Ambiguity and the historical equity premium
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Collard, Fabrice; Mukerji, Sujoy; Sheppard, Kevin; Tallon, Jean-Marc Article Ambiguity and the historical equity premium Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Collard, Fabrice; Mukerji, Sujoy; Sheppard, Kevin; Tallon, Jean-Marc (2018) : Ambiguity and the historical equity premium, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 9, Iss. 2, pp. 945-993, https://doi.org/10.3982/QE708 This Version is available at: https://hdl.handle.net/10419/217119 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Quantitative Economics 9 (2018), 945–993 1759-7331/20180945 Ambiguity and the historical equity premium Fabrice Collard Toulouse School of Economics, CNRS, University of Toulouse Capitole Sujoy Mukerji School of Economics and Finance, Queen Mary University of London Kevin Sheppard Department of Economics, University of Oxford Jean-Marc Tallon Paris School of Economics, CNRS This paper assesses the quantitative impact of ambiguity on historically observed financial asset returns and growth rates. The single agent, in a dynamic exchange economy, treats the conditional uncertainty about the consumption and dividends next period as ambiguous. We calibrate the agent’s ambiguity aversion to match only the first moment of the risk-free rate in data and measure the uncertainty each period conditional on the actual, observed history of (U.S.) macroeconomic growth outcomes. Ambiguity aversion accentuates the effect of conditional uncertainty endogenously in a dynamic way, depending on the history; for example, it increases during recessions. We show the model implied time series of asset returns substantially match the first and second conditional moments of observed return dynamics. In particular, we find the time-series properties of our model generated equity premium, which may be regarded as an index measure of revealed uncertainty, relates closely to those of the macroeconomic uncertainty indices developed recently in Jurado, Ludvigson, and Ng (2015) and Carriero, Clark, and Marcellino (forthcoming). Keywords. Ambiguity aversion, asset pricing, equity premium puzzle, timevarying uncertainty, uncertainty shocks. JEL classification. C63, D81, E21, G12. Fabrice Collard: [email protected] Sujoy Mukerji: [email protected] Kevin Sheppard: [email protected] Jean-Marc Tallon: [email protected] We thank three referees and the editor for thoughtful comments and suggestions. We thank R. Bansal, P. Beaudry, H. Bhamra, J. Borovicka, T. Cogley, H. Chen, H. d’Albis, V. Gala, C. Gollier, L. Hansen, P. Klibanoff, H. Liu, T. Ramadorai, and R. Uppal for helpful discussions. We also thank seminar and conference participants at Adam Smith Asset Pricing conference, AEA, RUD, Northwestern (MEDS), Warwick, Leicester, Transatlantic Theory Workshop, EUI (Florence), UBC, Workshop on Ambiguity and Robustness in Macroeconomics and Finance (Becker-Friedman Inst.). Tallon thanks support from the Investissement d’Avenir Program (ANR-10-LABX-93). ©2018 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE708
946 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) 1. Introduction This paper seeks to assess the quantitative impact of ambiguity on financial asset returns and prices, in particular their dynamic paths, conditioned on observed historical growth rates. Ambiguity refers to uncertainty about the “true” probability distribution governing future consumption and dividend outcomes. The decision maker’s ambiguity attitude determines how and to what extent such uncertainty affects his choices. Our goals are two-fold: to connect the macroeconomic uncertainty as it obtained on the path of history to the movements in asset returns and prices along that path and to assess quantitatively the role of ambiguity sensitivity in that connection. To serve these goals, we incorporate two components in our analysis. One, we only consider conditional uncertainty at information sets adapted to the path of observed historical macroeconomic growth rates, as opposed to counterfactual, simulated sample paths. Two, our model of agent’s preferences departs from standard expected utility solely by allowing for sensitivity to ambiguity; take that away, and the agent’s preferences reduce to standard expected utility. These two components, together with the demonstration that they alone are sufficient to substantially explain a range of asset return dynamics, distinguish the contribution in this paper. Ambiguity-averse agents are inclined to choose actions whose consequences are more robust to the perceived ambiguity, for example, a portfolio position whose (ex ante) value is relatively less affected by the uncertainty about probability distribution governing the future payoffs.1An important reason why ambiguity may be pervasive in economic and financial decision making is model uncertainty. For example, a typical professional investor may have different forecasting models for the same variable or different parameter estimates for the same model, all of which are plausible on the basis of historical data. If the models make distinct (probabilistic) forecasts about key variables of interest, it is natural to seek a portfolio that accounts for differences in the agent’s outcome across the range of forecasts rather than optimizing exclusively to the forecast from a single model as argued, for example, in Hansen (2007). This paper considers a standard single agent, Lucas-tree, pure-exchange economy with two less standard assumptions. First, the agent’s belief about the consumption and dividend process is ambiguous, that is, in each period, he is uncertain about the exact probability distribution governing the realization of consumption and dividends in the following period. Furthermore, this belief is dynamic, evolving as the agent learns from history. Second, the agent’s preferences are ambiguity-sensitive, modeled using the smooth ambiguity model of Klibanoff, Marinacci, and Mukerji (2005), Klibanoff, Marinacci, and Mukerji (2009) (henceforth KMM2005, KMM2009). The assumed source of the ambiguity in the agent’s beliefs is the occurrence of periodic, temporary changes in the probability distribution governing next period’s growth 1See Dow and Werlang (1992), Epstein and Wang (1994), Mukerji and Tallon (2001), Caballero and Krishnamurthy (2008), Chen, Ju, and Miao (2014), Gollier (2011), Boyle, Garlappi, Uppal, and Wang (2011), Hansen and Sargent (2010), Maccheroni, Marinacci, and Ruffino (2013), and Uhlig (2010), inter alia.Surveys of the related literature may be found in Gilboa and Marinacci (2016)(decisiontheory)andMukerji and Tallon (2004) (applications).
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 947 outcome due to the effect of the business cycle. These transient deviations are assumed to be governed by an autoregressive (AR(1)) latent variable. The agent is, however, unsure about the value of the persistence parameter of the AR(1)process since, even with a largesampleofgrowthrates,itisdifficulttodistinguishthecasewherethelatentgrowth state is highly volatile but moderately persistent, from the case where the state is less volatile but highly persistent. Uncertainty about persistence, in turn makes it harder to estimate the evolving location of the latent variable precisely. Furthermore, depending on the observed history, the imprecision of the estimate of the location will vary over time, making the uncertainty about the probability distribution governing next period’s growth vary over time. The ambiguity-averse agent’s robustness concerns generate, endogenously, doubt and pessimism, to use the language of Abel (2002). The portfolio choice of the ambiguityaverse agent in the model may be understood as that of an expected utility agent with an “as if” (probabilistic) belief that is more uncertain and pessimistic than the one obtained by objective inference, in the standard fashion, from data. Moreover, the endogenous accentuation of doubt depends on the observed history and the level of ambiguity aversion, making the severity of the effect of uncertainty endogenously time-varying.For instance, after a negative shock that follows a series of “normal” ones, the agent behaves as if the uncertainty is more severe and more persistent than what is implied by pure Bayesian inference (and the opposite, if it were a positive shock that broke the normal sequence). The level of ambiguity aversion is calibrated to match the average risk free rate (no other moment is used); all other parameters are either inferred/estimated from the history or fixed at values widely used in the literature. We present two kinds of results on model implied conditional moments of rates of return and price-dividend ratio: (time-)averages of the moments over the sample period (1978–2011) and time series of the moments over the same sample period, all based on conditional uncertainty at information sets reconcilable with historical growth data. We compare the level, volatility, and dynamics of the model implied rates of return and price-dividend ratio to their counterparts in U.S. data. The model generated conditional equity premium is a measure of conditional macroeconomic uncertainty as revealed by the behavior of the agent in the model. We show its time-series properties match those of the purely statistical index of macroeconomic uncertainty, recently developed in Jurado, Ludvigson, and Ng (2015)andCarriero, Clark, and Marcellino (forthcoming). Our model gives a theory of why an agent makes decisions following a positive shock that (endogenously) underplays the uncertainty and its persistence, while following a negative shock, behaves as if amoresevereand a more persistent shock were in play, thus explaining a key feature of the index and related findings of the recent literature on uncertainty shocks. In particular, the countercyclical persistence of equity premium and (revealed) uncertainty speaks directly to the mechanism of ambiguity aversion in our model. Altogether, our contribution is to demonstrate that model/parameter uncertainty and learning coupled with ambiguity aversion, by themselves, create a quantitatively plausible and intuitively meaningful mechanism for explaining the relationship between macroeconomic uncertainty and the dynamics of equity prices and returns.
948 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) The time-averaged conditional moments predicted by the model match data moments as well as the best matches in the literature (e.g., in Collin-Dufresne, Johannes, and Lochstoer (2016) and papers cited therein). Our more distinctive results are those on the predicted time series of conditional moments statistics. Two key stylized facts our model matches are the countercyclicality of conditional equity premium and the procyclicality of conditional (excess) return volatility. Models in the literature have found it hard to explain these facts without introducing at least one of the following elements: (a) some exogenously time varying uncertainty, such as, time dependent, stochastic volatility; (b) aversion to later resolution of risk via an intertemporal elasticity of substitution (IES) that is significantly greater than unity; (c) habit formation; all elements that are not part of our mechanism.2A reason to be interested in the mechanism posited in the present paper alongside these “best performing” alternatives in the recent literature is that the alternatives rest on assumptions that have been empirically questioned, and hence cannot be regarded as the “last word” on the subject. At the same time, the first findings on the estimation and calibrations of ambiguity aversion in the context of asset pricing are promising. The route of relying on exogenously posited stochastic volatility of aggregate consumption has been questioned because “the evidence for heteroskedasticity in aggregate consumption is fairly weak,”(Campbell (2000)). In a similar vein, Lettau and Ludvigson (2010)andLudvigson (2012) in their surveys argue that the evidence for stochastic volatility suggests it has neither the intertemporal shape nor the size required for models based on stochastic volatility to fit facts about inter-temporal variation in return moments. A more fundamental difference between stochastic volatility based asset pricing models and ours is that in the former there is no explanation, as such, of the variation in volatility: in those models agents are more uncertain when they believe they are in a state where future economic shocks are assumed (exogenously) to be more volatile. In contrast, our model gives a theory why an agent makes decisions following a positive shock that underplays the inferred uncertainty and, after a negative shock that follows a series of “normal” ones, behaves as if the uncertainty is more severe and persistent, than pure Bayesian inference would suggest. It is well documented that the empirical evidence on whether IES is greater than 1is very mixed (see discussions, e.g., in Beeler and Campbell (2012)andBansal, Kiku, and Yaron (2012).) Furthermore, recently, Epstein, Farhi, and Strzalecki (2014) argue using a calibration exercise, that the IES >1values applied in the recent asset pricing literature imply a very implausible premium for early resolution of uncertainty. While we do not know of conclusive direct evidence for or against habit formation, there is some evidence against the key underlying mechanism. Neither in data (nor in the model in the present paper) does lagged consumption growth predict the future price-dividend ratio, 2Bansal and Yaron (2004) incorporate (a) and (b); Campbell and Cochrane (1999)have(c);Drechsler (2013) incorporates model uncertainty, learning, ambiguity aversion (a) and (b); Collin-Dufresne, Johannes, and Lochstoer (2016), model uncertainty, learning and (b); Ju and Miao (2012) and Hansen and Sargent (2010) incorporate model uncertainty, learning, ambiguity aversion and (b). We discuss more details of this related literature in Section 5.
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 949 while in the habit-formation model it predicts the future price-dividend with an R2of over 40%. A recent study, Gallant, Jahan-Parvar, and Liu (2015), which uses macroeconomic and financial data to estimate the size of ambiguity aversion (as a parameter in a consumption-based asset pricing model based on an elaborated version of the smooth ambiguity model), finds that the estimate “suggests ample scope for ambiguity aversion” to explain asset pricing facts. In the present paper, in Section 4, we conduct a calibration exercise to argue that the size of the ambiguity aversion parameter we apply has very plausible implications for uncertainty premia. We view the preceding discussion about alternative models and ours as not an argument for considering the approach taken here to be the best, but as showing that it merits careful study and development. The rest of the paper is organized as follows. Section 2introduces the relevant details of smooth ambiguity preferences, describes and analyzes the amended Lucas tree economy, assuming a general form of beliefs. In a subsection, we describe and motivate the specific model of ambiguous beliefs we adopt. Section 3first outlines the numerical solution method we employ, then identifies the key qualitative mechanisms at work in our model and finally presents and explains the quantitative implications of our model for asset prices and returns in the light of the mechanisms identified. In Section 4, using a thought experiment, we show that a decision maker with preferences and beliefs calibrated to match those of our agent’s will demand a total uncertainty premium (for the Lucas tree) that is well within the bounds of the amounts widely considered as plausible. Section 5discusses the more closely related literature. A final section concludes. The Appendix gathers several items, including, details of parameter values used in the model, details of the model including the specification of beliefs, how they are updated, and the formulae for rates of return. 2. The model 2.1 Agent’s preferences: Recursive smooth ambiguity We follow KMM2009, which develops a dynamic, recursive version of the smooth ambiguity model in KMM2005. In KMM2009, the basis of the dynamic model is the state space, the set of all observation paths generated by an event tree, a graph of decision/observation nodes. The root node of the tree, s0, branches out into a set of immediate successor nodes, s1≡(s0s1)where s1∈S1, the set of possible observations at time t=1; and, so on. The decision maker (DM) chooses between consumption plans f, each of which associates a payoff to a node stin the event tree. The DM is uncertain about which stochastic process governs the probabilities on the event tree. The domain of this uncertainty is given by a parameter space Θθ, the set of unobservable parameters, over which the DM makes inference at each st. We denote by πθ(st+1|st)the probability under likelihood distribution πθthat the next observation will be st+1,giventhat node stis reached. The decisions maker’s prior on Θis denoted by μ. KMM2009 give assumptions such that recursive smooth ambiguity preferences over plans fat a node st are updated and represented as: Vst(f) =ufst+βφ−1Θ φSt+1 V(stst+1)(f)dπθst+1|stdμθ|st(1)
950 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) where Vst(f) is a recursively defined (direct) value function, ucharacterizes attitude to risk, βis a discount factor, φis a function characterizing the decision maker’s ambiguity attitude, while μ(·|st)denotes the Bayesian posterior. A concave φcharacterizes ambiguity aversion, which is defined to be an aversion to mean preserving spreads in the distribution over expected utility values. In general, the model does not impose reduction between the second-order belief μand the first-order probabilities πθ’s; reduction only applies when φis affine, representing an ambiguity neutral Bayesian expected utility maximizer. Ambiguity aversion in this model is equivalent to the DM behaving as more risk averse when choosing between bets on θthan when choosing between objective lotteries. That is, the DM strictly prefers a lottery which yields a unit payoff with objective probability m(and 0with probability 1−m) to a (same stakes) bet on an event T⊂Θ, where μ(T) =mand also strictly prefers the complementary lottery to the bet on the complementary event.3The behavior is exactly analogous to the modal behavior in the Ellsberg two-urn example: preference for betting on a draw from the urn with a known 50 :50 mix over betting on a draw from the urn with unknown mix. Hence, the secondorder measure μcannot be calibrated with a lottery; behaviorally, μis not treated as an objective probability. The standard interpretation is that the DM views his belief about events such as Tto be less reliable than an objective probability. 2.2 A Lucas-tree economy and Euler equations with general beliefs There is an infinitely-lived agent, with recursive smooth ambiguity preferences, consuming a single good. He can trade in a short lived risk-free asset, whose holding and price at time tare denoted btand Pf t, respectively. There is also an asset (whose quantity is normalized to one unit) that yields a stochastic dividend at each period, Dt.The asset with uncertain dividend (the “risky” asset) has a price Ptat time t, and its holding is denoted et. Consumption at time tis denoted Ct.AsinBansal and Yaron (2004) and Campbell (1996), we will assume that dividend and consumption follow different stochastic processes, thus departing from the original Lucas tree economy. The gap between consumption and dividend is due to some (exogenously given) labor income lt.4 Equilibrium will require that at each time Ct=lt+Dt. Next, we derive Euler equations that define equilibrium prices in this economy. At anode{CτDτ}t τ=1,letμtdenote the second-order belief, on parameters in Θdefining first-order probability distributions on immediate successors (Ct+1Dt+1). Beliefs are updated as a function of the observed realizations of the consumption and dividend signals according to Bayes’ law. Wealth at time t+1is Wt+1=(Pt+1+Dt+1)et+bt+ lt+1, and the budget constraint in period tis given by Ct=Wt−Ptet−Pf tbt.Theagent’s maximization problem may be described in terms of a recursive Bellman equation given by: J(Wtμt)=max Ctbtet u(Ct)+βφ−1EμtφEπθJ(Wt+1μt+1)(2) 3See Section Din the Appendix for details. 4In other words, we assume directly a stochastic process for Ctand Dt, leaving labor income ltimplicit.
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 951 subject to the budget constraint and the law of motion of the two “state” variables (wealth and beliefs), where J(Wtμt)denotes a recursively defined indirect value function (as opposed to the direct value function in equation (1)). An equilibrium of this economy is given by {(PτPf τeτbτCτ)}∞ τ=1such that the consumption and asset holding processes solve the maximization program and the market clears, that is, et=1, bt=0,Ct=Dt+ltat each t. First-order conditions are given by: βΥtEμtξt(θ)Eπθu(Ct+1)=Pf tu(Ct) (3) βΥtEμtξt(θ)Eπθ(Pt+1+Dt+1)u(Ct+1)=Ptu(Ct)(4) where Υt=Eμt[φ(Eπθ(J(Wt+1μt+1)))]×(φ−1)[Eμt(φ(Eπθ(J(Wt+1μt+1))))]and ξt(θ) =φEπθJ(Wt+1μt+1) EμtφEπθJ(Wt+1μt+1)(5) The function ξtis a Radon–Nikodym derivative effecting a node specific change of measure, or “distortion,” on the posterior μt, akin to martingale distortions arising in robust control problems considered by Hansen and Sargent. The distortion is a function of the continuation expected values obtained at successor nodes. In this paper, we assume φ(x) =−exp(−αx)/α, where the parameter αrepresents ambiguity attitude. This specification simplifies the expressions significantly, since we now have Υt=1.Itisalso assumed that u(x) =x1−γ 1−γ. With these specifications, the Euler equations are as follows: βRf tEμtξt(θ)Eπθexp(−γgt+1)=1(6) βEμtξt(θ)EπθRt+1exp(−γgt+1)=1(7) ⇔βEμtξt(θ)Eπθexp(zt+1)+1 exp(zt)exp(dt+1−γgt+1)=1(8) where zt=ln(Pt Dt),gt+1=ln(Ct+1 Ct),dt+1=ln(Dt+1 Dt), the logarithm of price-dividend ratio, rates of growth of consumption and dividend, respectively, while Rf t=1 Pf t ,Rt+1= Pt+1+Dt+1 Ptdenote the risk-free and risky rates of return. Remark 1. Observe these Euler equations look identical to ones obtained in a standard Bayesian model except for the inclusion of the distortion function, ξt. The distortion, in the case of ambiguity aversion, increases the (posterior) weight on likelihoods πθ with lower expected continuation values, Eπθ(J(Wt+1μt+1). One could splice together the one-period ahead predictive distributions, [ξt(θ) ×μt(θ)]⊗πθ(gt+1dt+1), and construct an overall “as if” unconditional probability distribution over the event tree which could be reinterpreted as coming from a Bayesian model. However, seen by itself, the constructed as if distribution cannot be linked to the given set of likelihoods {πθ}θ∈Θ; indeed, typically, it is not possible to obtain the constructed distribution by starting at the initial node with a different prior μ 1=μ1on Θwith μ t,t>1, obtained by updating in the usual way. Hence, an understanding of the role of ambiguity aversion in the
952 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) modeling exercise is that it provides a link between the subjective as if distribution and a specification of beliefs about possible data generating the processes ({μt}t{πθ}θ∈Θ); beliefs which, in principle, can be objectively reconciled with data. Remark 2. If φ(·)were different from an exponential, for example, a power function, then Υt= 1, in general, and hence in such a case the difference between these Euler equations and the standard set would not simply be the change of measure term ξt(θ). Thus it is down to our choice of the specification of φ(·)and of u(·)that we may interpret our Euler equations as arising from an agent we see in standard macro-finance models (with the preference over consumption given by a power function who has nonstandard (though Savage–Bayes’ rational) beliefs which may be justified by appealing to robustness/model uncertainty concerns. This way we can embed our model within that standard literature and, very much in terms of that literature, motivate and explain its point of departure and intuition. Given the specification, the point of departure is just the nonstandard beliefs that can be motivated entirely in terms of robustness concerns, arguably very reasonable, even normatively compelling, given the model and parameter uncertainty faced by a typical agent in the real world. Furthermore, the fact that the (nonstandard part of) beliefs is entirely shaped by the history dependent ξt(θ) is the key that will allow us to make transparent (as will be seen in Section 3.2)thetwokeymechanisms driving the results, the higher time averages and the endogenously dynamic fluctuation of returns. A drawback of this specification is that our value function misses a homogeneity property; note the dependence on (Wt)in (2) and, equivalently, on (Ct)in (10). Thus, the curse of dimensionality makes numerical analysis of the decision maker’s dynamic programming problem more complicated. Numerically, we already have a relatively high dimension problem. If we were to use a power function specification for ambiguity preference, wealth in (2) and consumption level in (10) will be factored out. This will not only reduce the dimension by 1, consumption level will drop out of all pricing equations. Our modeling choice reflects our belief that the two advantages of the adopted specification in providing economic motivation and intuition, outweigh the disadvantage of the ensuing numerical complication. Furthermore, dispensing with homogeneity, a departure from standard practice, required us to be innovative with our numerical method; these innovations might prove useful in future research. 2.3 Beliefs and how they are applied in the evaluation of the Lucas tree 2.3.1 Description We now describe the specific belief about the Lucas tree economy that we apply in our analysis. It is assumed the agent believes the growth rate of consumption (gt) and dividends (dt) are partly driven by a common latent state, xt,which evolves according to an AR(1)process with persistence ρ. While it is assumed there is a single persistence parameter operating through history, the agent is unsure what it is, believing there are two possible values of the parameter, high (ρh)orlow(ρl). At time t, the agent puts probability ηton persistence being low and (1−ηt)on persistence being
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 959 Table 1. Accuracy of the numerical solution. This table reports the measure of accuracy for the Euler equation. In each case, αwas set such that the model generates a risk-free rate of 15%. Known Persistence Unknown Persistence γαE 1E2E∞αE 1E2E∞ 201151 −498 −818 −452 1775 −363 −563 −334 25724 −554 −929 −509 1135 −407 −650 −377 30421 −866 −1559 −805 665 −578 −993 −548 performs well for the data we use. Results for both models are reported in Table 1and show that the approximation is accurate. Let us first consider a special case of our model where ηt=0, “known persistence,” that is, the agent acts as if he knew the persistence parameter ρwere equal to 085.In this case, taking γ=2for example, an agent who uses the approximate solution based on consumption claims would make, on average, a one dollar mistake for every $95500 invested in the assets, while the maximal error would be of the same order. In the general case of the model, with unknown persistence, the performances of the approximation slightly deteriorate. This accuracy loss is essentially due to the structure of the problem. When persistence is known, the model is almost log linear, and our approximation performs remarkably well. In the full model, the quasi log linearity is lost as we have to compose probabilities of each model. Increasing the degree of the polynomials yields some (marginal) improvements but (i) leave the results almost unchanged and (ii) comes at a substantial computational cost. We therefore kept the degrees of the polynomials as they are. 3.2 Understanding the mechanism of ambiguity aversion A good way to understand the key channels through which ambiguity aversion affects asset returns in our model is by understanding how the distortion function, ξ,shownin equation (5) shapes the “as if” belief of the agent, that is, the (probabilistic) belief which supports the action chosen by the agent in equilibrium. We identify two main mechanisms. The first works through the endogenous pessimism and added doubt that the “as if” belief embodies, at any one point in time, compared to the belief of an agent with rational expectations based on the processes underlying the specified belief model. The second mechanism is an endogenous accentuation of the cyclical variation in uncertainty. 3.2.1 Endogenous pessimism and doubt The intuition behind the first channel can be more transparently understood in the special case of the model of beliefs where there is no uncertainty about the persistence (e.g., η0=0). Under this assumption, the argument (xltηt)drops out of the value function described in (10), and the distortion is given as
960 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) (suppressing “k” subscripts):19 ξt(xt|Ct xt;α) ≡exp−αExtV(C t+1; xt+1) E xtexp−αExtV(C t+1; xt+1)(11) The effect of ξtis to create an “as if” posterior on xt, that is, a distorted posterior, ˜μt≡ξt(xt)⊗N(ˆ xtΩ),where xtis the filtered value at time t. In the case of ambiguity aversion, that is, α>0, it is evident from equation (11)that ˜μtputs relatively greater probability mass (compared to μt)onxt’s that generate probability distributions associated with lower expected continuation values, Ext(V (Ct+1; xt+1)). The distorted posterior gives rise to an “as if” conditional one-step-ahead distribution on growth which we call the twisted (predictive) distribution: gt+1∼ξt(xt)⊗N(ˆ xtΩ)⊗Nρxt+¯ gσ2 x+σ2 g(12) When ξt(xt)=1, the formula (12) describes the belief of a Savage–Bayes’ rational (or, equivalently, ambiguity neutral) agent, a useful benchmark. Such an agent, whom we dub “Bayesian,” is uncertain about xtwith belief about growth described by a mixture of normals. The twisted distribution, on the other hand, describes the predictive “as if” belief of an ambiguity-sensitive agent. Another useful benchmark is the predictive belief of an agent with “rational expectations,” narrowly defined. This distribution, N(ρˆ xt+¯ gσ2 x+σ2 g), arises from a posterior that is degenerate on ˆ xt. As Figure 1shows, compared to the rational expectations distribution, the twisted distribution has a lower mean and alargerspread.Abel (2002) argued that one can account for the observed equity premium and the risk-free rate by invoking pessimism and doubt in an otherwise standard asset pricing model. Pessimism is deemed, by Abel, as a subjective distribution on growth that is first order stochastically dominated by the “objective” distribution; doubt, corresponds to a subjective distribution that is a mean preserving spread of the objective distribution. Evidently, an ambiguity-averse agent’s conditional (“as if”) beliefs, in effect, incorporate endogenously both these elements while the Bayesian agent only incorporates the doubt. These observations will be the key to understanding our results on time averages of conditional returns moments. 3.2.2 Endogenous accentuation of cyclical variation in uncertainty To understand the second mechanism, we return to the beliefs model without the restriction of η0=0. Learning about persistence leads to time-varying mixing of the two processes through ηt. This produces a posterior predictive belief about consumption growth which is heteroskedastic across time, even though in each process (with a given persistence) the 19Henceforth, we shall write ξtas a function of direct continuation value V(·)instead of the indirect value, J(Wt+1μt+1). In a single agent economy consumption is exogenously determined, and so it is possible to solve for the continuation value at any node on the event tree without solving for the equilibrium prices first.
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 961 Figure 1. Beliefs and “as-if” beliefs. The agent’s “as-if” belief about the conditional distribution of consumption growth with no uncertainty about the latent state (R.E.), with uncertainty about the latent state but without ambiguity aversion (Bayesian) and with ambiguity aversion about the uncertainty of the latent state (Twisted). The distributions were computed using ρ=085, and the level of consumption and latent state as the average over 1978–2011. growth distribution is homoskedastic. The mean and variance of the mixture distribution on the latent state are ˆ xt=ηtˆ xlt +(1−ηt)ˆ xht(13) Vart(xt)=ηtΩl+(1−ηt)Ωh+ηt(1−ηt)( ˆ xht −ˆ xlt)2(14) It is as if the agent has two forecasting models. When the history is such that both models explain that history just as well (i.e., ηtis close to 05) and yet their core forecasts markedly disagree (i.e., (ˆ xht −ˆ xlt)2is large), the uncertainty, as shown by the variance, rises. In contrast in the case with η0=0, what happens over time to the posterior is that its mean ˆ xtmay change but not its variance, ensuring a homoskedastic predictive distribution.20 The endogenously time varying uncertainty in our model, due to learning about the persistence, creates a potential for uncertainty shocks, sudden sharp increases in uncertainty about consumption growth. One way an uncertainty shock can come about is as follows. A sequence of moderately positive growth realizations, being quite consistent with high and low persistent processes, brings ηtclose to 1/2.Ifoneormore negative realizations arise after such a sequence, (ˆ xht −ˆ xlt)2increases, thus increasing Vart(xt). Ambiguity aversion exacerbates the time variation of the Savage–Bayes uncertainty by endogenously accentuating that uncertainty asymmetrically between positive and negative shocks, creating “as if” uncertainty shocks that are far sharper than what is reflected by the dynamics of Vart(xt). 20The time-varying heteroskedasticity generated endogenously in our model is a forecast uncertainty, of beliefs, empirically driven by the history of growth outcomes and consistent with a stationary volatility of consumption shocks.
962 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) To see how, consider the following. The distorted posterior is a mixture of two component distorted posteriors, ξk t⊗ηt⊗N(ˆ xktΩk)for k=h l,whereξk tis as in eq. (28) in Section B.1.2 in the Appendix.Let ˜ xkt denote the mean of a distorted component posterior, ξk t⊗N(ˆ xktΩk). Due to the greater persistence, the aggregate uncertainty around ˆ xht—captured by Ωh—is larger than that around ˆ xlt. Since the distortion function is proportional to a negative exponential, it has more bite on a distribution which has more probability mass on the left tail by whipping up that mass even more; hence, we have ˆ xht −˜ xht >ˆ xlt −˜ xlt. Which means that (ˆ xht −ˆ xlt)2>(˜ xht −˜ xlt)2when ˆ xht >ˆ xlt (as would be, following a positive shock) and (ˆ xht −ˆ xlt)2<(˜ xht −˜ xlt)2when ˆ xht <ˆ xlt (following a negative shock). Hence, when ˆ xht <ˆ xlt, the components of the mixture yielding the “as if” posterior are further apart compared to the components of the Bayesian posterior (and, conversely, when ˆ xht >ˆ xlt). This has two implications. One, Vart(xt), the variance of the distorted posterior21 understates that of the Bayesian posterior following a positive shock, and exaggerates it following a negative shock, making it more pronouncedly countercyclical than Vart(xt). Two, the distorted posterior demonstrates a significant negative skewness compared to the Bayesian posterior in recessionary periods, but not in good times. The left panel in Figure 2shows how ˆ xht and ˆ xlt have moved with the business cycle. The right panel compares the variance of the posterior and the variance of the distorted posterior showing that the latter greatly amplifies movements in the former, especially at downturns. Figure 2also shows that in 1992 ˆ xht <ˆ xlt while in 1999 ˆ xht > ˆ xlt, though |ˆ xht −ˆ xlt|were similar in these two years. Figure 3demonstrates how much more significant the effect of the distortion was on the posterior in the latter year. The following argument focused on the uncertainty about ρkoffers another, and perhaps pithier, intuition. The ambiguity-averse agent behaves as if he forecasts consumption growth putting more weight (compared to the Bayesian posterior) on the “worst case” persistence, that is, the ρkthat minimizes the expected continuation utility. When consumption growth is below the mean, the worst case persistence parameter is ρh, suggesting that we will remain below the mean for a long time. In contrast, when Figure 2. Explaining time-varying ambiguity: The left panel shows the filtered latent variables assuming that the high ( ˆ xht )andlow(ˆ xlt) persistence as the DGP. The right panel graphs the conditional variance of the latent state variable (Vart(xt)) and the “as if” conditional variance ( Vart(xt)). In both panels, the gray line shows the HP–filtered consumption growth, indicating the business cycle. 21See Section B.1.1 in the Appendix for an analytical expression.
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 963 Figure 3. Time-varying distortion. The two panels plot beliefs about the latent state without ambiguity aversion (Bayesian) and with ambiguity aversion. The left panel shows a “bad” year where ˆ xht <ˆ xlt, and the right panel shows a “good” year where ˆ xht >ˆ xlt. consumption growth is above the mean, the worse case is that the persistence is ρl,so we revert quickly to the mean. Thus, the ambiguity-averse agent, endogenously behaves as if the uncertainty is more persistent and severe following negative shocks than in normal times (even though ηt≃1/2). These insights about the asymmetric reaction to good and bad news will be key to understanding how ambiguity aversion affects conditional returns and their variation over time, in particular, over the business cycle. 3.3 Comparing model implications with data We use annual data on real per-capita consumption Ctand estimates of xkt corresponding to the filtration imposed by the observed history of growth of real consumption and of real dividends to obtain a time series of model implied conditional moments of the annual rates of return using our numerical solution technique (see Section 1 in the supplementary material found on the journal website http://qeconomics.org/supp/708/ supplement.pdf.) We compute the model implied price-dividend ratio applying the relationship Rt+1=exp(pt+1−dt+1)+1 exp(pt−dt)exp(dt+1−dt)(15) where dtis taken from the historical data, Rt+1and pt+1are computed from the model, and the recursion is started from the actual price-dividend ratio in 1977 (t=0). Throughout the exercise, the level of ambiguity aversion was calibrated so that the average riskfree rate was 15%. We present and discuss two kinds of results on model implied conditional moments of rates of return and price-dividend ratio: averages of the moments over the sample period, 1978–2011 in Section 3.3.1, and time series (and time-series properties) of the moments over the same sample period in Section 3.3.2. In Section 3.3.3,wecompare the time series of our model implied equity premium with the leading macroeconomic uncertainty index in the literature.
964 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) 3.3.1 Time averages of moments Table 2reports the model implied conditional moments of returns and price-dividend ratio, time averaged over the sample period.22 The panels in Figure 4show the comparative statics of ambiguity aversion and risk aversion on the conditional rates of return. The model’s match of the first moments of returns is quite perfect and second moments are predicted to a large extent. Table 3reports on some further robustness checks. In particular, it checks for the consequences from alternative learning assumptions that would be implied by splitting the sample differently. As was explained, our benchmark calculations are obtained by splitting the 1930–2011 sample between a “learning period,” 1930–1977, where the timeseries parameters were estimated, and the remaining period where the model was evaluated, that is, the benchmark split is (1930–1977; 1978–2011). Table 3considers three alternative splits: (1930–1968; 1969–2011), (1930–1959; 1960–2011), (1930–1950; 1951– 2011). In addition, the Table 3allows for alternative pairs of values of high and low persistence parameter. As we can see, findings on average of conditional moments of rates of return are remarkably robust to alternative learning assumptions implied by the different sample splits. To help us understand these results (which were obtained numerically), we consider analytical approximations23 for the rates of return for the case where persistence is known, for example, with ηt=0. The risk-free rate is approximated as rf t=−lnβ+γg +γρ xt−γ2 2σ2 x+σ2 g+ρ2 Vart(xt)(16) where ˜ xtis the mean of the distorted posterior at time t. An increase in ambiguity aversion, α, decreases xtmaking the agent behave as if he were expecting a lower endowment income in future states. Implying, a rise in demand fortherisk-freeasset(a“flighttoquality,”astermedbyCaballero and Krishnamurthy (2008)) driving up its equilibrium price and lowering the risk-free rate. The accentuation of doubt, working through Vart(xt)reinforces the effect. This is a key effect of ambiguity aversion. Note, when α>0the term γρ xtacts to dampen the effect of γg, making the comparative static of γon the risk-free rate very different, qualitatively and quantitatively, depending on whether α>0or α=0, as a comparison of the middle and right panels of Figure 4shows. Hence, it is not possible to replicate the effect of ambiguity aversion by turning it off and simply varying γ. The first moment of the risky rate is approximated as Etrt=Const1+ρ(γ −ψ) ˜ xt+ψρ ˆ xt−ρ2 2(γ −ψ)2Const2 Vart(xt)(17) where Et≡Eˆ xtExtdescribes the conditional expectation of a Savage–Bayes’ rational observer/analyst who observes these prices and uses the same information as the agent to 22When trying to infer how ambiguity aversion affects returns/prices from the entries in Table 2,bearin mind αis calibrated so that rfis equal to 15%.So,whenγis changed, αdoes also to ensure calibration. From Table 2, if one wants to infer anything about a change in ambiguity aversion alone, then one can compare what happens in the Bayesian case (γ=25and α=0) with our benchmark case (γ=25,α=113). 23See Appendix Cfor details of the derivation.
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 965 Table 2. The top panel contains the average of the predicted conditional moments of rates of return (on dividend claim) for different values of γand calibrated α. Immediately below is a series of robustness checks where the parameter in the left-most column was changed from the basic specification (ρh=085,ρl=03,ψ=3,β=0975), taking γ=25as part of the baseline specification. The bottom panel contains the time-averaged model implied price/dividend ratio statistics over the period 1978–2011. AC1 and AC2 denote the firstand second-order autocorrelation of p−d. Returns and Volatility γ α E(r) E(r −rf)σ(r f)σ(r)σ(r−rf) Data 808 668 220 165161 10315661 508 120 222222 20178736 585 258 230230 25113797 646 329 2355 236 30665 866 714 396 2417 242 Robustness Checks ρh=090 25730 788 636 383 235236 ρl=025 25111798 648 305 237237 ψ=250 25113758 607 315 236235 β=0965 25130915 762 344 238238 β=097 25122856 705 336 237237 Bayesian 25≈0762 062 170 231232 Price-Dividend Ratio γαE(P/D) σ(P/D) E(p −d) σ(p −d) AC1 AC2 Data 45513 19954 3724 0445 0803 0759 10315293434 337 015 051 048 20178323592 346 019 065 060 25113440145373 034 085 078 30665 529222388 043 088 081 Robustness Checks ρh=090 25730 429137371 033 084 078 ρl=025 25111443148374 035 085 078 ψ=2525113396111364 029 082 075 β=0965 25130599281398 049 089 082 β=097 25122513206386 042 088 081 Bayesian 25≈0400115365 030 082 075 Bayesian, β=097 25≈0463165377 037 086 079 predict dividend at t+1.Const 1and Const2collect terms which are constant across time and not affected by ambiguity aversion. An increase in αhas two countervailing effects. The first effect, given by ργ xt, was also present in the expression for the risk-free rate;
966 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) Figure 4. Comparative statics. In the left panel, αvaries with γfixed at 25. In the middle panel, αwas fixed at 113and γvaries. The average comparative statics are constructed by first computing the comparative statics for each year using the filtered values ˆ xtand then averaging across t=19782011. The right panel depicts the Bayesian case, that is, with α≈0.(Thegraphscorrespond to our model with unknown persistence.) the intuition here is analogous. The second effect is in the term −ρψ xt.Asαincreases xt decreases, hence decreasing the (“as if”) expected future dividend payoff from the asset causing the agent to want to pay less for the asset. With γ≤3and ψ=3,aswehavehere, Table 3. Robustness of returns moments to the sample split. Returns and Volatility (ρlρh) γ α E(r) E(r −rf)σ(r f)σ(r)σ(r−rf) Data 808 668 220 165161 1969–2011 evaluation period (030085)200 17100 7475 5981 2398 23056 23056 (030085)250 10900 8084 6575 3061 23661 23653 (030085)300 6400 8775 7277 3699 24352 24329 (025085)250 11050 8096 6602 2937 23766 23730 (030090)250 7390 8036 6534 3420 23881 23842 1960–2011 evaluation period (030085)200 15050 7064 5555 2276 24412 24409 (030085)250 9500 7592 6089 2863 25088 25074 (030085)300 5535 8196 6690 3430 25845 25815 (025085)250 9150 7634 6128 2671 25244 25204 (030090)250 6030 7630 6121 3105 25401 25354 1951–2011 evaluation period (030085)200 24270 6270 4763 4612 27183 27464 (030085)250 15100 6888 5389 5594 28150 28490 (030085)300 8695 7587 6085 6473 29254 29634 (025085)250 15100 6957 5456 4738 28327 28538 (030090)250 7840 7075 5574 3859 28711 28760
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 967 the second effect dominates (very slightly) and equilibrium risky rate varies positively (but quite minimally) with ambiguity aversion. The approximation for the equity premium may be written as Etrt−rf t=Const3+ψρ( ˆ xt−˜ xt)+ρ2 2γ2−(ψ −γ)2Const2 Vart(xt) (18) where we have explicitly left the two terms which are affected by ambiguity aversion, (ˆ xt−˜ xt)and Vart(xt). The first term shows that the premium increases with ambiguity aversion (the difference ( xt− xt)increases when αis increased) and the magnitude of this effect is accentuated by persistence and leverage. A doubt factor also comes into play (principally) through its effect on the risk-free rate, discussed earlier. Since, the riskfree rate is conditionally nonstochastic, the conditional volatility of equity premium coincides with that of the risky rate. The overwhelming factor fixing the (average) conditional volatility of risky return is the volatility of the dividend claim, in turn determined by the volatility of the latent state multiplied by ψand ρ. To summarize, ambiguity aversion gets the first moment of equity premium right by holding the risk-free rate down while affecting the risky rate only very marginally. The volatility comes from two sources, the uncertainty about the latent state accentuated by the uncertainty about the persistence and the leverage factor. 3.3.2 Time series profiles of conditional rates of return and price-dividend ratio Perhaps the more distinctive results of the analysis in this paper concerns the time series of conditional moments. These are largely driven by dynamics of the “as if” belief explained in Section 3.2.2. Figure 5demonstrates this quite vividly in the case of the equity premium, especially the crucial role of the uncertainty about persistence. Studies have estimated conditional moments of equity premium on historical data, notably Whitelaw (1994)andLettau and Ludvigson (2010). The former summarizes a key finding as follows (p. 526; in the quote “expected return” is the conditional first moment of equity returns in excess of risk-free rate): “The expected return seems to reach a maximum at the trough of the business cycle and reach a minimum before, or at, the peak of the business cycle. Expected returns appear Figure 5. Movements in variance and model implied equity premium. Panel (a) shows the conditional equity premium and the conditional variance of the “as-if” posterior from the model with known persistence, ρ=085. Panel (b) shows the same as well as the variance of the undistorted posterior for the model with unknown persistence. Vertical dashed lines indicate years featuring a recession.
968 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) Figure 6. The two panels depict the conditional variance of (excess) returns and Covt(zt+1 dt+1), implied by the model, demonstrating the close link between the dynamics of the two. Vertical dashed lines indicate years featuring a recession. to decrease during economic expansions and increase during economic contractions. In contrast, the conditional volatility appears to reach a maximum earlier in the business cycle, at or slightly after the peak in the cycle, and to reach a minimum just after the business cycle trough.” Figures 5(b) and 6(a) show how well the series predicted by our model match the above quote. Equity premium, as predicted by the model, is countercyclical; its correlation with H-P filtered consumption growth is −059.Whitelaw (1994) estimated the contemporaneous correlation between the first and second (conditional) moments (of equity premium) to be −034; based on the data considered for this paper, which pertains to a different time period and frequency, the correlation of the same two statistics in our model is −086. What accounts for the pro-cyclical volatility of returns in our model? Starting from the standard approximation for the risky rate (equation (33) in Section Cin the Appendix), its variance may be seen to be composed as Vart(rt)≃κ2 1Vart(zt+1)+Vart(dt+1)+2κ1Covt(zt+1dt+1) (In our data, κ1=098.) It turns out the time averaged variance is completely swamped by the term Vart(dt+1)(Vart(rt)=00555,Vart(dt+1)=00541 and Vart(zt+1)=117e−4). However, as seen from Figure 6, the dynamics of Vart(rt)are very largely determined by Covt(zt+1dt+1). To see an intuition why this covariance is negative and even more so in recessionary times, note that the belief about dt+1is determined by the Bayesian posterior, with mean ˆ xt, while zt+1is guided by ˜ xt, the mean of the distorted posterior. As explained in Section 3.2.2,˜ xtis below ˆ xt, and even more so and less mean reverting (i.e., more persistent) than ˆ xtin recessions. Hence, in recessions there is a bigger measure of events where dt+1realizes above its mean while zt+1stays below its mean. The price-dividend ratio is function of the agent’s view of the longer term prospects while the dividend is just the outcome in the next period; the former may remain relatively downbeat and sluggish, especially in recessionary times, despite a positive outcome of the latter. Together, the countercyclical variation of the mean and the increase in volatility during recessions leads to countercyclical variation of the conditional Sharpe ratio,
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 975 Table 9. Countercyclical persistence: Columns 2 and 3 show estimates, corresponding to the time series indicated in column 1, of the AR(1)parameter and between parenthesis its standard deviation and the associated Student-tstatistic in years with and without recessionary episodes, respectively; the final columns show the p-value of the test for statistical significance of the difference in estimates in columns 2 and 3 and the associated level of significance. The final row of the table shows these numbers for the series obtained from the model with ambiguity neutrality (i.e., α≃0). Persistence in Years p-Value Level of w/ Recession w/o Recession Test Significance (1) (2) (1)–(2) Model Cond. Eqty. Prm. 076 055 0004 100% (012;619)(012;475) JLN Uncertainty index 089 048 0272 73% (022;406)(026;184) Bayes case Eq. Prm. (α≃0)068 069 0436 –% (013;518)(013;516) is symmetric with respect to the sign of shock. It is ambiguity aversion that is responsible for the asymmetric behavioral response to good and bad news and for increasing the (“as if”) belief on high persistence in recessionary periods, the key mechanism in our model. Could these features obtain in a model with stochastic volatility but no ambiguity aversion? As discussed earlier, investigations have shown that the evident consumption volatility in data has neither the right variation over time nor the size needed to explain the observed time variation in equity premium and the Sharpe ratio. Carriero, Clark, and Marcellino (forthcoming) (CCM henceforth) construct a measure of macro-uncertainty based on a large vector autoregression with stochastic volatility driven by common factors representing macroeconomic uncertainty. The index reflects changes in both the conditional mean and volatility of the variables. An advantage of this approach over JLN is that, the authors argue, it reduces the risk of biases and Figure 9. Dynamic correlations of (log) price/dividend ratio with JLN uncertainty index. Note: The graphs report the correlations corr(JLNt(p−d)t+k),fork=−88, where JLNtis the JLN uncertainty index and (p −d)t+kis the log of the price/dividend ratio (in data and model implied) evaluated at various leads and lags.
976 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) endogeneity problems stemming from measurement errors and omitted variables. As panels (c) and (d) of Figure 8show, compared to the JLN measure this index and our model implied conditional equity premium are even more closely related: the correlation is 073. both in levels and in differences. Recently, Orlik and Veldkamp (2014) have constructed a measure of macroeconomic uncertainty which also comes with a theory why such uncertainty is more countercyclical than stochastic volatility alone. In their model the agent does not know the true distribution of macroeconomic outcomes, but estimates its parameters in the way of a Bayesian econometrician using real time (GDP) data. They measure uncertainty as the conditional standard deviation of GDP growth, which captures uncertainty about the distributions’ estimated parameters. When the forecasting model admits only normallydistributed outcomes, they find small, acyclical changes in uncertainty. But when the forecasting model is enlarged in a specific way, so that agents also estimate parameters that regulate skewness, uncertainty fluctuations become more pronouncedly countercyclical. However, they find the uncertainty diminishes secularly and significantly due to the learning of the parameters. To rectify this, they add an exogenously specified stochastic volatility component which, like in Bansal and Yaron (2004), has a persistence that is independent of the business cycle. They report that their measure has a correlation of 031 with the JLN uncertainty index (recall, for our model this correlation is 058). 4. Assessing the calibrated value of ambiguity aversion Here, we discuss a way of assessing the plausibility of the calibrated levels of ambiguity aversion in terms of implied individual (as opposed to market) behavior. In standard analysis of the equity premium question, the value of (relative) risk aversion parameter is motivated by using a thought experiment; the typical question being how much an agent would pay to avoid a given risk. Arguably, neither the question nor the intuitive answer refers to the expected utility model, or any formal model of decision making for that matter. We now consider as a thought experiment the implied uncertainty premium of an individual investor with preferences and dynamic belief, precisely like the agent in our model, evaluating a Lucas’ tree prospect. We find the investor is willing to pay an overall uncertainty premium (a sum of the risk premium and the ambiguity premium) that is well within the bounds of what is regarded as intuitively plausible per the standard intuition and analysis. Our thought experiment consists of an offer at time tto our Lucas’ economy agent, with preference parameters (γαβ), to replace the uncertain consumption prospect he faces with a fixed consumption in each period, now and for ever. Define the consumption certainty equivalent,c(γαβ;ct),tobethecthat makes the agent indifferent, given information at t, between the plan (ccc) and his endowed stochastic consumption plan (ctct+1).Hence,c(γα =0β;ct)is the certainty equivalent for the Bayesian agent and c(γ =0α=0β;ct)is the certainty equivalent for a risk neutral agent [hence, sum of discounted conditional expectations.] The risk premium is R(γ 0β;ct)≡c(00β;ct)−c(γ0β;ct),andtheambiguity premium is
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 977 Table 10. Uncertainty premia in the thought experiment:We report the time average of γ(γαβ;ct)computed at each ton thesamplepath. β=0975 β=0965 γ20253025 α1775 1135 665 1300 γ(γαβ) 348793 351117 352169 376459 A(γαβ;ct)≡c(γ0β;ct)−c(γαβ;ct). Then the total uncertainty premium paid by our agent with preference parameters (γαβ)is given by U(γαβ;ct)≡R(γαβ;ct)+A(γαβ;ct) Hence, when α=0,U(γ0β;ct)=R(γ0β;ct). Finally, define γ(γαβ;ct), to be the value of the relative risk aversion parameter which solves the following equation: Rγα=0β;ct=U(γαβ;ct)⇔cγ0β;ct=c(γαβ;ct) (19) On the left and right of the first equality in (19) we have, respectively, the uncertainty premium of an ambiguity neutral and the uncertainty premium of an ambiguity averse agent, both facing the same uncertain prospect as the agent in our model. Table 10 reports calculations with γ=2253and αset to the corresponding calibrated values used in our model. Hence, our agent is calibrated to pay as much uncertainty premium (in total) as a standard expected utility agent with relative risk aversion around 35.Almost every equity premium study in the literature considers this amount of uncertainty premium very much within the range of plausibility in the context of a financial economy (Mehra and Prescott (1985), for example, had argued on this basis that γ≤10 was plausible). In this sense, the calibrated uncertainty attitude parameters, taken together,make a plausible preference configuration for an individual DM in a financial economy. The fact that our ambiguity averse agent is paying the same overall uncertainty premium as an expected utility maximizer with risk aversion 35may seem odd given how large the average model implied equity premium we find. Note, however, that the (conditional) equity premium is calculated, as is standard, by taking the expectation of risky rate (implied by the model) with respect to the posterior predictive distribution, as a Savage–Bayes’ ambiguity neutral outside observer would evaluate it. If we were to compute Er −rfas our ambiguity averse agent would, it would be much smaller given the pessimism of his evaluation functional. Thus, the equity premium the agent perceives he is paying is consistent with the uncertainty premium values implied by the values of γin the Table 10.
978 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) 5. Related literature We describe next how the analysis here relates to other explanations in the literature (of the observed behavior of equity premium) based on aggregate uncertainty in representative agent frameworks. Bansal and Yaron (2004) pioneered the use of the (basic) model of beliefs we apply to show how long run risk (LRR) and aversion to such risk (while allowing a Kreps and Porteus (1978)/Epstein and Zin (1989)/Weil (1989) like separation of IES from risk aversion) could explain aspects of the observed equity premium. The changes we introduce are: (1) letting the belief about the latent state be the full Bayes’ posterior, instead of degenerate, probability-one-belief on the filtered state; (2) letting the agent be uncertain about the value of the persistence parameter; (3) letting the agent preferences treat (1) and (2) as ambiguity without separation of IES from risk aversion. We show these changes are sufficient to yield a model of beliefs where the (endogenously accentuated) uncertainty varies enough over time, without resorting to an exogenously specified stochastic volatility. Bansal and Yaron’s Case II model assumed an exogenous stochastic volatility. In our model, notice the volatilities, σxk,σdk,σgk, are different conditional on the value of persistence, ρk, but since there is “one true model” for all time, the volatility is nonstochastic, per se. Our agent is agnostic about the value of persistence and never puts probability one on either value, k=h l. However, as has been explained in Section 3.2.2, depending on history, and because of ambiguity aversion, the agent amplifies the posterior probability mass on one or the other value, therefore, creating an endogenously accentuated stochastic volatility, that is, the uncertainty about the value of σ’s are accentuated endogenously. In Hansen and Sargent (2010), countercyclical risk prices are driven by a representative investor’s robust model averaging and a preference for early resolution of uncertainty. The investor carries along two difficult-to-distinguish models of consumption growth, one asserting i.i.d. log consumption growth, the other asserting that the growth in log consumption is a process with a slowly moving conditional mean. The investor uses observations on consumption growth to update a Bayesian prior over these two models, starting from an initial prior probability of 05. Each period, the agent expresses his specification distrust by pessimistically exponentially twisting a posterior over the two baseline models. That leads the investor to interpret good news as temporary and bad news as persistent, causing him to put countercyclical uncertainty components into the equilibrium price of risk. Our framework is inspired by Hansen and Sargent (2010).Wherewedepartistherole of ambiguity in the driving mechanism and in the quantitative match obtained. Their agent believes the economy evolves according to a model like we have here but processes belief differently, by applying two “risk-sensitivity operators.” The first operator, which may be interpreted as a Kreps and Porteus (1978) style preference for earlier resolution of risk, applies to the evaluation (of the consumption plan) conditional on each of the two values of ρ. The other operator may be interpreted as a KMM2005 style smooth ambiguity aversion transformation where the agent’s second order uncertainty is a twopoint (Bernoulli) belief, where each point in the support is the conditional evaluation
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 979 given a ρ. Hence, while uncertainty about the two values of ρis treated as ambiguity, the uncertainty about the latent state, given ρ,isnot processed as ambiguity, unlike in our model. Thus, the results they obtain have their origin both in ambiguity aversion and an IES >1.28 Ju and Miao (2012) use a modified smooth ambiguity framework to assess the effect of ambiguity on dynamics of asset prices. In the model of beliefs, there the latent state variable driving the (mean) growth rate in the economy may take only two possible values. The preference model also incorporates an IES effect, in addition to ambiguity aversion, with the IES parameter set at 15. They produce statistics on unconditional moments of returns and prices, by averaging across simulated, counterfactual paths, which match data well. They also report, using graphs, model implied conditional returns and prices along the observed, historical sample path; here, their model is evidently less successful. As panel B in their Figure 3 shows, throughout the post-war period the (second-order) belief has been almost completely stuck (virtually Dirac) on the same latent (high-growth) state. Hence, the results we obtain about predicted time series of moments of conditional returns (even countercyclical equity premium) could not be obtained in their model if actual history were applied.29 The part of Collin-Dufresne, Johannes, and Lochstoer (2016) most closely related to ours applies model/parameter uncertainty and Bayesian learning in a framework where the beliefs about the growth process is anchored to an uncertainty about whether the true process is LRR or i.i.d. They show that even a small probability of the LRR model being the true model leads to significant increase in the risk premium compared to the case in which consumption growth is known to be i.i.d. They also show that this uncertainty creates countercyclical fluctuations in the equity premium. However, as we underlined in the Introduction, the driving force in the agent’s preferences is an IES >1 (they consider values 15and 2, together with a relative risk aversion of 10). The mechanism at work is thus different from ours, as ambiguity aversion plays no role in their model. Drechsler (2013) introduced ambiguity aversion alongside model uncertainty and an IES >1. He obtained good matches of time-average returns moments. He used a maxmin approach in which the set of priors, that represents uncertainty, varies over time in an exogenous way calibrated to an uncertainty index. Bidder and Dew-Becker (2016) is similar, in that they modeled ambiguity aversion using a worst case scenario, à la Gilboa and Schmeidler’s maxmin expected utility model. The worst case model is literally the homoskedastic version of Bansal and Yaron’s (2004) long-run risk model. They also apply an Epstein and Zin style IES effect, in addition to ambiguity aversion. 28We implemented, on our data set, an amended version their preference model with simply the second (KMM style) operator on the two-point belief but excluding the other, Krep–Porteus style operator. We find the predicted time-averaged equity premium (conditional on actual history) is about 06% and that the conditional equity premium has a negative correlation with the JLN index. 29Recently, Strzalecki (2013) has shown that it is theoretically possible that recursive ambiguity frameworks have some preference for early resolution inseparably mixed in with ambiguity aversion. Compared to the model in the present paper what is different about the preferences in Ju and Miao (2012) and Hansen and Sargent (2010) is that those include separate components explicitly adding preference for early resolution above and beyond what may be already mixed in with ambiguity aversion.
980 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) Veronesi (1999) constructed and theoretically analyzed a dynamic, rational expectations, expected utility representative agent model of asset pricing where beliefs are based on two hidden states (each specifying a mean growth rate) and showed that it implies time-varying expected returns and prices. However, it is a theoretical exercise and does not show what actual values and magnitudes are implied along information paths based on observed history. David and Veronesi (2013) studied time-varying uncertainty but not the equity premium per se. In their model, agents must learn which regime the economy is in through signals about growth and inflation. The learning mechanism relies on (possibly small) money illusion. Gollier (2011) showed analytically, using a (static) smooth ambiguity model, that an increase in ambiguity aversion may not, in general, increase the equity premium, thereby making a good case for empirical investigation of the question. Abel (2002), Cecchetti, Lam, and Nelson (2000), Giordani and Soderlind (2006), Jouini and Napp (2006), showed that exogenously introducing pessimism and doubt in beliefs can generate a realistic equity premium and risk-free rate. Our results are driven by similar elements of pessimism and doubt, but in our framework these arise endogenously. Barro (2006), and Weitzman (2007) showed that rare risks and/or heavy tails may contribute to the large equity premium and low risk-free rate observed in the data. Our contribution focuses on “common” uncertainty near the current growth rate rather than on “rare” uncertainty, and so is easier to relate to observed consumption data. Constantinides (1990)andCampbell and Cochrane (1999) studied models with habits in consumption which can match the level, variation, and countercyclicality of the equity premia, though, as we have observed, in these models consumption growth predicts the price-dividend ratio, unlike in the data and in our model. Habits effectively allow the risk aversion to vary endogenously over the business cycle. The crucial difference to our paper is that we have constant aversion (to ambiguity and risk) but our agent faces time-varying uncertainty and it is variation in that uncertainty, rather than variation in the aversion to it, which causes the returns and premia to vary. 6. Concluding remarks Our model applied uncertainty and learning about persistent hidden states describing the cyclical component, and about the level of persistence; treating both these uncertainties as ambiguous and incorporating a level of ambiguity aversion calibrated to match the average risk-free rate. The uncertainty and learning compatible with a Bayesian agent (but not with rational expectations), explain quite substantially the average volatility of returns and prices, and also the level of risky rate. Ambiguity aversion was important in explaining the levels of risk-free rate and equity premium, and for shaping the dynamics of all the variables, especially the first and second moments (conditional) equity premium through the channel of an endogenously accentuated “as if” uncertainty. Our results show that observed levels and movements of moments of asset returns can be explained on the basis of aggregate macroeconomic risk, conditional on the actual history of aggregate output growth reports. That both first and second moments
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 981 of conditional excess returns have the cyclical properties that match the data is a significant finding. As was the finding that the model implied conditional equity premium matches the time-series properties of the JLN macroeconomic uncertainty index, thereby giving a theory of uncertainty shocks and the countercyclical nature of their severity and persistence. Thus, consistent with JLN’s conjecture, we do find that Knightian uncertainty can provide a good explanation of dynamics of macroeconomic uncertainty. Finally, it is worth appreciating the minimality of the departure from expected utility that was sufficient to capture so many aspects of returns data. These observations are very suggestive of the potential for this approach in domains of macrofinance research where effects of endogenously time-varying uncertainty are of interest. In terms of future work, an interesting next step would be to replace the exchange economy with a production economy. Such a model would allow us to explore at least two important issues. First, it would allow us to understand the effect of ambiguity on output decisions (rather than just asset prices). Secondly, in turn, this would shed light on how uncertainty shocks, when they are endogenously accentuated by ambiguity aversion, contribute to the business cycle properties of the economy (see Backus, Ferriere, and Zin (2015) for a first discussion of these issues.) Appendix A: Data and estimation of parameters of the stochastic models Equity returns are computed using the CRSP value-weighted index. Dividend growth is imputed using the difference in the returns on the value-weighted index with and without dividends multiplied by the market value. The risk-free rate was taken from Ken French’s data library. Consumption is defined as the sum of services and non-durable consumption and was taken from BEA Table 1.1. Population was taken from BEA Table 2.2. Both per-capita consumption growth and dividend growth were converted to real terms using the average CPI for the year taken from the BLS. Annual data was available from 1930 until 2011, a total of 82 observations. Turning to preference parameters, in all cases the ambiguity aversion parameter α was calibrated to produce a real risk-free rate of 15%,averagedovert=19782011, which is the average observed rate in that period. No other moments were used in the choice of α. The relative risk aversion parameter γwas allowed to range between 1 (log utility) and 3, regarded as plausible in macroeconomic models (Ljungqvist and Sargent (2004, p. 426)); the “baseline” calibration set γ=25.30 The discount factor β was set to 0975, which corresponds to the discount rate used in BY. To check for robustness, we varied a number of the key non-estimated parameters, including ρ=09, β∈{0965097098},andψ=25. The long-run risk model was fit to annual data using maximum likelihood. Parameter estimates are shown in Table 11. All parameters, except ρand ψwere estimated using data 1930–1977. The mean of consumption and dividends, ¯ gand ¯ d, respectively 30If the two smooth ambiguity preferences do not share the same risk attitude, it is not necessarily true that a more concave φmeans more ambiguity aversion. Hence αis meaningfully calibrated given avalue of γ;notindependentofγ.
982 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) Table 11. Parameter estimates (standard errors below in parentheses) using annual data and the long-run risk model, shown above, using data from 1930 until 1977. All variance estimates and their standard errors have been multiplied by 100. ψ=3ψ=25 Parameter ρ=025 ρ=03ρ=085 ρ=09ρ=03ρ=085 ¯ g192 (0302)192 (0302)192 (0302)192 (0302)192 (0302)192 (0302) ¯ d231 (221)231 (221)231 (221)231 (221)202 (221)202 (221) σ2 g0048 (0016)0046 (0016)0025 (0010)0020 (0007)0047 (0017)0026 (0008) σ2 d449 (0893)451 (0892)475 (0909)473 (0902)464 (0914)481 (0918) σ2 x0054 (0013)0054 (0013)0051 (0019)0059 (0021)0054 (0013)0050 (0021) were set to their values in the period 1930–1977. The variances of the latent state process, consumption growth, and dividend growth were estimated using the Kalman filter. The dividend leverage parameter, ψ,wassetto3as in BY, although Constantinides and Ghosh (2010) estimated it to be slightly lower, close to the value we use for robustness checks (ψ=25). Appendix B: Details of the model B.1 Beliefs and the direct value function The agent believes that the stochastic evolution of the economy follows a persistent latent state process given by a BY type specification with either a low persistence (ρl)or a high persistence (ρh), but does not know for sure which. That is, he believes either of the models described in equation (9) represent the true data generating process. Define xkt ≡E[xkt |gk1gktdk1dkt],k=lh, to denote the filtered xat time t conditional on the observed history of growth rates (of consumption and dividend), if the history were interpreted and beliefs updated using a Kalman filter which takes the model with ρ=ρkas the data generating process. At any node on the growth path, at a time t, the agent’s beliefs may be summarized by the tuple ( xlt xhtηt),wherethefirst two elements show the beliefs about the latent state variable conditional on alternative assumptions about the true data generating process (low or high persistence, respectively) while the last element shows the posterior belief that the true data generating process is the low persistence model. We denote by x(i) kt+1,i=lh,k=lh,theagent’s forecast for the (one period ahead) update to his belief about the filtered xif the growth outcome next period (along with the previous history) were interpreted using a Kalman filter, which takes the model with ρ=ρkas the data generating process, when the data is actually generated by the ipersistence model. The direct value function obtains as follows: V(C t xlt xhtηt)=(1−β)C1−γ t 1−γ
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 983 −β αlnηt∞ −∞ exp−α∞ −∞ VCtexp(glt+1) x(l) lt+1(εlt+1) x(l) ht+1(εlt+1)η(l) t+1(εlt+1)dF(εlt+1)dF(xlt)(20) +(1−ηt)∞ −∞ exp−α∞ −∞ VCtexp(ght+1) x(h) lt+1(εht+1) x(h) ht+1(εht+1) η(h) t+1(εht+1)dF(εht+1)dF(xht) where εlt+1=[εxlt+1εdlt+1εglt+1]is a 3by 1vector of standard normal shocks (and so is εht+1)andηtis the posterior probability at time tthat the model with ρlisthedatagenerating process. F(εlt+1)and F(εlt+1)are both trivariate independent standard normal distributions. F(xkt),k=lh, is a normal distribution with mean xkt and variance Ωk, where Ωkis defined below. The updates for x(i) kt+1are obtained as follows: x(l) lt+1(εlt+1)=ρlˆ xlt +Klν(l) lt+1(21) x(l) ht+1(εlt+1)=ρhˆ xht +Khν(l) ht+1(22) x(h) lt+1(εht+1)=ρlˆ xlt +Klν(h) lt+1(23) x(h) ht+1(εht+1)=ρhˆ xht +Khν(h) ht+1(24) where ν(i) kt+1,(i) =(l) or (i) =(h) and k=lh, denote the “surprises.” For example, when the DGP is (i) =(l) and the filter uses ρk,k=h, the surprise is defined ν(l) ht+1=glt+1−¯ g−ρhˆ xht dlt+1−¯ d−ψρhˆ xht=¯ g−¯ g+ρlxlt −ρhˆ xht +σxlεxlt+1+σglεglt+1 ¯ d−¯ d+ψρlxlt −ψρhˆ xht +ψσxlεxlt+1+σdlεdlt+1 The Kalman gain parameters, Kk,k=lh, depending on whether low or high persistence model is assumed to be the true model, respectively, are Kk=ρkΩk1ψˆ F−1 kwhere ˆ Fk=Ωk+σ2 gkψΩk ψΩkψΩk+σ2 dk Finally, Ωk,k=lh, is defined as the solution to Ωk=ρ2 kΩk−ρ2 kΩ2 k1ψˆ F−1 k1ψ+σ2 xk The Bayes update of ηtis obtained as follows: η(l) t+1(εlt+1)=ηtLν(l) lt+1ˆ Fl ηtLν(l) lt+1ˆ Fl+(1−ηt)Lν(l) ht+1ˆ Fh(25) η(h) t+1(εht+1)=ηtLν(h) lt+1ˆ Fl ηtLν(h) lt+1ˆ Fl+(1−ηt)Lν(h) ht+1ˆ Fh(26)
984 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) where the likelihood is Lν(i) jt+1ˆ Fj=1 2π|ˆ Fj|exp−ν(i) jt+1ˆ F−1 jν(i) jt+1 2where i=lh and j=lh B.1.1 Mean and variance of the distorted posterior The mean of the distorted (or, “as if”) posterior is given by xt=ηt∞ −∞ (xlt)ξ(l) t(Ct xlt xhtηt)dF(xlt) +(1−ηt)∞ −∞ (xht)ξ(h) t(Ct xlt xhtηt)dF(xht ) (27) and the variance, by Vart(xt)≡ηt∞ −∞x2 ltξ(l) t(Ct xlt xhtηt)dF(xlt) +(1−ηt)∞ −∞x2 htξ(h) t(Ct xlt xhtηt)dF(xht)− x2 t B.1.2 The rates of return The risky rate of return is a function of four state variables, Ct, xlt, xht,ηt, just like Vand ξt. In the sequel, it should be clear that variables in t+1are evaluated using the relevant stochastic components. Let Ckt+1=Ctexp(gkt+1),k=lh. The risk rate, Rt, will satisfy βηt∞ −∞ ξ(l) t(Ct xlt xhtηt)∞ −∞ RtClt+1 x(l) lt+1 x(l) ht+1η(l) t+1 ×uexp(glt+1)dF(εlt+1)dF(xlt) +β(1−ηt)∞ −∞ ξ(h) t(Ct xlt xhtηt)∞ −∞ RtCht+1 x(h) lt+1 x(h) ht+1η(h) t+1 ×uexp(ght+1)dF(εht+1)dF(xht)=1 where ξ(l) t(Ct xlt xhtηt)= φ∞ −∞ VClt+1 x(l) lt+1 x(l) ht+1η(l) t+1dF(εlt+1) Ψ(28) and ξ(h) t(Ct xlt xhtηt)= φ∞ −∞ VCht+1 x(h) lt+1 x(h) ht+1η(h) t+1dF(εht+1) Ψ(29)
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 991 Cecchetti, S., P. S. Lam, and C. M. Nelson (2000), “Asset pricing with distorted beliefs: Are equity returns too good to be true?” The American Economic Review, 90, 787–805. [980] Chen, H., N. Ju, and J. Miao (2014), “Dynamic asset allocation with ambiguous return predictability.” Review of Economic Dynamics, 17 (4), 799–823. [946] Cochrane, J. (1999), “New facts in finance.” Economic Perspectives, Federal Reserve Bank of Chicago,23(3).[971] Cochrane, J. (2008), “The dog that did not bark: A defense of return predictability.” Review of Financial Studies, 21 (4), 1533–1575. [971] Collin-Dufresne, P., M. Johannes, and L. Lochstoer (2016), “Parameter learning in general equilibrium: Asset pricing implications.” American Economic Review, 106 (3), 664– 698. [948,979] Constantinides, G. (1990), “Habit formation: A resolution of the equity premium puzzle.” The Journal of Political Economy, 98 (3), 519–543. [980] Constantinides, G. and A. Ghosh (2010), “Asset pricing tests with long run risks in consumption growth.” Working Paper 16618, Chicago Booth GSB and NBER. [955,956,982] David, A. and P. Veronesi (2013), “What ties return volatilities to price valuations and fundamentals?” Journal of Political Economy, 121 (4), 682–746. [980] Dow, J. and S. d. C. Werlang (1992), “Uncertainty aversion, risk aversion, and the optimal choice of portfolio.” Econometrica: Journal of the Econometric Society, 60 (1), 197–204. [946] Drechsler, I. (2013), “Uncertainty, time-varying fear, and asset prices.” The Journal of Finance, 68 (5), 1843–1889. [948,979] Epstein, L., E. Farhi, and T. Strzalecki (2014), “How much would you pay to resolve longrun risk?” American Economic Review, 104 (2), 2680–2697. [948] Epstein, L. and T. Wang (1994), “Intertemporal asset pricing under Knightian uncertainty.” Econometrica: Journal of the Econometric Society, 62 (2), 283–322. [946] Epstein, L. and S. Zin (1989), “Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework.” Econometrica: Journal of the Econometric Society, 57, 937–969. [978] Gallant, A. R., M. Jahan-Parvar, and H. Liu (2015), “Measuring ambiguity aversion.” Discussion paper, Department of Economics, Penn State University. [949] Gilboa, I. and M. Marinacci (2016), “Ambiguity and the Bayesian paradigm.” In Readings in Formal Epistemology, 385–439, Springer. [946] Giordani, P. and P. Soderlind (2006), “Is there evidence of pessimism and doubt in subjective distributions? Implications for the equity premium puzzle.” Journal of Economic Dynamics and Control, 30 (6), 1027–1043. [980]
992 Collard, Mukerji, Sheppard, and Tallon Quantitative Economics 9 (2018) Gollier, C. (2011), “Portfolio choices and asset prices: The comparative statics of ambiguity aversion.” TheReviewofEconomicStudies, 78 (4), 1329–1344. [946,980] Goyal, A. and I. Welch (2008), “A comprehensive look at the empirical performance of equity premium prediction.” Review of Financial Studies, 21 (4), 1455–1508. [971] Guvenen, F. (2009), “A parsimonious macroeconomic model for asset pricing.” Econometrica, 77 (6), 1711–1750. [970] Hamilton, J. (1989), “A new approach to economic analysis of nonstationary time series.” Econometrica, 57 (2), 357–384. [954] Hansen, L. (2007), “Beliefs, doubts and learning: Valuing macroeconomic risk.” American Economic Review, 97 (2), 1–30. [946] Hansen, L. and T. Sargent (2010), “Fragile beliefs and the price of uncertainty.” Quantitative Economics, 1 (1), 129–162. [946,948,956,978,979] Jouini, E. and C. Napp (2006), “Heterogeneous beliefs and asset pricing in discrete time: An analysis of pessimism and doubt.” Journal of Economic Dynamics and Control, 30, 1233–1260. [980] Ju, N. and J. Miao (2012), “Ambiguity, learning, and asset returns.” Econometrica,80(2), 559–591. [948,979] Judd, K. (1992), “Projection methods for solving aggregate growth models.” Journal of Economic Theory, 58 (2), 410–452. [957,958] Judd, K. (1998), Numerical Methods in Economics. MIT Press, Cambridge, MA. [958] Jurado, K., S. C. Ludvigson, and S. Ng (2015), “Measuring uncertainty.” American Economic Review, 105 (3), 1177–1216. [945,947,973] Klibanoff, P., M. Marinacci, and S. Mukerji (2005), “A smooth model of decision making under ambiguity.” Econometrica, 73 (6), 1849–1892. [946] Klibanoff, P., M. Marinacci, and S. Mukerji (2009), “Recursive smooth ambiguity preferences.” Journal of Economic Theory, 144 (3), 930–976. [946] Klibanoff, P., M. Marinacci, and S. Mukerji (2012), “On the smooth ambiguity model: Areply.”Econometrica, 80 (3), 1303–1321. [989] Koijen, R. and S. Van Nieuwerburgh (2011), “Predictability of returns and cash flows.” Annual Review of Financial Economics, 3, 467–491. [971] Kraay, A. and J. Ventura (2007), “The dot-com bubble, the bush deficits, and the U.S. current account.” In G7 Current Account Imbalances: Sustainability and Adjustment. 457– 496, National Bureau of Economic Research Inc. [970] Kreps, D. and E. Porteus (1978), “Temporal resolution of uncertainty and dynamic choice theory.” Econometrica, 46 (1), 185–200. [978] Lettau, M. and S. C. Ludvigson (2010), “Measuring and model variation in the risk-return trade-off.” In Handbook of Financial Econometrics, 617–690, Elsevier. B.V. [948,967,969]
Quantitative Economics 9 (2018) Ambiguity and the historical equity premium 993 Ljungqvist, L. and T. Sargent (2004), Recursive Macroeconomic Theory. The MIT Press. [981] Ludvigson, S. C. (2012), “Advances in consumption-based asset pricing: Empirical tests.” In Handbook of the Economics of Finance, 799–906, Elsevier. B.V. [948] Maccheroni, F., M. Marinacci, and D. Ruffino (2013), “Alpha as ambiguity: Robust mean– variance portfolio analysis.” Econometrica, 81, 1075–1113. [946] Mehra, R. and E. Prescott (1985), “The equity premium: A puzzle.” Journal of Monetary Economics, 15 (2), 145–161. [977] Mukerji, S. and J.-M. Tallon (2004), “An overview of economic applications of David Schmeidler’s models of decision making under uncertainty.” In Uncertainty in Economic Theory: Essays in Honor of David Schmeidler’s 65th Birthday (I. Gilboa, ed.), Routledge. [946] Mukerji, S. and J. M. Tallon (2001), “Ambiguity aversion and incompleteness of financial markets.” Review of Economic Studies, 68 (4), 883–904. [946] Orlik, A. and L. Veldkamp (2014), “Understanding uncertainty shocks and the role of the black swan.” Working Paper 20445, NBER. Available at http://www.nber.org/papers/ w20445.[976] Pohl, W., K. Schmedders, and O. Wilms (2015), “Higher-order effects in asset-pricing models with long-run risks.” Discussion Paper 14-68, Swiss Finance Institute. [955] Shephard, N. and A. Harvey (1990), “On the probability of estimating a deterministic component in the local level model.” Journal of Time Series Analysis, 11 (4), 339–347. [954,955] Strzalecki, T. (2013), “Temporal resolution of uncertainty and recursive models of ambiguity aversion.” Econometrica, 81 (3), 1039–1074. [979] Uhlig, H. (2010), “A model of a systemic bank run.” Journal of Monetary Economics,57 (1), 78–96. [946] Veronesi, P. (1999), “Stock market overreactions to bad news in good times: A rational expectations equilibrium model.” Review of Financial Studies, 12 (5), 975–1007. [979,980] Weil, P. (1989), “The equity premium puzzle and the risk-free rate puzzle.” Journal of Monetary Economics, 24 (3), 401–421. [978] Weitzman, M. (2007), “Subjective expectations and asset-return puzzles.” The American Economic Review, 97 (4), 1102–1130. [980] Whitelaw, R. F. (1994), “Time variations and covariations in the expectation and volatility of stock market returns.” Journal of Finance,49 (2), 515–541. [967,968] Co-editor Karl Schmedders handled this manuscript. Manuscript received 22 April, 2016; final version accepted 23 November, 2017; available online 16 January, 2018.