Measuring Ambiguity Aversion
Abstract:
1 Introduction
In this paper, we confront the smooth ambiguity aversion model of Klibanoff, et al. (2005,2009), (henceforth, KMM), in its generalized form advanced by Hayashi & Miao (2011) and Ju & Miao (2012), with data to close two existing gaps in the literature. First, we use macroeconomic and financial data to estimate the size of ambiguity aversion together with other structural parameters in a representative-agent consumption-based asset pricing model with smooth ambiguity aversion preferences. Second, based on the estimated model, we investigate asset pricing implications of smooth ambiguity aversion. Given the rising popularity of smooth ambiguity preferences in economics and finance, it is important to characterize this model's empirical strengths as well as its shortcomings. One crucial feature of smooth ambiguity aversion is the separation of ambiguity and ambiguity aversion, where the former is a characteristic of the representative agent's subjective beliefs, while the latter derives from the agent's tastes. This study provides a fully market data-based estimation of a dynamic asset pricing model with these preferences. Our structural estimation results suggest that ambiguity aversion is important in matching salient features of asset returns in the U.S. data. Our study shows that ignoring ambiguity aversion in estimation of structural models of financial data leads to inadequate characterization of the market dynamics.
The benchmark asset pricing model that we adopt in the estimation is the model developed by Ju & Miao (2012). In this model, aggregate consumption growth follows a Markov switching process with an unobservable state. Mean growth rates of consumption depend on the state. The agent can learn about the state through observing the past consumption data. Ambiguity arises in that the agent may find it difficult to form forecasts of the mean growth rate. Because the underlying state evolves according to a Markov chain, learning cannot resolve this ambiguity over time. The agent is not only risk averse in the usual sense but also ambiguity averse in that he dislikes a mean-preserving-spread in the continuation value implied by the agent's belief about the unobservable state. As a result, compared with a risk-averse agent, the ambiguity-averse agent effectively assigns more weight to bad states that are associated with lower continuation value. Ju and Miao show that the utility function that permits a three-way separation among risk aversion, ambiguity aversion and the the intertemporal elasticity of substitution (IES) is successful in matching moments of asset returns in the U.S. data. Throughout the paper, we call the model without ambiguity aversion "alternative" model. In this alternative model, the representative agent is endowed with Epstein & Zin (1989) recursive utility preferences.
Similar to other macro-finance applications, we face sparsity of data. As has become standard in the macro-finance empirical literature, we use prior information and a Bayesian estimation methodology to overcome data sparsity. Specifically, we use the "General Scientific Models" (henceforth, GSM) Bayesian estimation method developed by Gallant & McCulloch (2009). GSM is the Bayesian counterpart to the classical "indirect inference" and "efficient method of moments" (hereafter, EMM) methods introduced by Gouriéroux, et al. (1993) and Gallant & Tauchen (1996,2010,1998). These are simulation-based inference methods that rely on an auxiliary model for implementation. GSM follows the logic of the EMM variant of indirect inference and relies on the theoretical results of Gallant & Long (1997) in its construction of a likelihood. A comparison of Aldrich & Gallant (2011) with Bansal, et al. (2007) displays the advantages of a Bayesian EMM approach relative to a frequentist EMM approach, particularly for the purpose of model comparison. An indirect inference approach 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 (see Section 4) 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.1 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 GARCH and leverage effects, the three-dimensional error distribution implied by the smooth ambiguity aversion model is skewed in all three components and has fat-tails for consumption growth and stock returns and thin tails for bond returns.
Our GSM Bayesian estimation suggests that estimates of the ambiguity aversion parameter are large and statistically significant. Ambiguity aversion in the estimated benchmark model, to a great extent, explains the high market price of risk implied by equity returns data and generates high equity premium. Ignoring ambiguity aversion leads to biased estimates of the risk aversion parameter. In addition, our estimates for the IES parameter are significantly greater than one. Our estimates for the IES parameter provide support for one of the main predictions of the long-run risk theory. According to the long-run risk literature, a high IES together with a moderate risk aversion coefficient imply that the agent prefers earlier resolution of uncertainty. We find that this demand for early resolution of uncertainty is robust to inclusion of ambiguity aversion, different model specifications, and data samples. Apart from estimating preference parameters, our GSM Bayesian estimation of the asset pricing model with learning indicates two distinct regimes for the mean growth rate of aggregate consumption, where the good regime is persistent while the bad regime is transitory. This result is consistent with many calibration studies using Markov switching models for consumption growth, for example, see Veronesi (1999) and Cecchetti, et al. (2000).
Related Literature: Two types of ambiguity preferences garner considerable attention in the literature: smooth ambiguity utility of KMM and multiple priors utility of Chen & Epstein (2002) (henceforth, MPU). In the multiple priors framework, the set of priors, which characterizes ambiguity (uncertain beliefs), also determines the degree of ambiguity aversion. However, smooth ambiguity preferences achieve the separation between ambiguity and ambiguity aversion. Thus, it is feasible to do comparative statics analysis by holding the set of relevant probability distributions constant while varying the degree of ambiguity aversion. Furthermore, asset pricing models with MPU are generally difficult to solve with refined processes of fundamentals because MPU features kinked preferences. In comparison with MPU, models with smooth ambiguity preferences are tractable in a wide range of applications.2
Klibanoff et al. (2005,2009) first introduced smooth ambiguity preferences. Hayashi & Miao (2011) generalized the preferences by disentangling risk aversion and the IES. Applications include endowment economy asset pricing (Ju & Miao (2012), Ruffino (2013), and Collard, et al. (2015)), production-based asset pricing (Jahan-Parvar & Liu (2014) and Backus, et al. (2015)), and portfolio choice (Gollier (2011), Maccheroni, et al. (2013), Chen, et al. (2014), and Guidolin & Liu (2014)), among others. These studies typically rely on calibration to examine impacts of ambiguity aversion. Popular calibration methods include the "detection-error probability" method of Anderson, et al. (2003) and Hansen (2007) (see Jahan-Parvar & Liu (2014) for an application to smooth ambiguity utility) and "thought experiments" similar to Halevy (2007) (see Ju & Miao (2012) and Chen et al. (2014) for applications). Structural estimation of dynamic models with ambiguity is still rare in the literature. To the best of our knowledge, our paper is the first to fully estimate a structural asset pricing model with smooth ambiguity utility.
A number of studies are closely related to ours. Jeong, et al. (2015) estimate an equilibrium asset pricing model where a representative agent has recursive MPU. Their estimation results suggest that fear of ambiguity on the true probability law governing fundamentals carries a premium. The ambiguity aversion parameter, which measures the size of the set of priors in the MPU framework, is both economically and statistically significant and remains stable across alternative specifications. Our paper is different from Jeong, et al. (2015) in two dimensions. First, we study smooth ambiguity utility, which enables us to obtain an estimate of ambiguity aversion as a preference parameter that clearly describes the agent's tastes, rather than beliefs. Second, our GSM method allows us to estimate preference parameters and parameters in the processes of fundamentals altogether. Park et al. employ a two-stage econometric methodology that first extracts the volatilities of market returns, consumption growth and labor income growth as latent factors and then estimates preference parameters and the magnitude of the set of priors.
Ilut & Schneider (2014) estimate a dynamic stochastic general equilibrium (DSGE) model where agents have MPU. Their estimation results suggest that time varying confidence in future total factor productivity explains a significant fraction of the business cycle fluctuations. Bianchi, et al. (2014) estimate a DSGE model with endogenous financial asset supply and ambiguity-averse agents. They show that time varying uncertainty about corporate profits explains high equity premium and excess volatility of equity prices observed in the U.S. data. Their estimated model can also replicate the joint dynamics of asset prices and real economic activity in the postwar data.
Empirical studies on reduced-form estimation of models with ambiguity aversion include Anderson, et al. (2009), Viale, et al. (2014), and Thimme & Völkert (2015). These papers show that ambiguity aversion is priced in the cross-section of expected returns. Anderson, et al. (2009) use survey of professional forecasts to construct the uncertainty measure and test model implications in the robust control framework. Viale, et al. (2014) rely on relative entropy to construct the ambiguity measure in the multiple priors setting. Fixing the IES at the calibrated value, Thimme & Völkert (2015) use the generalized method of moments (GMM) to estimate the ambiguity aversion parameter. Both Viale, et al. (2014) and Thimme & Völkert (2015) formulate the stochastic discount factor (SDF) under ambiguity using reduced-form regression methods. Ahn, et al. (2014) use experimental data to estimate ambiguity aversion in static portfolio choice settings.
The rest of the paper proceeds as follows. Section 2 describes the data used for estimation. Section 3 presents the consumption-based asset pricing model with generalized recursive smooth ambiguity preferences developed by Ju & Miao (2012). Section 4 discusses the estimation methodology and presents our empirical findings. Section 5 presents model comparison results, forecasts, and asset pricing implications. Section 6 concludes.
2 Data
Throughout this paper, lower case denotes the logarithm of an upper case quantity; e.g., $$ c_{t}=\ln (C_{t}),$$ where Ct is the observed consumption in period $$ t,$$ and $$ d_{t}=\ln (D_{t}),$$ where Dt 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 ( $$ r^e_{t}=\ln{(P^e_{t}+D_{t})}-\ln P^e_{t-1}$$ ) in the analysis, where $$ P^e_{t}$$ is the stock price in period t.
We use real annual data from 1929 to 2013 and real quarterly data from the second quarter of 1947 to the second quarter of 2014 for the purpose of inference respectively. For the annual (quarterly) sample, we use the sample period 1929-1949 (1947:Q2-1955:Q2) to provide initial lags for the recursive parts of our estimation and the sample period 1950-2013 (1955:Q3-2014:Q2) for estimation and diagnostics. Our measure for the risk-free rate is one-year U.S. Treasury Bill rates for annual data and 3-months U.S. Treasury Bill rates for quarterly data. Our proxy for risky asset returns is the value-weighted returns on CRSP-Compustat stock universe. We use the sum of nondurable and services consumption from Bureau of Economic Analysis (BEA) and deflate the series using the appropriate price deflator (also provided by the BEA). We use mid-year population data to obtain per capita consumption values.
As noted in Garner, et al. (2006) and Andreski, et al. (2014), there are notable discrepancies among measures of consumption based on collection methods used and released by different agencies. Thus, throughout the paper, we assume a 5% measurement error in the level of real per capita consumption.3 We assume a linear error structure. That is, $$ C_t = C^*_t + u_t$$ where Ct is the observed value, $$ C^*_t$$ is the true value, and ut is the measurement error term. We have $$ c_t = \ln(C_t)=\ln(C^*_t+u_t)$$ and $$ \Delta c_t =\ln(C_t/C_{t-1}).$$ Table 1 presents the summary statistics of samples. The p-values of Jarque & Bera (1980) test of normality imply that the assumption of normality is rejected for risk-free rate and log consumption growth series, but it cannot be rejected for aggregate market returns and excess returns at annual frequency. Annual data plots are shown in Figure 1.
3 The Model
The intuitive notions behind any consumption-based asset pricing model are that agents receive income (wage, interest, and dividends) which they use to purchase consumption goods. Agents reallocate their consumption over time by trading stocks that pay random dividends and bonds that pay interest with certainty. This is done for consumption smoothing over time (for example, insurance against unemployment, saving for retirement, $$ \cdots$$ ). The budget constraint implies that the purchase of consumption, bonds, and stocks cannot exceed income in any period. Agents are endowed with a utility function that depends on the entire consumption path. The first-order conditions of their utility maximization deliver an intertemporal relation of prices of stocks and bonds.
We consider the representative-agent model of Ju & Miao (2012) as our benchmark model. Among all tradable assets, we focus on the risky asset that pays aggregate dividends Dt and the one-period risk-free bond with zero net supply. Aggregate consumption follows the process
| $$\displaystyle \Delta c_{t+1}\equiv \ln \left( \frac{C_{t+1}}{C_{t}}\right) =\kappa _{z_{t+1}}+\sigma_{c} \epsilon_{t+1},$$ | (1) |
where $$ \epsilon _{t}$$ is an $$ i.i.d.$$ standard normal random variable, and $$ z_{t+1}$$ follows a two-state Markov chain with state 1 being the good state and state 2 being the bad state ( $$ \kappa_{1}>\kappa_{2}$$ ). The transition matrix is given by
| $$\displaystyle \mathbb{P}=\left[ \begin{array}{cc} p_{11} & 1-p_{11} \\ 1-p_{22} & p_{22}\end{array}\right],$$ |
where pij denotes the probability of switching from state i to state j.
Because aggregate dividends are more volatile than aggregate consumption, we model dividends and consumption separately, see Bansal & Yaron (2004). The dividend growth process is given by and an idiosyncratic component,
| $$\displaystyle \Delta d_{t+1}\equiv \ln \left( \frac{D_{t+1}}{D_{t}}\right) =\lambda \Delta c_{t+1}+g_{d}+\tilde{\sigma}_{d}\varepsilon_{d,t+1},$$ | (2) |
where $$ \varepsilon_{d,t+1}$$ is an $$ i.i.d.$$ standard normal random variable that is independent of all other shocks in the model. The parameter $$ \lambda $$ can be interpreted as the leverage ratio (see Abel (1999)). The parameters gd and $$ \sigma_{d}$$ can be pinned down by calibrating the process to the mean and volatility of dividend growth. We set the mean dividend growth rate to the unconditional mean of consumption growth implied by the Markov-switching model. In addition, we denote the volatility of dividend growth by $$ \sigma_{d}$$ and estimate this parameter using historical data on equity returns and the GSM Bayesian method.
The agent cannot observe the regimes of expected consumption growth but can learn about the state (zt) through observing the past consumption data. The agent also knows the parameters of the model, namely, $$ \left\{ \kappa _{1},\kappa _{2},p_{11},p_{22},\sigma_{c},\lambda ,g_{d},\sigma _{d}\right\}$$ . The agent updates beliefs $$ \mu _{t}=\Pr \left( z_{t+1}\vert\Omega _{t}\right)$$ according to Bayes' rule:
| $$\displaystyle \mu _{t+1}=\frac{p_{11}f\left( \Delta c_{t+1}\vert 1\right) \mu _{t}+(1-p_{22})f\left( \Delta c_{t+1}\vert 2\right) \left( 1-\mu _{t}\right) }{f\left( \Delta c_{t+1}\vert 1\right) \mu _{t}+f\left( \Delta c_{t+1}\vert 2\right) \left( 1-\mu _{t}\right) },$$ | (3) |
where $$ f\left( \Delta c_{t+1}\vert i\right) ,$$$$ i=1,2$$ is the normal density function of consumption growth conditional on state i.
The agent's preferences are represented by the generalized recursive smooth ambiguity utility function,
| $$\displaystyle V_{t}(C)$$ | $$\displaystyle =$$ | $$\displaystyle \left[ (1-\beta )C_{t}^{1-1/\psi }+\beta \left\{ \mathcal{R}_{t}\left( V_{t+1}\left( C\right) \right) \right\} ^{1-1/\psi }\right] ^{\frac{1}{1-1/\psi }},$$ | (4) |
| $$\displaystyle \mathcal{R}_{t}\left( V_{t+1}\left( C\right) \right)$$ | $$\displaystyle =$$ | $$\displaystyle \left( \mathbb{E}_{\mu _{t}}\left[ \left( \mathbb{E}_{z_{t+1},t}\left[ V_{t+1}^{1-\gamma }\left( C\right) \right] \right) ^{\frac{1-\eta }{1-\gamma }}\right] \right) ^{\frac{1}{1-\eta }},$$ | (5) |
where $$ \beta\in(0,1)$$ is the subjective discount factor, $$ \psi$$ is the IES parameter, $$ \gamma$$ is the coefficient of relative risk aversion, and $$ \eta$$ is the ambiguity aversion parameter and must satisfy $$ \eta>\gamma$$ to maintain ambiguity aversion in the utility function. Equation (5) characterizes the certainty equivalent of future continuation value, which is the key ingredient that distinguishes this utility function from Epstein & Zin (1989) recursive utility. In Equation (5), the expectation operator $$ \mathbb{E}_{z_{t+1},t}\left[ \cdot \right] $$ is with respect to the distribution of consumption conditioning on the next period's state $$ z_{t+1}$$ , and the expectation operator $$ \mathbb{E}_{\mu _{t}}$$ is with respect to the posterior beliefs about the unobservable state.
Under this utility function, the SDF is given by (see Hayashi & Miao (2011) for a derivation)
| $$\displaystyle M_{z_{t+1},t+1}=\beta \left( \frac{C_{t+1}}{C_{t}}\right) ^{-1/\psi }\left( \frac{V_{t+1}}{\mathcal{R}_{t}\left( V_{t+1}\right) }\right) ^{1/\psi -\gamma }\left( \frac{\left( \mathbb{E}_{z_{t+1},t}\left[ V_{t+1}^{1-\gamma }\right] \right) ^{\frac{1}{1-\gamma }}}{\mathcal{R}_{t}\left( V_{t+1}\right) }\right) ^{-\left( \eta -\gamma \right) }.$$ | (6) |
The last multiplicative term in Equation (6) is due to ambiguity aversion. It makes the SDF more countercyclical than in the case with Epstein-Zin's recursive preferences. Numerically, we can show that $$ M_{z_{t+1},t+1}$$ tends to be higher if $$ z_{t+1}$$ appears to be state 2 (the bad state). In addition, the last term in Equation (6) induces additional variation in the SDF (compared with Epstein-Zin SDF) and leads to a high market price of risk, defined as $$ \sigma(M)/\mathbb{E}(M)$$ .
Stock returns, defined by $$ R_{t+1}^{e}=\frac{P_{t+1}^{e}+D_{t+1}}{P_{t}^{e}}$$ , satisfy the Euler equation
The risk-free rate, $$ R_{f,t}$$ , is the reciprocal of the expectation of the SDF:
| $$\displaystyle R^f_{t}=\frac{1}{\mathbb{E}_{\mu_t, t}\left[ M_{z_{t+1},t+1}\right] }.$$ |
We can rewrite the Euler equation as
| $$\displaystyle 0=\tilde{\mu}_{t}\mathbb{E}_{1,t}\left[ M_{z_{t+1},t+1}^{EZ}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right] +\left( 1-\tilde{\mu}_{t}\right) \mathbb{E}_{2,t}\left[ M_{z_{t+1},t+1}^{EZ}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right] ,$$ |
where $$ M_{z_{t+1},t+1}^{EZ}$$ can be interpreted as the SDF under Epstein-Zin recursive utility:
| $$\displaystyle M_{z_{t+1},t+1}^{EZ}=\beta \left( \frac{C_{t+1}}{C_{t}}\right) ^{-\frac{1}{\psi}}\left( \frac{V_{t+1}}{\mathcal{R}_{t}\left( V_{t+1}\right) }\right) ^{\frac{1}{\psi} -\gamma },$$ |
and $$ \tilde{\mu}_{t}$$ can be interpreted as ambiguity distorted beliefs and represented by:
| $$\displaystyle \tilde{\mu}_t = \frac{\mu_t\left(\mathbb{E}_{1,t}\left[V^{1-\gamma}_{t+1}\right]\right)^{-\frac{\eta-\gamma}{1-\gamma}}}{\mu_t\left(\mathbb{E}_{1,t}\left[V^{1-\gamma}_{t+1}\right]\right)^{-\frac{\eta-\gamma}{1-\gamma}}+(1-\mu_t)\left(\mathbb{E}_{2,t}\left[V^{1-\gamma}_{t+1}\right]\right)^{-\frac{\eta-\gamma}{1-\gamma}}}.$$ | (8) |
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 5.3.
We follow Ju & Miao (2012) and use the projection method with Chebyshev polynomials to solve the model. The model has to be solved for each set of parameter values simulated in the GSM method. We did experiments to solve the model for a number of combinations of parameter values and found that the solution method is robust. Specifically, homogeneity in utility preferences implies $$ V_{t}\left( C\right) =G\left( \mu _{t}\right) C_{t}$$ , and $$ G\left( \mu _{t}\right)$$ satisfies the following functional equation
| $$\displaystyle G\left( \mu _{t}\right) =\left[ (1-\beta )+\beta \left( \mathbb{E}_{\mu _{t}}\left[ \left( \mathbb{E}_{z_{t+1},t}\left[ G\left( \mu _{t}\right) ^{1-\gamma }\left( \frac{C_{t+1}}{C_{t}}\right) ^{1-\gamma }\right] \right) ^{\frac{1-\eta }{1-\gamma }}\right] \right) ^{\frac{1-1/\psi }{1-\eta }}\right] ^{\frac{1}{1-1/\psi }}.$$ |
To solve for the value function, we approximate $$ G\left( \mu _{t}\right)$$ using Chebyshev polynomials in the state variable $$ \mu_t$$ . The approximation takes the form
| $$\displaystyle G\left( \mu \right) \simeq \sum_{k=0}^{p}\phi _{j}T_{j}\left( c\left( \mu \right) \right) ,$$ |
where p is the order of Chebyshev polynomials, Tj with j=1,...,p are Chebyshev polynomials, and $$ c\left( \mu \right)$$ maps the sate variable $$ \mu$$ onto the interval $$ \left[ -1,1\right]$$ . We then choose a set of collocation points for $$ \mu$$ and solve for the coefficients $$ \left\{ \phi _{j}\right\} _{j=0,...,p}$$ using a nonlinear equations solver. The expectation $$ \mathbb{E}_{z_{t+1},t}\left[ \cdot \right] $$ is approximated using Gauss-Hermite quadrature.
The equilibrium price-dividend ratio is a functional of the state variable, $$ \frac{P_{t}^{e}}{D_{t}}=\varphi \left( \mu _{t}\right)$$ . To solve for the price-dividend ratio, we rewrite the Euler equation as
| $$\displaystyle \varphi \left( \mu _{t}\right) =\mathbb{E}_{t}\left[ M_{z_{t+1},t+1}\left( 1+\varphi \left( \mu _{t+1}\right) \right) \frac{D_{t+1}}{D_{t}}\right] .$$ |
The price-dividend ratio can also be approximated using Chebyshev polynomials in $$ \mu_t$$ . Since the SDF $$ M_{z_{t+1},t+1}$$ can be easily written as a functional of $$ G\left( \mu _{t+1}\right)$$ and consumption growth $$ \Delta c_t = \ln\left(C_{t+1}/C_{t}\right)$$ , we can solve for the price-dividend ratio in a similar way as we solve for the value function. We simulate logarithmic values of consumption growth, stock returns and risk-free rates $$ \left\{ \Delta c_{t+1},r_{t+1}^{e},r_{t+1}^{f}\right\} _{t=1}^{T}.$$
If
$$ \eta =\gamma $$ , then the agent is ambiguity neutral and has the familiar Kreps & Porteus (1978) and Epstein & Zin (1989)
preferences:
| $$\displaystyle V_{t}(C)$$ | $$\displaystyle =$$ | $$\displaystyle \left[ (1-\beta )C_{t}^{1-1/\psi }+\beta \left\{ \mathcal{R}_{t}\left( V_{t+1}\left( C\right) \right) \right\} ^{1-1/\psi }\right] ^{\frac{1}{1-1/\psi }},$$ | |
| $$\displaystyle \mathcal{R}_{t}\left( V_{t+1}\left( C\right) \right)$$ | $$\displaystyle =$$ | $$\displaystyle \mathbb{E}_{t}\left[ V_{t+1}^{1-\gamma}\left( C\right) \right]^{\frac{1}{1-\gamma }}.$$ |
We consider this model as the alternative model for estimation. The model is solved and simulated using the projection method described above.
4 Estimation of Model Parameters
To estimate model parameters we use a Bayesian method proposed by Gallant & McCulloch (2009), abbreviated GM hereafter, that they termed General Statistical Models (GSM). The GSM methodology was refined in Aldrich & Gallant (2011), abbreviated AG hereafter. The discussion here incorporates those refinements and is to a considerable extent a paraphrase of AG. The symbols $$ \zeta$$ , $$ \theta$$ , etc. that appear in this section are general vectors of statistical parameters and are not instances of the model parameters of Section 3.
Let the transition density of a structural model be denoted by
where $$ x_{t-1}=(y_{t-1},\ldots,y_{t-L})$$ if Markovian and $$ x_{t-1}=(y_{t-1},\ldots,y_{1})$$ if not. As a result, xt-1 serves as a shorthand for lag-lengths that are generally greater than 1. Thus, transition densities may depend on L-lags of the data (if Markovian) or the entire history of observations (if non-Markovian). There are two structural models under consideration in this application: the benchmark model and the alternative model, described in Section 3.
We presume that there is no straightforward algorithm for computing the likelihood but that we can simulate data from $$ p(\cdot\vert\cdot,\theta)$$ for a given $$ \theta$$ . We presume that simulations from the structural model are ergodic. We assume that there is a transition density
and a map
such that
| $$\displaystyle p(y_t \vert x_{t-1}, \theta) = f(y_t \vert x_{t-1},g(\theta)), \quad \quad \theta \in \Theta.$$ | (12) |
We assume that $$ f(y_t \vert x_{t-1},\zeta)$$ and its gradient $$ (\partial / \partial \zeta) f(y_t \vert x_{t-1},\zeta)$$ are easy to evaluate. f is called the auxiliary model and g is called the implied map. When Equation (12) holds, f is said to nest p. Whenever we need the likelihood $$ \prod^n_{t=1}p(y_t\vert x_{t-1},\theta)$$ , we use
where $$ \{y_t,x_{t-1}\}^n_{t=1}$$ are the data and n is the sample size. After substituting $$ {\cal L}(\theta)$$ for $$ \prod^n_{t=1}p(y_t\vert x_{t-1},\theta)$$ , standard Bayesian MCMC methods become applicable. That is, we have a likelihood $$ {\cal L}(\theta)$$ from Equation (13) and a prior $$ \pi(\theta)$$ from Subsection 4.4 and need nothing beyond that to implement Bayesian methods by means of MCMC. A good introduction to these methods is Gamerman & Lopes (2006).
The difficulty in implementing GM's propsal is to compute the implied map g accurately enough that the accept/reject decision in an MCMC chain (Step 5 in the algorithm below) is correct when f is a nonlinear model. The algorithm proposed by AG to address this difficulty is described next.
Given $$ \theta$$ , $$ \zeta=g(\theta)$$ is computed by minimizing Kullback-Leibler divergence
| $$\displaystyle g: \theta \mapsto argmax\sum_{t=1}^{N} \log f(\hat y_t \,\vert\, \hat x_{t-1},\zeta).$$ | (14) |
We use N=1000 in the results reported below. Results (posterior mean, posterior standard deviation, etc.) are not sensitive to N; doubling N makes no difference other than doubling computational time. It is essential that the same seed be used to start these simulations so that the same $$ \theta$$ always produces the same simulation.
GM run a Markov chain $$ \{\zeta_t\}^K_{t=1}$$ of length K to compute $$ \hat \zeta$$ that solves expression (14). There are two other Markov chains discussed below so, to help distinguish among them, this chain is called the $$ \zeta$$ -subchain. While the $$ \zeta$$ -subchain must be run to provide the scaling for the model assessment method that GM propose, the $$ \hat \zeta$$ that corresponds to the maximum of $$ \sum_{t=1}^{N} \log f(\hat y_t \,\vert\, \hat x_{t-1},\zeta)$$ over the $$ \zeta$$ -subchain is not a sufficiently accurate evaluation of $$ g(\theta)$$ for our auxiliary model. This is mainly because our auxiliary model uses a multivariate specification of the generalized autoregressive conditional heteroscedasticity (GARCH) of Bollerslev (1986) that Engle & Kroner (1995) call BEKK. Likelihoods incorporating BEKK are notoriously difficult to optimize. AG use $$ \hat \zeta$$ as a starting value and maximize the expression (14) using the BFGS algorithm, see Fletcher (1987). This also is not a sufficiently accurate evaluation of $$ g(\theta)$$ . A second refinement is necessary. The second refinement is embedded within the MCMC chain $$ \{\theta_t\}^R_{t-1}$$ of length R that is used to compute the posterior distribution of $$ \theta$$ . It is called the $$ \theta$$ -chain. Its computation proceeds as follows.
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 (13) and the prior, $$ \pi(\theta)$$ , is described in Section 4.4.
The prior may require quantities computed from the simulation $$ \{\hat{y}_t, \hat{x}_{t-1}\}^N_{t-1}$$ that are used in computing Equation (13). In particular, quantities computed in this fashion can be viewed as the evaluation of a functional of the structural model of the form $$ p(\cdot\vert\cdot, \theta)\mapsto \varrho,$$ where $$ \varrho\in\mathbf{P}$$ . Thus, the prior is a function of the form $$ \pi(\theta, \varrho)$$ . But since the functional $$ \varrho$$ is a composite function with $$ \theta\mapsto p(\cdot\vert\cdot, \theta)\mapsto\varrho$$ , $$ \pi(\theta, \varrho)$$ is essentially a function of $$ \theta$$ alone. Thus, we only use $$ \pi(\theta, \varrho)$$ notation when attention to the subsidiary computation $$ p(\cdot\vert\cdot, \theta)\mapsto\varrho$$ 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. 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 below 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 in Table 2 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 $$ \pi(\theta)$$ . Because we never need to normalize $$ \pi(\theta)$$ this does not matter. Similarly for the joint distribution $$ f(y\vert x,g(\theta))\pi(\theta)$$ considered as a function of $$ \theta$$ . However, $$ f(y\vert x,\zeta)$$ must be normalized such that $$ \int\!f(y\vert x,\zeta)\,dy = 1$$ to ensure that the implied map expressed in (14) is computed correctly.
The algorithm for the $$ \theta$$ -chain is as follows. Given a current $$ \theta^o$$ and the corresponding $$ \zeta^o = g(\theta^o)$$ , obtain the next pair $$ (\theta^{\,\prime},\zeta^{\,\prime})$$ as follows:
- Draw $$ \theta^*$$ according to $$ q(\theta^o,\theta^*)$$ .
- Draw $$ \{\hat y_t, \hat x_{t-1}\}_{t=1}^N$$ according to $$ p(y_t \vert x_{t-1}, \theta^*)$$ .
- Compute $$ \zeta^* = g(\theta^*)$$ and the functional $$ \varrho^*$$ from the simulation $$ \{\hat y_t, \hat x_{t-1}\}_{t=1}^N$$ .
- Compute $$ \alpha = \min\left(1,\frac {\mathcal{L}(\theta^*)\,\pi(\theta^*,\varrho^*)\,q(\theta^*,\,\theta^o)} {\mathcal{L}(\theta^o)\,\pi(\theta^o,\varrho^o)\,q(\theta^o,\theta^*)}\right)$$ .
- With probability $$ \alpha$$ , set $$ (\theta^{\,\prime},\zeta^{\,\prime}) = (\theta^*,\zeta^*)$$ , otherwise set $$ (\theta^\prime,\zeta^{\,\prime}) = (\theta^o,\zeta^o)$$ .
It is at step 3 that AG made an important modification to the algorithm proposed by GM. At that point one has putative pairs $$ (\theta^*,\zeta^*)$$ and $$ (\theta^o,\zeta^o)$$ and corresponding simulations $$ \{\hat y^*_t, \hat x^*_{t-1}\}_{t=1}^N$$ and $$ \{\hat y^o_t, \hat x^o_{t-1}\}_{t=1}^N$$ . AG use $$ \zeta^*$$ as a start and recompute $$ \zeta^o$$ using the BFGS algorithm, obtaining $$ \hat \zeta^o$$ . If
4.1 Relative Model Comparison
Relative model comparison is standard Bayesian inference. The posterior probabilities of the models with and without ambiguity aversion are computed using the Newton & Raftery (1994) $$ {\hat p}^4$$ method for computing the marginal likelihood from an MCMC chain when assigning equal prior probability to each model. The advantage of that method is that knowledge of the normalizing constants of the likelihood $$ \mathcal{L}(\theta)$$ and the prior $$ \pi(\theta)$$ 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 both models. Otherwise the normalizing constant of $$ \mathcal{L}(\theta)$$ would be required. One divides the marginal density for each model by the sum for both models to get the probabilities for relative model assessment. Or, because we are only comparing two models, one can equally as well use the ratio of the two probabilities, i.e., the odds ratio.
4.2 Forecasts
A forecast is a functional $$ \Upsilon:f(\cdot\vert\cdot,\zeta)\mapsto$$ $$ \upsilon$$ of the auxiliary model that can be computed from $$ f(\cdot\vert\cdot,\zeta)$$ either analytically or by simulation. Due to the map $$ \zeta=g(\theta),$$ we view such a forecast as both a forecast from the structural model and as a function of $$ \theta$$ . Viewing it as a function of $$ \theta$$ , we can compute $$ \upsilon$$ at each draw in the posterior MCMC chain for $$ \theta$$ which results in an MCMC chain for $$ \upsilon$$ . From the latter chain the mean and standard deviation of $$ \upsilon$$ can be computed. The same quantities can be computed for draws from the prior. Examples are in Figure 6.
4.3 The Auxiliary Model
The observed data are yt for $$ t=1,\ldots,n$$ , where yt is a vector of dimension three in our application. We use the notation $$ x_{t-1}=\{y_{t-1},\cdots, y_{t-L}\}$$ , if the auxiliary model is Markovian, and $$ x_{t-1}=\{y_{t-1},\cdots, y_{1}\}$$ if it is not.4 Either way, xt-1 serves as a shorthand for lagged values of yt vector. In this application, the auxiliary model is not Markovian due to the recursion in expression (17). The data are modeled as
which is the location function of a one-lag vector auto-regressive (VAR) specification, and $$ R_{x_{t-1}}$$ is the Cholesky factor of
| $$\displaystyle \Sigma _{x_{t-1}}$$ | $$\displaystyle =$$ | $$\displaystyle R_{0}R^{\prime }_{0}$$ | (16) |
| $$\displaystyle + Q\Sigma _{x_{t-2}}Q^{\prime }$$ | (17) | ||
| $$\displaystyle + P(y_{t-1}-\mu _{x_{t-2}})(y_{t-1}-\mu _{x_{t-2}})^{\prime }P^{\prime }$$ | (18) | ||
| $$\displaystyle + \max [0,V(y_{t-1}-\mu _{x_{t-2}})]\max[0,V(y_{t-1}-\mu _{x_{t-2}})]^{\prime }.$$ | (19) |
In computations, $$ \max(0,x)$$ in expression (19), which is applied element-wise, is replaced by a twice differentiable cubic spline approximation that plots slightly above $$ \max(0,x)$$ over (0.00,0.10) and coincides elsewhere.
The density $$ h(z)$$ of the $$ i.i.d.~ z_{t}$$ is the square of a Hermite polynomial times a normal density, the idea being that the class of such h is dense in Hellenger norm and can therefore approximate a density to within arbitrary accuracy in Kullback-Leibler distance, see Gallant & Nychka (1987). Such approximations are often called sieves; Gallant and Nychka term this particular sieve semi-nonparametric maximum likelihood estimator, or SNP. The density $$ h(z)$$ 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 xt-1. This has the effect of adding a nonlinear term to the location function (15) and the variance function (16). It also causes the higher moments of $$ h(z)$$ to depend on xt-1 as well. The SNP auxiliary model is determined statistically by adding terms as indicated by the Bayesian information criteria (BIC) protocol for selecting the terms that comprise a sieve, see Schwarz (1978).
In our specification, R0 is an upper triangular matrix, P and V are diagonal matrices, and Q is scalar. The degree of the SNP $$ h(z)$$ density is four. The constant term of the SNP density does not depend on the past.
The auxiliary model chosen for our analysis, based on BIC, has 1 lag in the conditional mean component, 1 lag in each of autoregressive conditional heteroscedasticity (ARCH) and generalized autoregressive conditional heteroscedasticity (GARCH) terms. The model admits leverage effect in the ARCH term. The auxiliary model has 37 estimated parameters.
The implied error distributions in GSM estimation can differ significantly from the error shocks used for solving the structural model. For example, we numerically solve the two structural models in Sections 3 assuming normal distributions for error terms in Equations (1) and (2). This is a simplifying assumption to ease numerical solutions. The error distributions of simulations from these models are non-Gaussian. For example, in addition to GARCH and leverage effects, the three-dimensional error distribution implied by the benchmark smooth ambiguity aversion model is skewed in all three components and has fat-tails for consumption growth and stock returns and thin tails for bond returns.
The auxiliary model is determined from simulations of the structural model so issues of data sparsity do not arise; one can make the simulation length N as large as necessary to determine the parameters of the auxiliary model accurately. As stated above, we used N = 1,000 and found that using larger values of N did not change results other than increase run times.
4.4 The Prior and Its Support
Both the benchmark and alternative models are richly parameterized. The benchmark model has 11 structural parameters, given by
| $$\displaystyle \pi \left( \theta \right) =\prod\limits_{i=1}^{n}N\left[ \left. \theta _{i}\right\vert \left( \theta _{i}^{\ast },\sigma _{\theta }^{2}\right) \right]$$ |
where n denotes the number of parameters.
For annual data, the prior location parameters of the benchmark model are
We constrain the subjective discount factor $$ \beta$$ to be between 0.00 and 1.00. We bound the coefficient of risk aversion $$ \gamma$$ to be above 0.00 and below 15.00, in line with the recommendation of Mehra & Prescott (1985) that risk aversion should be moderate. Fully parameterized Kreps & Porteus (1978) and Epstein & Zin (1989) preferences imply a separation between risk aversion and the IES; therefore we impose $$ \gamma \neq 1/\psi$$ .5 Following the long-run risk literature (e.g., Bansal & Yaron (2004)), we impose $$ \psi>1.00$$ such that persistent movements in expected consumption growth are significantly priced in equity returns. The upper bound for $$ \psi$$ is set as $$ \psi<5.00$$ . Relaxing this bound has little impact on our estimation results.
We bound $$ \eta$$ between 2.00 and 100.00. We impose $$ \eta>\gamma$$ in the estimation of the benchmark model. Hayashi & Miao (2011) and Ju & Miao (2012) furnish detailed discussions of this requirement. Briefly, with $$ \eta =\gamma $$ , compound predictive probability distributions are reduced to an ordinary predictive probability distribution, removing ambiguity from the model. When estimating the alternative model, we impose the restriction $$ \eta =\gamma $$ to obtain ambiguity neutrality.
Following consumption-based asset pricing models (e.g., Abel (1999) and Bansal & Yaron (2004)) and based on empirical findings of Aldrich & Gallant (2011), we require positive leverage in the model. To this end, we impose $$ \lambda>1.00e{-7}$$ for the leverage parameter. For parameters in the consumption growth process, we impose $$ 0.93960<p_{11}< 0.99962$$ , $$ 0.2514<p_{22}<0.7806$$ , $$ 0.01596<\kappa_1<0.02906$$ , $$ -0.1055<\kappa_2<-0.0302$$ , and $$ 0.02646<\sigma_{c}<0.03608$$ . These bounds are adopted based on the parameter estimates and the associated standard errors in the Markov switching model for consumption growth, which are reported in Ju & Miao (2012) and Cecchetti, et al. (2000). The bound for the volatility of dividend growth is $$ 0.06542<\sigma_{d}< 0.1746$$ , set according to the estimate and standard error provided by Bansal & Yaron (2004). The prior information for annual estimation is summarized in the first three columns of Table 2.
For estimation based on quarterly data, we appropriately rescale the prior (the location and scale parameters, and the support conditions as well) in the annual estimation. The prior specifications for the preference parameters $$ \beta,\gamma$$ and $$ \psi$$ and for the leverage parameter $$ \lambda $$ remain unchanged. The location parameter for ambiguity aversion, $$ \eta$$ , is adjusted to yield a sizable equity premium when the benchmark model is evaluated at the rescaled location parameters. The prior information for quarterly estimation is summarized in the first three columns of Table 3.
Our prior specification and support conditions help the GSM Bayesian estimation identify parameter estimates of both benchmark and alternative models. As will be discussed below, even after combining our loose priors with support conditions, the estimation procedure and data are important for the identification of key parameters.
4.5 Empirical Results
We plot the prior and posterior densities of the structural parameters of the benchmark and alternative models in Figures 2-5. The plots show considerable shifts in both location and scale, suggesting that the estimation procedure and data have a strong influence on our estimation results. This observation is reassuring because an important concern in Bayesian estimation is identification of parameter estimates. In other words, one wants to know the relative contribution of priors and support conditions versus the contribution of the data. It is clear from Figures 2-5 that for almost all of the estimated parameters, the posterior densities shift significantly compared to the prior densities. Moreover, this observation is true for estimations of the benchmark and alternative models at both annual and quarterly frequencies. Our discussion below focuses on posterior densities in the annual estimation. The reader can apply the same logic and line of reasoning to quarterly estimation results.
Figure 2 reveals that the identification of the ambiguity aversion parameter $$ \eta$$ , which is the key preference parameter in the model, is strong in the annual estimation. Both the location and scale change dramatically as a consequence of the estimation procedure. In addition, it can be seen from Figures 2 and 3 that the identification of other preference parameters including $$ \beta,\gamma$$ and $$ \psi$$ also appears to be notable for both the benchmark and alternative models, though the posterior density of the $$ \gamma$$ estimate is moderately more dispersed in the benchmark model estimation.
Posterior densities of model parameters governing the dynamics of consumption and dividends also indicate that our estimation procedure has an important impact on the identification of those parameters, with the help of the priors and support conditions described in Section 4.4. The posterior densities of the estimated transition probabilities are more concentrated in the benchmark model estimation than in the alternative model estimation. We note that in Figure 2, the low mean growth rate of consumption, $$ \kappa_{2}$$ , has a very tight posterior density. This result is due to the inclusion of ambiguity aversion in the benchmark model. The ambiguity-averse agent distorts beliefs toward the bad regime. As a result, the low mean growth rate largely determines the impact of ambiguity aversion on the SDF and therefore equity premium. This feature of the model is manifested in the estimation results in that both the identification of $$ \eta$$ and that of $$ \kappa_{2}$$ are strong in the benchmark model estimation. By contrast, the posterior density of the high mean growth rate, $$ \kappa_{1}$$ , is tight in the estimation of the alternative model, as shown in Figure 3. Other parameters including the leverage parameter ($$ \lambda $$ ), the volatility of consumption growth innovation ( $$ \sigma_{c}$$ ) and the volatility of dividend growth ( $$ \sigma_{d}$$ ) have posterior densities significantly different from the corresponding prior densities.
The estimated moments of the model parameters are summarized in Table 2 and Table 3, for the annual and quarterly samples, respectively. We report means, modes and standard deviations of the parameters in the benchmark model featuring ambiguity aversion and in the alternative model with recursive utility. The posterior mean and mode estimates of the subjective discount factor $$ \beta$$ are stable across the benchmark and alternative models for the annual and quarterly samples. The estimates are below 1 in all cases, which is consistent with values adopted by many calibration studies. Moreover, they are reasonably close to estimates reported by Aldrich & Gallant (2011) and Bansal et al. (2007) and also to the GMM estimate of Yogo (2006). Thus, they do not cause any concern for us and imply precise estimation of the target parameter.
In contrast to the discount factor parameter, estimates of the risk aversion parameter, $$ \gamma$$ , are sensitive to the presence of ambiguity aversion. For the annual estimation, the posterior mean and mode of $$ \gamma$$ in the alternative model are significantly larger than the corresponding estimates in the benchmark model. For instance, the posterior mean of $$ \gamma$$ is 1.62 in the benchmark model, as opposed to 6.32 in the alternative model. When quarterly data are used for estimation, the posterior mean and mode of $$ \gamma$$ in alternative model are an order of magnitude larger than those for the benchmark model. This result is plausible given the calibration studies of Ju & Miao (2012) and Collard, et al. (2015), who show that with smooth ambiguity aversion, low risk aversion is required to account for high and time varying equity premium. The result is also related to the findings of Jeong et al. (2015) for their estimation of the recursive utility model and the multiple priors model, where aggregate wealth consists of financial wealth only. Jeong et al. (2015) report estimates of $$ \gamma$$ ranging between 0.20 to 2.90 in the multiple priors model, while the $$ \gamma$$ estimate is 4.90 in the recursive utility model. In comparison with Aldrich & Gallant (2011), the estimates of $$ \gamma$$ in the benchmark model with ambiguity aversion are smaller than their estimates for both habit formation and long-run risk models, but similar to their prospect theory-based results.
The posterior mode and mean of the ambiguity aversion parameter $$ \eta$$ are 29.80 and 30.33 for the annual sample and 57.08 and 55.93 for the quarterly sample.6 The standard deviation of the posterior distribution of $$ \eta$$ is consistently low. Taken together, tightly estimated values for $$ \eta$$ and the impact of modeling ambiguity aversion on estimation of risk aversion parameter, $$ \gamma$$ , strongly imply that ambiguity aversion does explain important features of asset returns in the data, namely low risk-free rates, high equity premium and volatile equity returns. Since all of the model parameters are estimated simultaneously by the GSM Bayesian estimation methodology, the posterior estimates of the ambiguity aversion parameter depend on the estimation results for other model parameters especially primitive parameters in the consumption growth process. Using the post-war data, our quarterly estimation generates parameter estimates in the consumption growth process that are quite different from the results in the annual estimation. This observation explains the difference between annual and quarterly estimates of the ambiguity aversion parameter. Typical values used in the calibration studies, for example, $$ \eta = 8.86$$ in Ju & Miao (2012) and $$ \eta = 19$$ in Jahan-Parvar & Liu (2014), provide a lower bound for our estimates.7
Ahn et al. (2014) conduct an experimental study on estimating smooth ambiguity aversion. Based on a static formulation, they report values of an ambiguity aversion parameter ranging between 0.00 and 2.00, with a mean value of 0.207. Their estimates of the ambiguity aversion parameter are statistically insignificant and are at least an order of magnitude smaller than our dynamic model-based estimates. We believe that ignoring intertemporal choice under ambiguity explains these differences in estimates of ambiguity aversion.8
There is an ongoing debate about the value of the IES parameter $$ \psi$$ in the asset pricing literature. This parameter is crucial for equilibrium asset pricing models to match macroeconomic and financial moments in the data. In the empirical literature, some studies (e.g., Hall (1988) and Ludvigson (1999)) find that the IES estimates are close to zero as implied by aggregate consumption data. Other studies find higher values using cohort- or household-level data (e.g., Attanasio & Weber (1993) and Vissing-Jorgensen (2002)). Attanasio & Vissing-Jorgensen (2003) find that the IES estimate for stockholders is typically above 1. Bansal & Yaron (2004) point out that the IES estimates will be under-estimated unless heteroscedasticity in aggregate consumption growth and asset returns is taken into account.
Our estimation strongly suggests an IES greater than unity, as advocated by the long-run risk literature. Tables 2 and 3 present the posterior mode and mean of the IES parameter, $$ \psi$$ , estimated in the annual and quarterly samples respectively. The posterior mode and mean estimates range from 2.50 to 5.00 across different models with small standard deviations. These estimates are larger than those reported by Aldrich & Gallant (2011), which are in the neighborhood of 1.50. Risk aversion and the IES both determine the representative agent's preference for the timing of resolution of uncertainty. If $$ \gamma > 1/\psi$$ , the agent prefers earlier resolution of uncertainty; see Epstein & Zin (1989) and Bansal & Yaron (2004). Given the high estimates of $$ \psi$$ , both benchmark and alternative models point to a representative agent who prefers an earlier resolution of uncertainty. Adding ambiguity aversion attenuates this preference moderately: Once ambiguity aversion is taken into account, the estimates of $$ \psi$$ are around 2.62 - 2.96 in annual estimates and 4.28 - 4.55 for quarterly data. The GSM estimation delivers stable estimates of the IES parameter. This is in contrast to the results of Jeong et al. (2015), where the $$ \psi$$ estimates range from 0.00 to $$ \infty$$ . In particular, when only financial wealth is used to proxy total wealth and ambiguity is represented by multiple priors, Jeong et al. (2015) obtain estimates of $$ \psi$$ that are equal to 0.68 with time-varying volatility and 11.16 with nonlinear stochastic volatility.
Table 2 and Table 3 present posterior mean and mode estimates of the primitive parameters in the consumption growth process for the annual and quarterly estimations respectively. The results indicate that the GSM estimation method can successfully identify two distinct regimes of consumption growth for both benchmark and alternative models. The difference between $$ \kappa_{1}$$ and $$ \kappa_{2}$$ estimates is sizable. The transition probability estimate of p11 is above 0.90 in all cases, while the estimate of p22 is about 0.30 - 0.40 in the annual estimation and about 0.70 - 0.80 in the quarterly estimation. This result suggests that the good regime is very persistent while the bad regime is transitory. All these estimates together with the estimates for volatility of the growth innovation, $$ \sigma_{c}$$ , have low standard deviations. Compared with empirical estimates reported by Cecchetti, et al. (2000), differences in several parameter estimates are noticeable. However, this is not surprising. Cecchetti, et al. (2000) fit a Markov switching model to consumption data only. Our GSM estimation uses both consumption data and asset returns data to estimate the model. Besides, we use different sample periods.
Dividend growth, $$ \Delta d_{t}$$ , is a latent variable in our estimation. In the benchmark annual estimation, the posterior estimates of the leverage parameter $$ \lambda $$ are moderately greater than 1. This is consistent with the argument of Abel (1999) that aggregate dividends are a levered claim on aggregate consumption. However, the estimates of $$ \lambda $$ are lower than the value used in the calibration of Ju & Miao (2012) where $$ \lambda=2.74$$ . In the alternative model estimation at the annual frequency, estimates of $$ \lambda $$ shown on Table 2 are closer to this value. In the quarterly estimation, the posterior mean estimates of $$ \lambda $$ are between 1 and 2. The volatility estimate of dividend growth is stable across different models and samples. Our estimates of $$ \lambda $$ and $$ \sigma_{d}$$ are not directly comparable to results of Aldrich & Gallant (2011) or Bansal et al. (2007) due to different specifications for modeling dividend growth. Specifically, Aldrich & Gallant (2011) and Bansal et al. (2007) estimate the long-run risk model featuring time variation in the volatility of fundamentals, while we rely on Markov-switching mean growth rates and learning to generate time-varying volatility of equity returns. However, our estimates of $$ \sigma_d$$ are close in magnitude to that in the estimated prospect theory model reported by Aldrich & Gallant (2011). Aldrich and Gallant posit constant volatility for the dividend growth process in the prospect theory model.
In summary, apart from estimates for the risk aversion parameter and ambiguity aversion parameter, estimates of other structural parameters in our study are remarkably stable and are generally comparable in magnitude to values reported by other empirical asset pricing studies. Thus, it is reasonable to believe that parameter estimates other than risk aversion and ambiguity aversion estimates have small influence on identification and model comparison when it comes to the model featuring smooth ambiguity aversion. In addition, ignoring ambiguity aversion can lead to biased estimates of the risk aversion parameter.
5.1 Relative Model Comparison
Relative model comparison is standard Bayesian inference as described in Subsection 4.1. The computed odds ratio is $$ 1/6.09e{−-85}$$ for the annual estimation and $$ 1/1.18e{−-36}$$ for the quarterly estimation, which strongly favors the benchmark model over the alternative model. This ratio implies that our benchmark model provides a better description of the available data. Given the logarithmic values of posterior evaluated at the mode for the benchmark and alternative models reported in Tables 2 and 3, it is also obvious that the benchmark model is the preferred model. One can gain a rough appreciation for what these odds ratios indicate from a frequentist perspective by disregarding the effects of the prior and support conditions and comparing the log posteriors shown in Tables 2 and 3 as if they were log likelihoods. For the annual comparison minus twice the log likelihood ratio gives a $$ \chi^2$$ -statistic equal to 260.00 on one degree of freedom and for the quarterly data 82.70 on one degree of freedom. The p-value for either is less than 0.0001.
5.2 Forecasts
Forecasts are constructed as described in Subsection 4.2. Prior forecasts (not shown) do not differ much between pre- and post-Great Recession periods. There are, however, differences between prior forecasts based on the benchmark model and the alternative model. The main difference is the disparity in the level of benchmark and alternative model-based forecasts of the short rate. The benchmark model forecasts a higher level for the short rate and a wider standard deviations than the alternative model does. The second difference is the slight increase in the consumption growth path forecasted by the benchmark model, against the drop forecasted by the alternative model.
Prior forecasts are not a measure of a model's success in predicting the data dynamics. For that purpose, we rely on posterior forecasts shown in Figure 6.9 As Figure 6 shows, the posterior forecasts of consumption growth differ across the pre- and post-Great Recession episodes. Both benchmark and alternative models forecast a drop in consumption growth for the pre-recession period while a slight increase for the post-recession period based on available information by the end of 2011. The posterior forecasts paths generated by both modes are on average similar. However, the benchmark model implies slightly more variation in consumption growth forecasts.
For the pre-recession period, the benchmark model forecasts a steeper drop in stock returns compared to the alternative model. For the post-recession period, the benchmark model yields posterior return forecasts that are lower than those generated by the alternative model. These results reflect the pessimism inherent in our benchmark model with ambiguity aversion. Based on data up to the Great Recession period, Aldrich & Gallant (2011) report posterior forecasts of stock returns generated from the long-run risk model. Their return forecasts are roughly 6% for the period 2009-2013. Given the recent past experience, this level appears to be high.
It is clear from Figure 6 that the benchmark model predicts overall lower short rates compared to the alternative model's forecasts. Moreover, the benchmark model predicts a drop in the short rate for both pre- and post-recession periods while the alternative model implies the opposite results.10 These results about the posterior forecasts echo the mechanism of earlier models on ambiguity (or uncertainty) that the induced higher precautionary savings motive tends to reduce the risk-free rate. Given recent announcements by various practitioners, academicians and former policy makers about the likelihood of interest rates reverting back to "old normal" levels, the posterior forecasts generated by our benchmark model seem reasonable.11
Given that Bayesian model comparison prefers the benchmark model over the alternative model, our results of posterior forecasts merit attention. The two models lead to very different dynamics for consumption growth and asset returns. If we indeed live in a world populated by ambiguity-averse agents, then policy and decision makers need to be aware of the inherently different attitudes and hence, market behavior of agents endowed with ambiguity aversion preferences, as opposed to those assumed in standard rational expectation models.
5.3 Asset Pricing Implications
In this section, we study the asset pricing implications of the benchmark model, using the estimated model parameters. Unlike calibration studies, our focus here is not to match unconditional moments of asset returns in the data as closely as possible. Instead, we want to assess the impact of ambiguity aversion on equity premium and the price of risk based on our estimated model rather than independently chosen parameter values. If the estimated benchmark model is reasonably successful in reproducing high price of risk and unconditional equity premium that are not explicitly targeted in our estimation, we view this outcome as confirmation that the dynamics of asset prices implied by our estimation are reasonably close to the underlying data generating process (DGP).
Table 4 presents key financial moments generated by both benchmark and alternative models when model parameters are set to their posterior mean values reported in Table 2.12 Although matching financial moments is not set an explicit target in our estimation, the estimated benchmark model implies moments of asset returns close to the data. All moments reported in 4 are annualized. We observe the following. First, under the benchmark model, the risk-free rate has a mean of about 1 percent and low volatility. Low volatility of the risk-free rate is due to the high estimate of the IES parameter, which implies strong intertemporal substitution effect, see Bansal & Yaron (2004). The mean risk-free rate implied by the alternative model is 1.44 percent and is much higher than the data, at 1.07 percent.
Second, while both benchmark and alternative models generate volatility of equity returns close to the data, the two models differ dramatically in terms of their performance in producing high equity premium. The mean equity premium implied by the benchmark model estimation is 7.31 percent, close to that in the annual sample at 7.47 percent. In contrast, the alternative model implies a mean equity premium of 1.36 percent. As shown by Bansal & Yaron (2004), without high risk aversion or time-varying uncertainty, the long-run risk model with Epstein and Zin preferences has difficulty in matching the mean equity premium. For that reason, Bansal and Yaron consider long-run risk in the mean of consumption growth and also set $$ \gamma=10$$ to match the mean equity premium. Since the estimated $$ \gamma$$ for the alternative model is smaller ( $$ \mathbb{E}(\gamma) = 6.32$$ and $$ \sigma(\gamma)=0.26$$ ) and the model abstracts from stochastic volatility, the mean equity premium implied by the alternative model is too low.
Third, the market price of risk, defined by $$ \sigma(M)/\mathbb{E}(M)$$ , is closely related to moments of equity returns via the Hansen-Jagannathan bound
| $$\displaystyle \left\vert \frac{\mathbb{E}\left( R^{e}_t-R^{f}_t\right) }{\mathbb{\sigma }\left( R^{e}_t-R^{f}_t\right) }\right\vert \leq \frac{\sigma \left( M_t\right) }{\mathbb{E}\left( M_t\right) }.$$ |
A reasonable model that can explain asset prices data well should deliver a SDF that satisfies the bound. The price of risk under the alternative model is 0.28, whereas the Sharpe ratio is 0.37 in the annual sample. It is obvious that the estimated alternative model violates the Hansen-Jagannathan bound. The estimated benchmark model generates $$ \sigma \left( M_t\right) /\mathbb{E}\left( M_t\right)=2.63$$ and thus satisfies the bound. In addition, the Sharpe ratio implied by the benchmark model is 0.42, close to the data at 0.37.
Figure 7 plots conditional equity premium, equity volatility, price of risk and the price-dividend ratio as functions of the state belief $$ \mu _{t}$$ , the posterior probability of the high mean regime of consumption growth. The results are similar to conditional moments plotted by Ju & Miao (2012) except that our results are based on estimated model parameters. Under the alternative model with Epstein and Zin's preferences, conditional equity premium, equity volatility and price of risk display humped shapes. The maximum of these conditional moments is attained when $$ \mu _{t}$$ is close to 0.50 due to high uncertainty induced by Bayesian learning. Ambiguity aversion increases conditional equity premium and price of risk significantly for values of $$ \mu _{t}$$ near its steady-state level implied by the estimated Markov-switching model. The intuition is that the agent distorts her beliefs pessimistically in the face of a shock to consumption growth and thus demands high risk premium. Because the high mean growth regime is persistent, the distribution of $$ \mu _{t}$$ is highly skewed toward 1.00. Thus, the impact of ambiguity aversion on conditional price of risk and equity premium is strong when $$ \mu _{t}$$ is close to its steady-state value. The pessimistic distortion also yields lower price-dividend ratios in the benchmark model, as shown in Panel D, Figure 7. Similar to Ju & Miao (2012), our estimated benchmark model can also reproduce the countercyclical pattern of equity premium and equity volatility. The simulation results are plotted in Figure 8. We observe that the distorted belief puts more probability weight on the bad regime. When shocks to consumption growth are large in magnitude, the distorted belief becomes even more pessimistic and conditional equity premium and equity volatility rise significantly. Thus, the model can reproduce volatility clustering, which is also captured by the auxiliary model used in our estimation.
Finally, an important question is: Do our structural estimations imply reasonable magnitudes for ambiguity aversion? To address this question, we use detection-error probabilities to assess the room for ambiguity aversion based on our estimation results. This exercise is meaningful because our estimation is grounded in the data and thus is informative about the behavior of economic agents and the dynamics of economic variables.
Detection-error probabilities are an approach developed by Anderson, et al. (2003) and Hansen & Sargent (2010) to assess the likelihood of making errors in selecting statistically "close" (in terms of relative entropy) data generating processes (DGP). In this study, the reference DGP refers to the Markov switching model specified in Equation (1). Without ambiguity aversion, the transition probabilities of the Markov chain are defined by p11 and p22. Ambiguity aversion implies distortion to the transition probabilities and thus gives rise to the distorted DGP. The Appendix shows that the reference DGP and the distorted DGP differ only in terms of transition probabilities. We adapt the approach of computing detection-error probabilities in Jahan-Parvar & Liu (2014) to the endowment economy in our study. This approach enables us to simulate artificial data from the reference and distorted DGPs and then evaluate the likelihood explicitly. Details of the algorithm are available in the Appendix.
A sizable detection-error probability ($$ p(\eta)$$ ) associated with a certain value of the ambiguity aversion parameter, $$ \eta$$ , implies that there is a large chance of making mistakes in distinguishing the reference DGP from the distorted DGP, and thus ample room exists for ambiguity aversion. Based on the estimated parameters of the benchmark model, the detection-error probability is 10.22% for the annual estimation and 13% for the quarterly estimation. Anderson, et al. (2003) advocates that a detection-error probability of about 10% suggests plausible extent for ambiguity. Thus, our estimated model parameters admit reasonably large scope for ambiguity aversion.
6 Conclusion
Smooth ambiguity preferences of Klibanoff et al. (2005,2009) have gained considerable popularity in recent years. This popularity is due to clear separation between ambiguity, which is a characteristic of the representative agent's subjective beliefs, and ambiguity aversion that derives from the agent's tastes. In this paper, we estimate the endowment equilibrium asset pricing model with smooth ambiguity preferences proposed by Ju & Miao (2012) using U.S. data and GSM Bayesian estimation methodology of Gallant & McCulloch (2009) to: (1) investigate the empirical properties of such an asset pricing model as an adequate characterization of the returns and consumption growth data and, (2) provide an empirical estimation of the ambiguity aversion parameter and its relationship with other structural parameters in the model. Our study contributes to the existing literature by providing a formal empirical investigation for adequacy of this class of preferences for economic modeling and presenting estimations for the structural parameters of this model. The estimated structural parameters are in line with theoretical expectations and are comparable with estimated parameters in related studies. With respect to measurement of ambiguity aversion, our results show a marked improvement over the existing literature. The existing empirical literature either provides measures of ambiguity (which is usually the size of the set of priors in the MPU framework) instead of ambiguity aversion of the agent, or implausible estimates (economically or statistically) for smooth ambiguity aversion parameter. Our study addresses both shortcoming in the extant literature.
We find that Bayesian model comparison strongly favors the benchmark model featuring a representative agent endowed with smooth ambiguity preferences, over the alternative model featuring Epstein-Zin's recursive preferences. Our estimates of the ambiguity aversion parameter are large and have important asset pricing implications for the market price of risk and equity premium. Detection-error probabilities computed using the estimated parameters imply ample scope for ambiguity aversion. Structural estimations ignoring ambiguity aversion may lead to biased estimates of the risk aversion parameter and are unable to explain the high market price of risk implied by financial data. Our estimates of the IES parameter are significantly greater than 1 and suggest a strong preference for earlier resolution of uncertainty, as is consistent with the long-run risk literature.
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7 Appendix: Detection-error Probabilities
- In constructing distorted transition probabilities, we consider a "full information model", where the agent is ambiguity averse but state zt is observable. In this case, the Euler equation is
$$\displaystyle 0=p_{11}\mathbb{E}_{1,t}\left[ M_{z_{t+1},t+1}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right] +\left( 1-p_{11}\right) \mathbb{E}_{2,t}\left[ M_{z_{t+1},t+1}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right]$$
for $$ z_{t}=1$$ and$$\displaystyle 0=\left( 1-p_{22}\right) \mathbb{E}_{1,t}\left[ M_{z_{t+1},t+1}\left( R_{e,t+1}-R_{f,t}\right) \right] +p_{22}\mathbb{E}_{2,t}\left[ M_{z_{t+1},t+1}\left( R_{e,t+1}-R_{f,t}\right) \right]$$
for $$ z_{t}=2$$ . The Euler equation can be rewritten as
0 $$\displaystyle =$$ $$\displaystyle \tilde{p}_{11}\mathbb{E}_{1,t}\left[ M_{z_{t+1},t+1}^{EZ}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right] +\left( 1-\tilde{p}_{11}\right) \mathbb{E}_{2,t}\left[ M_{z_{t+1},t+1}^{EZ}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right]$$ 0 $$\displaystyle =$$ $$\displaystyle \left( 1-\tilde{p}_{22}\right) \mathbb{E}_{1,t}\left[ M_{z_{t+1},t+1}^{EZ}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right] +\tilde{p}_{22}\mathbb{E}_{2,t}\left[ M_{z_{t+1},t+1}^{EZ}\left( R^{e}_{t+1}-R^{f}_{t}\right) \right]$$
where $$ M_{z_{t+1},t+1}^{EZ}$$ is the SDF under recursive utility without ambiguity aversion, and $$ \tilde{p}_{11}$$ and $$ \tilde{p}_{22}$$ are distorted transition probabilities and are given by
$$\displaystyle \tilde{p}_{11}$$ $$\displaystyle =$$ $$\displaystyle \frac{p_{11}}{p_{11}+\left( 1-p_{11}\right) \left( \dfrac{\mathbb{E}_{2}\left[ V_{z_{t+1},t+1}^{1-\gamma }\right] }{\mathbb{E}_{1}\left[ V_{z_{t+1},t+1}^{1-\gamma }\right] }\right) ^{-\frac{\eta -\gamma }{1-\gamma }}},$$ (20) $$\displaystyle \tilde{p}_{22}$$ $$\displaystyle =$$ $$\displaystyle \frac{p_{22}}{\left( 1-p_{22}\right) \left( \dfrac{\mathbb{E}_{1}\left[ V_{z_{t+1},t+1}^{1-\gamma }\right] }{\mathbb{E}_{2}\left[ V_{z_{t+1},t+1}^{1-\gamma }\right] }\right) ^{-\frac{\eta -\gamma }{1-\gamma }}+p_{22}},$$ (21)
where $$ V_{z_{t},t}$$ , ($$ z_{t}=1,2$$ ) are solutions to the following value function under full information:
$$\displaystyle V_{z_{t},t}(C)$$ $$\displaystyle =$$ $$\displaystyle \left[ (1-\beta )C_{t}^{1-\frac{1}{\psi }}+\beta \left\{ \mathcal{R}_{z_{t}}\left( V_{z_{t+1},t+1}\left( C\right) \right) \right\} ^{1-\frac{1}{\psi }}\right] ^{\frac{1}{1-\frac{1}{\psi }}},$$ $$\displaystyle \mathcal{R}_{z_{t}}\left( V_{z_{t+1},t+1}\left( C\right) \right)$$ $$\displaystyle =$$ $$\displaystyle \left( \mathbb{E}_{z_{t}}\left[ \left( \mathbb{E}_{z_{t+1},t}\left[ V_{z_{t+1},t+1}^{1-\gamma }\left( C\right) \right] \right) ^{\frac{1-\eta }{1-\gamma }}\right] \right) ^{\frac{1}{1-\eta }}.$$ - The numerical algorithm of calculating detection-error probabilities takes the following steps:
- Repeatedly draw $$ \left\{ \Delta c_{t}\right\} _{t=1}^{T}$$ under the reference data generating process (DGP), which is the two-state Markov switching model with transition probabilities p11 and p22.
- Evaluate the log likelihood function under the reference DGP by computing
$$\displaystyle \ln L_{T}^{r}=\sum_{t=1}^{T}\ln \left\{ \sum_{z_{t}=1}^{2}f\left( \Delta c_{t}\vert z_{t}\right) \Pr \left( z_{t}\vert\Omega _{t-1}\right) \right\}$$
where $$ \pi _{t-1}=\Pr \left(z_{t}=1\vert\Omega _{t-1}\right)$$ are filtered probabilities implied by the Markov switching model. - Evaluate the log likelihood function under the distorted DGP by computing
$$\displaystyle \ln L_{T}^{d}=\sum_{t=1}^{T}\ln \left\{ \sum_{z_{t}=1}^{2}f\left( \Delta c_{t}\vert z_{t}\right) \widetilde{\Pr \left( z_{t}\vert\Omega _{t-1}\right) }\right\}$$
where $$ \widetilde{\Pr \left( z_{t}\vert\Omega _{t-1}\right) }$$ are the filtered probabilities that are obtained by applying the distorted transition probabilities $$ \tilde{p}_{11,t}$$ and $$ \tilde{p}_{22,t}$$ (in place of the constant transition probabilities p11 and p22) to the Markov switching model's filter. - Compute the fraction of simulations for which $$ \ln \left( \frac{L_{T}^{d}}{L_{T}^{r}}\right) >0$$ and denote it as pr. The fraction approximates the probability that the econometrician believes that the distorted DGP generated the data, while the data are actually generated by the reference DGP.
- Do a symmetrical computation and simulate
$$ \left\{ \Delta c_{t}\right\} _{t=1}^{T}$$ under the distorted DGP. Compute the fraction of simulations for which
$$ \ln \left( \frac{L_{T}^{r}}{L_{T}^{d}}\right)>0$$ and denote it as pd. This fraction approximates the probability that the reference DGP generated the
data when actually the distorted DGP generates the data.
Assuming an equal prior on the reference and the distorted DGP, the detection error probability is defined by (see Anderson et al. (2003)):
In the approximation, we set T=100 years and simulate 20,000 samples of artificial data.
Table 1: Summary Statistics of the Data
This table reports summary statistics for annual (1929-2013) and quarterly (1947:Q2-2014:Q2) U.S. data. 1-year Treasury Bill rate ($$ r^f_{t}$$ ), aggregate equity returns ($$ r^e_{t}$$ ), excess returns ( $$ r^e_{t} - r^f_{t}$$ ), and real, per capita, log consumption growth ( $$ \Delta c_t$$ ) are expressed in percentages. Mean and standard deviation of quarterly data are annualized. The row titled "$$ J-B$$ test" reports the p-values of Jarque & Bera (1980) test of normality.| $$ r^e_{t}$$ | $$ r^f_{t}$$ | $$ r^e_{t} - r^f_{t}$$ | $$ \Delta c_t$$ | |
|---|---|---|---|---|
| Mean | 8.54 | 1.07 | 7.47 | 1.85 |
| St Dev | 20.35 | 0.06 | 20.35 | 2.15 |
| Skewness | -0.29 | 0.60 | -0.29 | -1.49 |
| Kurtosis | -0.72 | 1.32 | -0.72 | 5.01 |
| J-B test | 0.3938 | 0.0133 | 0.4012 | 0.0001 |
1947:Q2-2014:Q2
| $$ r^e_{t}$$ | $$ r^f_{t}$$ | $$ r^e_{t} - r^f_{t}$$ | $$ \Delta c_t$$ | |
|---|---|---|---|---|
| Mean | 8.76 | 1.05 | 7.71 | 1.91 |
| St Dev | 16.43 | 0.02 | 16.44 | 1.02 |
| Skewness | -0.57 | 0.99 | -0.57 | -0.42 |
| Kurtosis | 1.90 | 1.27 | 1.90 | 1.11 |
| J-B test | 0.0013 | 0.0001 | 0.0013 | 0.0017 |
Table 2: GSM Annual Estimation Results
This table presents priors and posteriors on mode, mean, and standard deviation of model parameters for the benchmark model featuring ambiguity aversion and the alternative model with Epstein-Zin's recursive utility. We impose $$ \eta =\gamma $$ for the alternative model estimation. Preference parameters ( $$ \beta, \gamma, \psi$$ and $$ \eta$$ ) represent subjective discount factor, coefficients of risk aversion, intertemporal elasticity of substitution, and ambiguity aversion respectively. p11 and p22 are transition probabilities in the Markov-switching model for consumption growth. $$ \kappa_1$$ and $$ \kappa_2$$ are good and bad state mean consumption growth rates, respectively. $$ \lambda $$ is the leverage parameter, and $$ \sigma_{c}$$ and $$ \sigma_{d}$$ are volatilities for consumption and dividend growth, respectively. "Log. Post." represents log posterior evaluated at the mode for the benchmark and alternative models. Estimation results are for annual data 1929-2013. In our GSM Bayesian estimation, we use the 1929-1949 data to prime the estimation procedure, and the 1950-2013 data to obtain the estimated parameters.| Prior Mode | Prior Mean | Prior Std. Dev. | Posterior Mode | Posterior Mean | Posterior Std. Dev. | |
|---|---|---|---|---|---|---|
| $$ \beta$$ | 0.9766 | 0.6267 | 0.2340 | 0.9428 | 0.9470 | 0.0034 |
| $$ \gamma$$ | 1.8750 | 1.9808 | 0.8972 | 1.6172 | 1.6264 | 0.6817 |
| $$ \psi$$ | 1.3750 | 1.7765 | 0.5147 | 2.6172 | 2.9646 | 0.4978 |
| $$ \eta$$ | 10.0000 | 9.3388 | 3.6839 | 29.7969 | 30.3285 | 1.3914 |
| p11 | 0.9805 | 0.9682 | 0.0179 | 0.9399 | 0.9411 | 0.0013 |
| p22 | 0.4688 | 0.5133 | 0.1393 | 0.2753 | 0.2733 | 0.0120 |
| $$ \kappa_{1}$$ | 0.0217 | 0.0225 | 0.0036 | 0.0231 | 0.0201 | 0.0023 |
| $$ \kappa_{2}$$ | -0.0615 | -0.0662 | 0.0199 | -0.0683 | -0.0662 | 0.0012 |
| $$ \lambda $$ | 2.8750 | 2.5191 | 0.8536 | 1.0781 | 1.2497 | 0.1930 |
| $$ \sigma_{c}$$ | 0.0308 | 0.0310 | 0.0028 | 0.0267 | 0.0268 | 0.0003 |
| $$ \sigma_{d}$$ | 0.1133 | 0.1263 | 0.0267 | 0.1742 | 0.1705 | 0.0039 |
| Log. Post. | -392.2910 |
Panel B: Alternative model
| Prior Mode | Prior Mean | Prior Std. Dev. | Posterior Mode | Posterior Mean | Posterior Std. Dev. | |
|---|---|---|---|---|---|---|
| $$ \beta$$ | 0.9551 | 0.5635 | 0.2373 | 0.9793 | 0.9819 | 0.0052 |
| $$ \gamma$$ | 5.0625 | 4.9505 | 2.4958 | 6.4570 | 6.3259 | 0.2561 |
| $$ \psi$$ | 1.6875 | 1.7740 | 0.5108 | 4.3828 | 4.0195 | 0.5753 |
| $$ \eta$$ | N/A | N/A | N/A | N/A | N/A | N/A |
| p11 | 0.9902 | 0.9706 | 0.0170 | 0.9435 | 0.9482 | 0.0090 |
| p22 | 0.5156 | 0.5107 | 0.1411 | 0.4470 | 0.3665 | 0.0841 |
| $$ \kappa_{1}$$ | 0.0229 | 0.0222 | 0.0038 | 0.0160 | 0.0166 | 0.0008 |
| $$ \kappa_{2}$$ | -0.0684 | -0.0672 | 0.0193 | -0.0472 | -0.0428 | 0.0136 |
| $$ \lambda $$ | 2.8750 | 2.5877 | 0.8896 | 3.3438 | 2.5988 | 0.8080 |
| $$ \sigma_{c}$$ | 0.0349 | 0.0312 | 0.0028 | 0.0344 | 0.0342 | 0.0022 |
| $$ \sigma_{d}$$ | 0.1133 | 0.1284 | 0.0271 | 0.1726 | 0.1661 | 0.0082 |
| Log. Post. | -522.2885 |
Table 3: GSM Quarterly Estimation Results
This table presents priors and posteriors on mode, mean, and standard deviation of model parameters for the benchmark model featuring ambiguity aversion and the alternative model with Epstein-Zin's recursive utility. We impose $$ \eta =\gamma $$ for the alternative model estimation. Preference parameters ( $$ \beta, \gamma, \psi$$ and $$ \eta$$ ) represent subjective discount factor, coefficients of risk aversion, intertemporal elasticity of substitution, and ambiguity aversion respectively. p11 and p22 are transition probabilities in the Markov-switching model for consumption growth. $$ \kappa_1$$ and $$ \kappa_2$$ are good and bad state mean consumption growth rates, respectively. $$ \lambda $$ is the leverage parameter, and $$ \sigma_{c}$$ and $$ \sigma_{d}$$ are volatilities for consumption and dividend growth, respectively. "Log. Post." represents log posterior evaluated at the mode for the benchmark and alternative models. Estimation results are for quarterly data 1947:Q2-2014:Q2 data. In our GSM Bayesian estimation, we use the 1947:Q2-1955:Q2 data to prime the estimation procedure, and the 1955:Q3-2014:Q2 data to obtain the estimated parameters.| Prior Mode | Prior Mean | Prior Std. Dev. | Posterior Mode | Posterior Mean | Posterior Std. Dev. | |
|---|---|---|---|---|---|---|
| $$ \beta$$ | 0.9883 | 0.7063 | 0.2462 | 0.9892 | 0.9897 | 0.0006 |
| $$ \gamma$$ | 0.8750 | 0.8422 | 0.3662 | 1.3672 | 1.3237 | 0.4015 |
| $$ \psi$$ | 4.0625 | 3.2063 | 1.0600 | 4.5547 | 4.2770 | 0.4572 |
| $$ \eta$$ | 66.0000 | 53.3760 | 22.3194 | 57.0781 | 55.9348 | 4.2912 |
| p11 | 0.9727 | 0.9563 | 0.0244 | 0.9994 | 0.9993 | 0.0003 |
| p22 | 0.6719 | 0.7029 | 0.0442 | 0.7677 | 0.7287 | 0.0370 |
| $$ \kappa_{1}$$ | 0.0056 | 0.0057 | 0.0021 | 0.0081 | 0.0077 | 0.0010 |
| $$ \kappa_{2}$$ | -0.0146 | -0.0155 | 0.0056 | -0.0198 | -0.0183 | 0.0012 |
| $$ \lambda $$ | 1.6250 | 1.8377 | 0.5345 | 2.1719 | 1.6436 | 0.7118 |
| $$ \sigma_{c}$$ | 0.0132 | 0.0131 | 0.0039 | 0.0169 | 0.0150 | 0.0028 |
| $$ \sigma_{d}$$ | 0.1211 | 0.0883 | 0.0223 | 0.0905 | 0.0843 | 0.0038 |
| Log. Post. | -1658.8368 |
Panel B: Alternative model
| Prior Mode | Prior Mean | Prior Std. Dev. | Posterior Mode | Posterior Mean | Posterior Std. Dev. | |
|---|---|---|---|---|---|---|
| $$ \beta$$ | 0.9102 | 0.6552 | 0.2423 | 0.9869 | 0.9875 | 0.0007 |
| $$ \gamma$$ | 27.8750 | 19.3914 | 10.7661 | 13.5234 | 15.0289 | 0.9939 |
| $$ \psi$$ | 3.5625 | 3.2937 | 0.9976 | 4.9141 | 4.9031 | 0.0950 |
| $$ \eta$$ | N/A | N/A | N/A | N/A | N/A | N/A |
| p11 | 0.9258 | 0.9584 | 0.0247 | 0.9997 | 0.9998 | 0.0001 |
| p22 | 0.6719 | 0.7038 | 0.0441 | 0.7743 | 0.7741 | 0.0019 |
| $$ \kappa_{1}$$ | 0.0051 | 0.0057 | 0.0020 | 0.0148 | 0.0147 | 0.0003 |
| $$ \kappa_{2}$$ | -0.0146 | -0.0153 | 0.0056 | -0.0250 | -0.0262 | 0.0007 |
| $$ \lambda $$ | 1.3750 | 1.8407 | 0.5327 | 1.0469 | 1.1008 | 0.0979 |
| $$ \sigma_{c}$$ | 0.0103 | 0.0132 | 0.0039 | 0.0188 | 0.0193 | 0.0011 |
| $$ \sigma_{d}$$ | 0.1133 | 0.0878 | 0.0224 | 0.0890 | 0.0850 | 0.0035 |
| Log. Post. | -1700.1807 |
Table 4: Financial Moments
This table presents unconditional financial moments generated from the estimated benchmark and alternative models using annual data. Model parameters are set at their posterior mean values reported in Table 2. $$ \mathbb{E}(r^f_t)$$ and $$ \mathbb{E}(r^e_t-r^f_t)$$ are mean risk-free rate and mean equity premium respectively, in percentage. $$ \sigma(r^f_t)$$ and $$ \sigma(r^e_t-r^f_t)$$ are volatilities of risk-free rates and excess returns respectively, in percentage. $$ \sigma(M_t)/\mathbb{E}(M_t)$$ is the market price of risk.| Data(1929--2013) | Benchmark model $$ p(\eta)=10.22\%$$ | Alternative model | |
|---|---|---|---|
| $$ \mathbb{E}(r^f_t)$$ | 1.07 | 0.98 | 1.44 |
| $$ \sigma(r^f_t)$$ | 0.06 | 0.09 | 0.16 |
| $$ \mathbb{E}(r^e_t-r^f_t)$$ | 7.47 | 7.31 | 1.36 |
| $$ \sigma(r^e_t-r^f_t)$$ | 20.35 | 17.41 | 17.66 |
| Sharpe ratio | 0.37 | 0.42 | 0.08 |
| $$ \sigma(M_t)/\mathbb{E}(M_t)$$ | N/A | 2.63 | 0.28 |
Figure 1: Risk Free Rate, Aggregate Equity Returns, Excess Returns, and Consumption Growth.
The figure shows, from top to bottom, annual returns of CRSP-Compustat value-weighted index returns, 1-year Treasury Bill rates, excess returns over 1-year T-Bill
rates, and annual real per-capita log consumption growth for the 1929-2013 period. Shaded areas represent NBER recessions.
This figure plots prior and posterior densities of the benchmark model parameters. The solid lines depict posterior densities and dotted lines depict prior
densities. The results are based on 1929-2013 annual data.
Figure 3: Prior and Posterior Densities of Estimated Parameters of the Alternative Model, Annual Data.
This figure plots prior and posterior densities of the alternative model parameters, where the restriction
$$ \eta =\gamma $$ is imposed. The solid lines depict posterior densities and dotted lines depict prior densities. The results are based on 1929-2013 annual data.
Figure 4: Prior and Posterior Densities of Estimated Parameters of the Benchmark Model, Quarterly Data.
This figure plots prior and posterior densities of the benchmark model parameters. The solid lines depict posterior densities and dotted lines depict
prior densities. The results are based on 1947-2014 quarterly data.
Figure 5: Prior and Posterior Densities of Estimated Parameters of the Alternative Model, Quarterly Data.
This figure plots prior and posterior densities of the alternative model parameters, where the restriction
$$ \eta =\gamma $$ is imposed. The solid lines depict posterior densities and dotted lines depict prior densities. The results are based on 1947-2014 quarterly data.
Figure 6: Posterior Forecasts.
This figure shows posterior forecasts for consumption growth, stock returns, and the short rate for the pre- and post-Great Recession periods. The plots are based on annual estimations of the benchmark and alternative models.
The dashed lines are the $$ \pm 1.96$$ posterior standard deviations.
Figure 7: Conditional Financial Moments.
This figure plots conditional financial moments implied by the benchmark and alternative models as functions of state belief $$ \mu _{t}$$ , i.e., the perceived probability of high mean consumption growth under Bayesian learning. The results are based on the GSM Bayesian estimation applied to annual data (1929-2013). Model parameters are set at their posterior mean estimates reported in Table 2.
Figure 8: Quantitative Implications of Ambiguity Aversion.
This figure plots simulated series of Bayesian-filtered and distorted state beliefs ($$ \mu _{t}$$ and
$$ \tilde{\mu}_{t}$$ ), conditional equity premium and equity volatility for the benchmark model with ambiguity aversion. The benchmark model parameters are set at their posterior mean estimates
reported in Table 2.
