A continuous-time representative-agent model with Epstein–Zin recursive utility and recursive multiple-priors ambiguity placed simultaneously on the drift and on Poisson jumps of consumption growth (Ruan and Zhang, 2021) reproduces the historical U.S. equity premium and risk-free rate with a relative risk-aversion coefficient as small as 2, provided the drift- and jump-ambiguity bounds (a,b)(a,b) are chosen by calibration. Whether those bounds have any empirical content is open: the model class has never been estimated, the closest estimator of multiple-priors ambiguity is a continuous-time model that explicitly excludes jump-contaminated data, and the author's own long-run-risks extension states that estimation of the ambiguity parameters remains an open empirical task. The problem is to establish econometric identification of the ambiguity bounds jointly with the risk-aversion, intertemporal-substitution, and jump parameters in this framework (or to prove non-identification), to implement structural estimation with valid inference, and to determine whether empirically justified bounds support the model's decomposition of the equity premium into drift- and jump-ambiguity components.

Background

The equity premium is the historically large average excess return of the aggregate stock market over short-term risk-free interest rates in the United States (roughly 6 percent per year over the last century). Mehra and Prescott (1985) showed that a standard representative-agent consumption-based model with time-separable power utility cannot jointly reproduce the high equity premium and the low risk-free rate unless the coefficient of relative risk aversion (RRA) is implausibly large (above 40 in the original calibration), and a large RRA in turn forces an unrealistically high risk-free rate — the equity premium and risk-free-rate puzzles. Epstein and Zin (1989) recursive utility separates the RRA coefficient from the elasticity of intertemporal substitution (EIS), the parameter governing how desired consumption growth responds to interest-rate changes, but recursive utility alone does not resolve the puzzles. A separate research program attributes the missing premium to ambiguity (Knightian uncertainty): the decision maker does not know the probability law of consumption growth and evaluates utility under the worst element of a set of priors. Chen and Epstein (2002) gave this recursive multiple-priors utility a continuous-time formulation, but with ambiguity on the diffusion drift only. Bansal and Yaron (2004) proposed the long-run risks (LRR) framework, in which a small but highly persistent component of expected consumption growth, interacting with recursive preferences, helps explain the puzzles.

Ruan and Zhang (2021) combine these ingredients: consumption follows a jump-diffusion

dCtCt=μ dt+σ dBt+L dNt,\frac{dC_t}{C_t}=\mu\,dt+\sigma\,dB_t+L\,dN_t,

where BtB_t is a Brownian motion (the continuous diffusive shock), NtN_t is a Poisson counting process with constant intensity λ\lambda that generates rare disasters (L<0L<0 is the proportional jump size in consumption), dividends are a levered claim on consumption with leverage ϕ>1\phi>1, and the representative agent has Epstein–Zin multiple-priors utility with ambiguity represented by a density-process generator dmt/mt=−a dBt+b [dNt−λdt]dm_t/m_t=-a\,dB_t+b\,[dN_t-\lambda dt] — the density process is the Radon–Nikodym derivative relating the worst-case prior to the reference probability law, so a>0a>0 shifts probability toward adverse diffusion outcomes and b>0b>0 toward more frequent disasters — where the bounds a>0a>0 and b>0b>0 measure ambiguity about the drift and about the jump intensity respectively. Under a constant log price–dividend ratio the model yields closed forms for the equity premium and the risk-free rate; the equity premium decomposes into diffusion-risk, jump-risk, drift-ambiguity, and jump-ambiguity parts,

EP=ϕ(γσ+a)σ−λ[(1+L)−γ(1+b)−1][(1+L)ϕ−1],EP=\phi(\gamma\sigma+a)\sigma-\lambda\big[(1+L)^{-\gamma}(1+b)-1\big]\big[(1+L)^{\phi}-1\big],

with γ\gamma the RRA coefficient. Calibrating (a,b)(a,b) by moment matching, the authors match the observed U.S. equity premium and risk-free rate with γ=2\gamma=2, where the no-ambiguity model requires γ≈37\gamma\approx37; the plausibility of the calibrated drift bound is defended by appeal to the structural estimates of a related continuous-time multiple-priors model by Jeong, Kim and Park (2015), who estimate a single constant ambiguity bound by a martingale-regression method. The authors themselves list what remains open: the model is deliberately parsimonious (constant volatility, constant jump intensity, constant log price–dividend ratio, which suppresses the role of the EIS), extensions to long-run-risks dynamics are left to future work, no structural estimation or econometric inference for (a,b)(a,b) is provided, and how to justify the magnitude of the jump-ambiguity bound — on which the quantitative importance of jump ambiguity relative to drift ambiguity hinges — is unresolved. The author's own follow-up, Ruan (2021), extends the framework to continuous-time Bansal–Yaron-type long-run risks with Chen–Epstein multiple priors and shows, again by calibration with a log-linear approximation, that large ambiguity on the long-run-risk component improves the fit of the equity premium and risk-free rate at low RRA; its stated limitations again leave the estimation of the ambiguity parameters from data open.

The adjacent estimation literature does not cover this model class. Jeong, Kim and Park (2015) estimate a constant ambiguity bound in a continuous-time multiple-priors model and explicitly note that their method eliminates jump-contaminated observations and that accommodating genuine jump processes would make the identification of ambiguity dependent on the assumed jump law. Ju and Miao (2012) structurally estimate an ambiguity-and-learning model in the smooth-ambiguity family — where unknown probabilities enter a smooth objective that is nonlinear in probabilities, rather than a set of priors — in discrete time, without jump ambiguity; Collard, Mukerji, Sheppard and Tallon (2018) study smooth ambiguity in a Lucas-tree economy with a persistent latent state and calibrate (rather than estimate) the ambiguity-aversion level; Suzuki (2022) adds a Poisson jump to continuous-time smooth ambiguity and matches the equity premium, risk-free rate, and variance risk premium, again purely by calibration. Structural estimation here means recovering the model's parameters from data by an explicitly probabilistic method with a stated identification argument and valid standard errors or confidence sets — as distinct from calibration, where parameter values are hand-picked to match target moments. No such estimation exists for continuous-time multiple-priors models with simultaneous drift and jump ambiguity, and in those models the drift bound aa enters the equity premium additively with the risk-aversion coefficient γ\gamma, which makes the empirical separation of ambiguity aversion from risk aversion, and the measurement of the jump bound bb, exactly the questions on which the credibility of this resolution of the equity premium puzzle depends.

Problem Statement

Within the continuous-time consumption-based framework of Ruan and Zhang (2021) — consumption following dCt/Ct=μ dt+σ dBt+L dNtdC_t/C_t=\mu\,dt+\sigma\,dB_t+L\,dN_t with constant drift μ\mu, diffusion volatility σ\sigma, and Poisson jumps of constant intensity λ\lambda and proportional size L<0L<0 (BtB_t a Brownian motion, NtN_t a Poisson counting process), dividends a levered claim on consumption, a representative agent with Epstein–Zin multiple-priors recursive utility, and constant ambiguity bounds (a,b)(a,b) on the drift and jump components entering through the density generator dmt/mt=−a dBt+b [dNt−λdt]dm_t/m_t=-a\,dB_t+b\,[dN_t-\lambda dt] — give the ambiguity parameters empirical content:

(1) Identification. Determine formally whether the ambiguity bounds (a,b)(a,b), jointly with the preference parameters (γ,ψ)(\gamma,\psi) (relative risk aversion and elasticity of intertemporal substitution) and the jump parameters (λ,L)(\lambda,L) (jump intensity and proportional jump size), are identified from observable data (time-aggregated consumption growth, the risk-free rate, and equity returns; option prices or survey expectations may be added as auxiliary data). Either prove identification, exhibiting the moments, instruments, or likelihood features that deliver it, or prove that the ambiguity bounds are not identified, with an explicit observational-equivalence construction: two admissible parameter configurations that differ in (a,b)(a,b) and induce the same joint law of observables in the baseline data and in any of the auxiliary data (option prices or survey expectations) permitted above.

(2) Estimation. Under identification, construct a structural estimator of the identified parameters and implement it on historical data, reporting estimates with valid confidence sets for the ambiguity bounds, with inference that accounts for the latent jumps and the worst-case measure selection inherent in multiple-priors utility.

(3) Discipline of the jump bound. State the criterion by which values of the ambiguity bounds count as empirically or decision-theoretically justified, and the measure by which the jump-ambiguity component is judged quantitatively important, anchoring both to an external benchmark or a stated, source-supported reference magnitude for the jump bound bb, and use the estimated (or otherwise justified) bounds under that criterion to evaluate the model's central quantitative claim: whether empirically defensible values of the jump-ambiguity bound make the jump-ambiguity component of the equity premium quantitatively important, or whether the model's resolution of the equity premium and risk-free-rate puzzles at low risk aversion rests on implausibly large ambiguity bounds.

These three components jointly decide whether the ambiguity-based resolution of the equity premium puzzle in this model class has empirical content; a non-identification result in (1) exhibits precisely what the calibration is free to choose. In that branch component (2) is void, and the task is resolved when component (3) is answered using bounds justified under its stated criterion.

References

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