start these simulations so that the same |$\theta$| always produces the same simulation.

GM run a Markov chain |$\{\omega_t\}^K_{t=1}$| of length |$K$| to compute |$\hat \omega$| that solves Expression (7). There are two other Markov chains discussed later and so this chain is called the |$\omega$|-subchain to distinguish among them. While the |$\omega$|-subchain must be run to provide the scaling for the model assessment method proposed by GM, the |$\hat \omega$| that corresponds to the maximum of |$\sum_{t=1}^{N} \log f(\hat y_t \,|\, \hat z_{t-1},\omega)$| over the |$\omega$|-subchain is not a sufficiently accurate evaluation of |$g(\theta)$| for our auxiliary model. This is mainly because our auxiliary model is a multivariate GARCH specification of Bollerslev (1986) that Engle and Kroner (1995) call BEKK. Likelihoods incorporating BEKK are notoriously difficult to optimize. AG use |$\hat \omega$| as a starting value and maximize Expression (7) using the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) algorithm; see Fletcher (1987). This is also not a sufficiently accurate evaluation of |$g(\theta)$|⁠. A second refinement is necessary. The second refinement is embedded within the MCMC chain |$\{\theta_t\}^H_{t-1}$| of length |$H$| that is used to compute the posterior distribution of |$\theta$|⁠. It is called the |$\theta$|-chain. The |$\theta$|-chain is generated using the Metropolis algorithm. The Metropolis algorithm is an iterative scheme that generates a Markov chain whose stationary distribution is the posterior of |$\theta$|⁠. To implement it, we require a likelihood, a prior, and transition density in |$\theta$| called the proposal density. The likelihood is Equation (6) and the prior, |$\xi(\theta)$|⁠, is described in Section 2.5.

The prior may require quantities computed from the simulation |$\{\hat{y}_t, \hat{z}_{t-1}\}^N_{t-1}$| that are used in computing Equation (6). In particular, quantities computed in this fashion can be viewed as the evaluation of a functional of the structural model of the form |$p(\cdot|\cdot, \theta)\mapsto \varrho,$| where |$\varrho\in\mathbf{P}$| and |$\mathbf{P}$| is the space of functionals of the form |$\theta\mapsto p(\cdot|\cdot, \theta)\mapsto\varrho$|⁠. Thus, the prior is a function of the form |$\xi(\theta, \varrho)$|⁠. But since the functional |$\varrho$| is a composite function with |$\theta\mapsto p(\cdot|\cdot, \theta)\mapsto\varrho$|⁠, |$\xi(\theta,\varrho)$| is essentially a function of |$\theta$| alone. Thus, we only use |$\xi(\theta,\varrho)$| notation when attention to the subsidiary computation |$p(\cdot|\cdot, \theta)\mapsto\varrho$| is required.

Let |$q$| denote the proposal density. For a given |$\theta$|⁠, |$q(\theta,\theta^*)$| defines a distribution of potential new values |$\theta^*$|⁠. We use a move-one-at-a-time, random-walk, proposal density that puts its mass on discrete, separated points, proportional to a normal density. Two aspects of the proposal scheme are worth noting. The first is that the wider the separation between the points in the support of |$q$|⁠, the less accurately |$g(\theta)$| needs to be computed for |$\alpha$| at step 5 of the algorithm for the |$\theta$|-chain to be correct. A practical constraint is that the separation cannot be much more than a standard deviation of the proposal density or the chain will eventually stick at some value of |$\theta$|⁠. Our separations are typically 1/2 of a standard deviation of the proposal density. In turn, the standard deviations of the proposal density are typically no more than the standard deviations of the prior distributions of structural parameters shown in Tables 3 to 8 and no less than one order of magnitude smaller. The second aspect worth noting is that the prior is putting mass on these discrete points in proportion to |$\xi(\theta)$|⁠. Because one does not have to normalize either the likelihood or the prior in an MCMC chain, normalization of densities does not matter for the computation of the chain, and similarly for the joint distribution |$f(y|z,g(\theta))\xi(\theta)$| considered as a function of |$\theta$|⁠. However, |$f(y|z,\omega)$| must be normalized such that |$\int\!f(y|x,\omega)\,dy = 1$| to ensure that the implied map expressed in Equation (7) is computed correctly.

The algorithm for the |$\theta$|-chain is as follows. Given a current |$\theta^o$| and the corresponding |$\omega^o = g(\theta^o)$|⁠, we obtain the next pair |$(\theta^{\,\prime},\omega^{\,\prime})$| as follows:

1. Draw |$\theta^*$| according to |$q(\theta^o,\theta^*)$|⁠.

2. Draw |$\{\hat y_t, \hat z_{t-1}\}_{t=1}^N$| according to |$p(y_t | z_{t-1}, \theta^*)$|⁠.

3. Compute |$\zeta^* = g(\theta^*)$| and the functional |$\varrho^*$| from the simulation |$\{\hat y_t, \hat z_{t-1}\}_{t=1}^N$|⁠.

4. Compute |$\alpha = \min\left(1,\frac {\mathcal{L}(\theta^*)\,\xi(\theta^*,\varrho^*)\,q(\theta^*,\,\theta^o)}{\mathcal{L}(\theta^o)\,\xi(\theta^o,\varrho^o)\,q(\theta^o,\theta^*)}\right)$|⁠.

5. With probability |$\alpha$|⁠, set |$(\theta^{\,\prime},\omega^{\,\prime}) = (\theta^*,\omega^*)$|⁠, otherwise set |$(\theta^\prime,\omega^{\,\prime}) = (\theta^o,\omega^o)$|⁠.

2.3 GSM: Estimation of the auxiliary model

The observed data are |$y_{t}$| for |$t=1,\ldots,n$|⁠, where |$y_{t}$| is a vector of dimension |$M$|⁠. The vector of observable variables used in estimation has four components: real equity returns, real interest rates, real per capita consumption growth rates, and real dividend growth rates. The symbols |$P, Q$|⁠, |$V$|⁠, etc. that appear in this section are general vectors (matrices) of statistical parameters and are not instances of the model parameters or functionals in Section 1.

The density |$h(\varepsilon)$| of the i.i.d. |$\varepsilon_{t}$| is the square of a Hermite polynomial times a normal density, the idea being that the class of such |$h$| is dense in Hellinger norm and can therefore approximate a density to within arbitrary accuracy in the Kullback-Leibler distance; see Gallant and Nychka (1987). Such approximations are often called sieves; Gallant and Nychka term this particular sieve semi-nonparametric or SNP.¹² The density |$h(\varepsilon)$| is the normal when the degree of the Hermite polynomial is zero. In addition, the constant term of the Hermite polynomial can be a linear function of |$z_{t-1}$|⁠. This has the effect of adding a nonlinear term to the location function in Equation (8) and the variance function in Equation (9). It also causes the higher moments of |$h(\varepsilon)$| to depend on |$z_{t-1}$| as well. The SNP auxiliary model is determined statistically by adding terms as indicated by the BIC protocol for selecting the terms that constitute a sieve; see Schwarz (1978).

In our specification, |$U_{0}$| is an upper triangular matrix, |$P$| and |$\tilde{V}$| are diagonal matrices, and |$Q$| is scalar. The degree of the SNP |$h(\varepsilon)$| density is four. The auxiliary model chosen for our analysis, based on the BIC, has one lag in the conditional mean component, one lag in each of ARCH and GARCH terms. Although the univariate analysis of stock price dynamics generally incorporates a leverage term, we find in our SNP estimation with four variables that this term is not necessary according to the BIC.

The auxiliary model in the SNP estimation has 51 parameters, of which 50 are estimated and one determined by a normalization rule. The error distributions implied by the auxiliary model differ significantly from the distributions of innovation shocks assumed in those structural models in Section 1. We numerically solve the structural models assuming normally distributed innovation shocks to consumption and dividend growth rates, as discussed in the Online Appendix. The error distributions of simulations from these models are markedly non-Gaussian. For example, in addition to GARCH effects, the four-dimensional error distribution implied by the AAMS model is skewed in all four components and has fat tails for consumption growth, dividend growth, and stock returns and thin tails for bond returns.

2.4 Relative model comparison

Relative model comparison is standard Bayesian inference. The posterior probabilities of the six structural models may be computed using the Newton and Raftery (1994),|${\hat p}^4$| method for computing the marginal likelihood from an MCMC chain when assigning equal prior probability to each model. An alternative is method |$f_5$| of Gamerman and Lopes (2006), Section 7.2.1. The advantage of these methods is that knowledge of the normalizing constants of the likelihood |$\mathcal{L}(\theta)$| and the prior |$\xi(\theta)$| are not required. We do not know these normalizing constants due to the imposition of support conditions. It is important, however, that the auxiliary model be the same for all models. Otherwise the normalizing constant of |$\mathcal{L}(\theta)$| would be required. One divides the marginal density for each model by the sum for all models to get the posterior probabilities for relative model assessment.

Unfortunately, these and similar methods require that the range of the likelihoods that occur in the MCMC be within the float limits of the computing equipment employed. This can be remedied by deleting draws with exceedingly small computed likelihood, which can be interpreted as a modification to the prior. However, not only is it hard to interpret a truncation prior of this sort, but also we found that the implied ordering of the models is sensitive to the truncation for both the |${\hat p}^4$| and |$f_5$| methods. Therefore, in the results reported in the next section we used the BIC for model selection.

2.5 The prior and its support

The prior support of the subjective discount factor (⁠|$\beta$|⁠), the coefficient of risk aversion (⁠|$\gamma$|⁠), and the EIS (⁠|$\psi$|⁠) parameter are set to |$0.9<\beta<0.995$|⁠, |$0.1<\gamma<100$|⁠, and |$0.1<\psi<10$|⁠, respectively. The subjective discount factor must be high enough to imply a reasonably low risk-free rate. The range |$0.9<\beta<0.995$| is wide compared with the prior on this parameter in Schorfheide, Song, and Yaron (2018). The support interval for |$\gamma$| that we use is much wider than the reasonable range |$1<\gamma<10$| suggested by Mehra and Prescott (1985). Different from calibration studies on long-run risks, we do not impose |$\psi>1$| but allow for the possibility of |$\psi<1$| and thus a preference for late resolution of uncertainty. For the ambiguity aversion parameter |$\eta$|⁠, the support interval is |$\gamma<\eta<200$|⁠. Again, this interval is wide given calibrated studies such as Ju and Miao (2012) and Jahan-Parvar and Liu (2014). Because the agent is ambiguity averse when |$\eta>\gamma$|⁠, we impose this condition in estimating models with smooth ambiguity utility. The location parameters for |$\beta, \gamma, \psi$|⁠, and |$\eta$| in the prior are set at values consistent with the extant calibration studies. The scale parameters for these preference parameters are set to large values to deliver loose priors.

For the EZMS, EZMSTV, AAMS, and AAMSTV models, we use the parameter estimates and the associated standard errors reported in Cecchetti, Lam, and Mark (2000) to determine the location and scale parameter values for parameters |$\mu_h, \mu_l, \sigma_{c}, p_{hh}$|⁠, and |$p_{ll}$| in the Markov-switching model of consumption growth. In the AAMSTV model with time-varying volatility, our parameter choices for location and scale of |$p^{\sigma}_{hh}$|⁠, |$p^{\sigma}_{ll}, \sigma_h$|⁠, and |$\sigma_l$| rely on estimates of Lettau, Ludvigson, and Wachter (2008). The location values of the dividend volatility parameter |$\sigma_{d}$| and the leverage parameter |$\lambda$| are determined by the calibration of Ju and Miao (2012). Following Abel (1999), we impose |$\lambda\geq1$| in the estimation. Estimation results of Bansal, Gallant, and Tauchen (2007), Aldrich and Gallant (2011), and Schorfheide, Song, and Yaron (2018) lead to values of |$\lambda$| in the |$[1.5, 4.5]$| range. We choose |$1\leq\lambda\leq6$| as the support interval.

For the AALRRSV and EZLRRSV models, we use the calibrated parameter values in Bansal, Kiku, and Yaron (2012) and priors postulated in Schorfheide, Song, and Yaron (2018) to choose the location and scale parameter values, and support intervals as well. For example, the location of the unconditional mean of consumption growth, |$\mu_c$|⁠, is set at 0.02 with a small scale parameter value. The location of the persistence parameter of the long-run risk component, |$\rho_{x}$|⁠, is set at 0.95 with a large scale parameter value of 0.2. The support interval for |$\rho_{x}$| is |$-0.99<\rho_{x}<0.99$|⁠. Similarly, other model parameters also have loose priors and wide support intervals as in Schorfheide, Song, and Yaron (2018).

2.6 Estimation results

We summarize estimation results in Tables 3 to 8.¹³ We plot the prior and posterior densities of the estimated structural parameters in Figures 3 to 8. These figures show considerable shifts in both location and scale between priors and posteriors, suggesting that the estimation procedure and data have a significant impact on estimation results. The impact of priors and support conditions is notable, but of second order of importance.

Figure 3

Prior and posterior densities of parameters of the AAMS model

This figure plots prior and posterior densities of parameters in the AAMS model. The solid lines depict posterior densities, and the dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015.

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Figure 4

Prior and posterior densities of parameters of the AAMSTV model

This figure plots prior and posterior densities of parameters in the AAMSTV model. The solid lines depict posterior densities, and the dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015.

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Figure 5

Prior and posterior densities of parameters of the AALRRSV model

This figure plots prior and posterior densities of parameters in the AALRRSV model. The solid lines depict posterior densities, and the dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015.

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Figure 6

Prior and posterior densities of parameters of the EZMS model

This figure plots prior and posterior densities of parameters in the EZMS model. The solid lines depict posterior densities, and the dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015.

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Figure 7

Prior and posterior densities of parameters of the EZMSTV model

This figure plots prior and posterior densities of parameters in the EZMSTV model. The solid lines depict posterior densities, and the dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015.

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Figure 8

Prior and posterior densities of parameters of the EZLRRSV model

This figure plots prior and posterior densities of parameters in the EZLRRSV model. The solid lines depict posterior densities, and the dotted lines depict prior densities. The results are based on the U.S. annual data for 1941–2015.

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Estimation results show that the posterior estimates of |$\beta$| are tightly bounded in all models and generally imply low risk-free rates. There is an ongoing debate about the value of the EIS parameter (⁠|$\psi$|⁠) in the macrofinance literature. This parameter is crucial for equilibrium asset pricing models to match macroeconomic and financial moments in the data, see; Bansal and Yaron (2004), Croce (2014), and Liu and Miao (2015), among others. Our estimation strongly suggests an EIS greater than 1 and thus a preference for early resolution of uncertainty. As shown in Tables 3 to 8, the posterior mean, median, and 90% credible intervals of |$\psi$| estimates are uniformly above 1 for all models presented in Section 1. The plots of the posterior density for |$\psi$| in Figures 3 to 8 also reveal that the posterior dispersion of this parameter over the MCMC chain is small. Jeong, Kim, and Park (2015) estimate the recursive multiple-prior utility model using asset prices data and obtain implausibly high estimates of |$\psi$| that are greater than 10. High estimates of |$\psi$| generated from our estimation imply low and stable risk-free rates (see Section 3). In a DSGE analysis with a broader scope, Bianchi, Ilut, and Schneider (2018) rely on the mechanism of time-varying ambiguity on operating costs to ease the tension between excess equity volatility and smooth risk-free rates.

The posterior estimates of |$\psi$| for the AAMS and EZMS models are high and comparable to the estimates in the long-run risk literature. The posterior mean, median, 5th, and 95th percentiles of |$\psi$| estimates are moderately higher in the EZMS model than in the AAMS model, with the posterior mean and median being above 2. The |$\psi$| estimates in the EZMS model are close to results obtained by Schorfheide, Song, and Yaron (2018) and Bansal, Kiku, and Yaron (2016). Our estimation results suggest that incorporating ambiguity in the model leads to lower estimates of |$\psi$|⁠. This is also evident from a comparison of estimates in the EZMSTV and EZLRRSV models and those in the AAMSTV and AALRRSV models. The posterior estimates of |$\psi$| are moderately lower in the AAMSTV model, and significantly lower in the AALRRSV than in the EZLRRSV model. Nevertheless, our estimates of |$\psi$| in the long-run risk model are still lower than those reported by Schorfheide, Song, and Yaron (2018). The discrepancy arises because (i) we use the projection method rather than log-linear approximation, (ii) we use the GSM Bayesian method for model estimation, and (iii) we use a different data sample.

Our estimation results strongly support asset pricing models with smooth ambiguity preferences. The posterior estimates of the ambiguity aversion parameter |$\eta$| are significantly large in the AAMS, AAMSTV, and AALRRSV models. Not surprisingly, the estimates obtained for the AAMS model are close to the calibrated value in Ju and Miao (2012) (⁠|$\eta=8.864$|⁠). The estimates of |$\eta$| are modestly higher when regime-switching volatility in consumption growth is incorporated in the estimation. We observe that the posterior mean and median of |$\eta$| are about 10 in the AAMSTV model but only about 7 in the AAMS model. It is important to note that the difference of |$\eta$| estimates in the two models does not stem from a moments-matching exercise. The GSM Bayesian estimation delivers estimates of parameters in the utility function and consumption growth process jointly. In comparison with results for the AAMS model, the moderately higher estimates of |$\eta$| in the AAMSTV model are likely due to lower estimates of the transition probability |$p^{\mu}_{ll}$|⁠, which implies less persistence of the contraction regime for this model.¹⁴ In the long-run risk setting, the GSM Bayesian estimation generates high posterior estimates of |$\eta$| with mean and median of about 23. These results suggest that empirical support for models with smooth ambiguity is robust to different specifications of consumption dynamics and that the extent of ambiguity aversion largely depend on other preference parameters and primitive parameters in the consumption and dividend growth processes. While the estimated degree of ambiguity aversion varies across several models, these estimates are all reasonable from the perspective of decision-making. One could conduct thought experiments as in Halevy (2007) and Ju and Miao (2012) to gauge reasonable values of the ambiguity aversion parameter.

Estimates of the coefficients of risk aversion (⁠|$\gamma$|⁠) importantly hinge on the presence of ambiguity aversion. Estimation results for the EZMS, EZMSTV, and EZLRRSV models show that the posterior mean and median of |$\gamma$| are high and the 5th and 95th percentiles imply fairly tight bounds for the estimate. In particular, the posterior estimates of |$\gamma$| in the estimated long-run risk model EZLRRSV are close to the results reported by Schorfheide, Song, and Yaron (2018) and Bansal, Kiku, and Yaron (2016). The posterior mean of |$\gamma$| is 8.4, and the associated 95th percentile is 10.4. These values are also close to the calibrated values in Bansal and Yaron (2004) and Bansal, Kiku, and Yaron (2012). On the other hand, the |$\gamma$| estimate is more dispersed in models with smooth ambiguity, that is, the AAMS, AAMSTV, and AALRRSV models, as is evident from slightly wider 90% credible intervals. In a related work, X. Chen, Favilukis, and Ludvigson (2013) estimate preference parameters of recursive utility using a semiparametric technique. Their estimated relative risk-aversion parameter ranges from 17 to 60.

In the GSM Bayesian estimation, primitive parameters in the consumption and dividend growth processes are jointly estimated with preference parameters. The AAMS, AAMSTV, EZMS, and EZMSTV models have Markov-switching consumption growth while models AALRRSV and EZLRRSV feature long-run risks. In the Markov-switching environment, our estimation method identifies a normal regime and a contraction regime for mean consumption growth. The posterior estimates of |$\mu_h$| are largely in line with the historical average annual consumption growth. For instance, the posterior mean and median of |$\mu_h$| in the AAMS model are about 2%. In addition, the posterior estimates of the transition probability |$p_{hh}$| (⁠|$p^{\mu}_{hh}$| in the AAMSTV and EZMSTV models) are close to 1 and thus indicate that this regime is very persistent. Furthermore, the estimates of low mean growth regime for these models indicate a relatively transitory contraction regime with lower estimates of the transition probability |$p_{ll}$| (⁠|$p^{\mu}_{ll}$| in the AAMSTV and EZMSTV models).

Note that we obtain these estimates from structural estimation of asset pricing models using data on both fundamentals and asset returns. The GSM Bayesian estimation takes into account equilibrium asset prices and yields estimated consumption dynamics that correspond to the agent’s subjective belief. Compared with estimates of the parameters of the Markov-switching model reported by calibration studies (for example, Cecchetti, Lam, and Mark (2000) and Ju and Miao (2012)), our estimates imply a “peso” version of the model. That is, the severe contraction state rarely realizes in the observed data or simulations due to its low likelihood (⁠|$1-p_{hh}$|⁠) implied by our estimation. However, because an agent cannot observe the mean growth state and is also aware of severity (⁠|$\mu_l$|⁠) and persistence (⁠|$p_{ll}$|⁠) of the contraction regime, the agent is always concerned about state uncertainty, and moreover, ambiguity aversion magnifies the impact of this concern. In addition, the posterior estimates of the low mean regime |$\mu_l$| seem low given the postwar economy, and the estimated persistence of this regime varies significantly across different models. These results suggest that apart from ambiguity on the mean growth state, additional sources of ambiguity about parameters of the Markov-switching model may coexist.

In estimating the AAMSTV and EZMSTV models, we find two distinct volatility regimes, both of which are persistent. This result is consistent with the findings of Lettau, Ludvigson, and Wachter (2008). However, the posterior estimates of the high volatility regime |$\sigma_{h}$| are somewhat high compared with the postwar consumption data. The estimates of |$\mu_l$| are more negative than the estimates for the AAMS model. Nevertheless, these estimates are consistent with the long sample of Shiller’s data.¹⁵ Again, additional sources of ambiguity may arise due to learning from the past experience or parameter uncertainty.¹⁶ For the AAMS, AAMSTV, and EZMSTV models, the leverage parameter |$\lambda$| and the dividend growth volatility |$\sigma_{d}$| estimates are reasonably close to the calibrated values considered by Abel (1999), Bansal and Yaron (2004), and Ju and Miao (2012). The posterior estimates of |$\lambda$| are roughly between 1.5 and 4, with a posterior mean of about 3 for the AAMS and AAMSTV models, and around 2 for the EZMSTV model. The estimates of |$\lambda$| and |$\sigma_{d}$| for the EZMS model are higher than those for the AAMS, AAMSTV, and EZMSTV models.