Recursive-preference re-estimation and Bayesian model comparison for the consumption CAPM in Norway, Sweden, and the UK: is near-risk-neutrality an artifact of the CRRA restriction?

A 2024 nonparametric-Bayesian estimation of the constant-relative-risk-aversion (CRRA) consumption capital asset pricing model (CCAPM) for Norway, Sweden, and the United Kingdom found wide, near-zero risk-aversion posteriors for Norway and Sweden, a significant risk aversion of about 6.2 for the UK, and model-implied equity premia below 1% — far below observed historical premia. Because CRRA utility forces the elasticity of intertemporal substitution (EIS) to equal the inverse of relative risk aversion, these inferences may be artifacts of that single-parameter restriction rather than features of European data. The open problem is to estimate the consumption CAPM for these three countries under recursive (Epstein–Zin–Weil) preferences that separate risk aversion from the EIS, formally compare the two preference specifications by Bayesian marginal likelihoods, and determine whether near-risk-neutrality and the trivially small implied premia survive, together with robustness to consumption and dividend data construction. No published work has performed this estimation or comparison for these countries.

Background

The consumption capital asset pricing model (CCAPM) prices any asset by the covariance of its return with a stochastic discount factor (SDF) — the representative consumer's marginal rate of substitution between today's and tomorrow's consumption. Under constant relative risk aversion (CRRA) utility $u(C)=C^{1-\gamma}/(1-\gamma)$, the SDF is $M_{t}=\beta(C_{t}/C_{t-1})^{-\gamma}$, where $\beta$ is the subjective discount factor and $\gamma$ is the coefficient of relative risk aversion. CRRA utility is time-separable expected utility, and this forces the elasticity of intertemporal substitution (EIS) — the sensitivity of planned consumption growth to the real interest rate — to equal exactly $1/\gamma$, so one parameter governs two conceptually distinct attitudes: dislike of risk and willingness to move consumption across time. Hansen and Singleton (1982) introduced the generalized method of moments (GMM) estimation of this model and rejected it on US data; Mehra and Prescott (1985) showed that the average US equity premium (about 6% per year over government bonds) cannot be reconciled with the model for any plausible $\gamma$, the equity premium puzzle. Two classic responses were (i) heterogeneous-agent models with market frictions, and (ii) richer preferences: recursive (Kreps–Porteus / Epstein–Zin, 1989) utility, which separates risk aversion from the EIS and yields an SDF depending on both consumption growth and the return on aggregate wealth (the Epstein–Zin–Weil framework). Epstein and Zin (1991) estimated this specification on US data and rejected the time-additive expected-utility restrictions; Bansal and Yaron (2004) showed that recursive preferences with EIS above one can resolve several asset-pricing puzzles when consumption risk is persistent; Chen, Favilukis and Ludvigson (2013), estimating Epstein–Zin–Weil preferences semi-parametrically on US quarterly data, found relative risk aversion between 17 and 60 and EIS between 1.1 and 2.0 — decisively different from $1/\gamma$. Long-run international evidence (Dimson, Marsh and Staunton, 2003, 2008, covering 16–17 countries including Sweden and the UK over the 20th century) shows historical equity premia that are lower than the US benchmark but still of several percent per year, so the gap between model-implied and observed premia is a worldwide, still-open phenomenon. Against this backdrop, Solibakke (2024) estimated the CRRA-CCAPM for Norway, Sweden, and the UK on quarterly data 1980–2022Q2 (non-durables-and-services consumption; dividend-inclusive all-share stock indices) using a nonparametric Bayesian method: moment conditions are handled by nonparametric Bayes subject to overidentified moment conditions (NPB; Gallant, 2022) with semi-nonparametric (SNP) time-series densities, and the posterior is sampled with the Markov chain Monte Carlo (MCMC) Surface Sampling Algorithm (SSA) of Zappa, Holmes-Cerfon and Goodman (2018), a Monte Carlo method for densities supported on lower-dimensional manifolds, which also delivers the normalization constant (the marginal data density, usable for model comparison). The estimated yearly discount factor was about 0.99 in all three countries; the risk-aversion posterior was significant only for the UK ($\gamma\approx6.18$), while Norway gave a wide posterior ($\gamma\approx3.28$, standard error 2.5, mass near zero) and Sweden an insignificant $\gamma\approx0.2$ extending to risk neutrality; implied EIS values ranged from 0.16 (UK) to 5 (Sweden); and the CRRA-implied equity premium was below 1% everywhere. The restriction EIS $=1/\gamma$ is exactly what makes these estimates mutually constrained: Sweden's near-risk-neutrality forces an implausibly high EIS of 5, and no country's estimate comes close to matching observed premia. Whether relaxing the restriction — the natural next step the paper's literature review frames but which no subsequent work has carried out for these countries — changes these cross-country inferences is the open question.

Problem Statement

For Norway, Sweden, and the United Kingdom, using quarterly consumption (non-durables and services) and dividend-inclusive stock-index data of the kind and span used in the source study (1980–2022Q2, extendable as available): (1) estimate the consumption CAPM under recursive Epstein–Zin–Weil preferences, whose stochastic discount factor separates relative risk aversion $\gamma$ from the elasticity of intertemporal substitution $\psi$ and involves the unobservable return on aggregate wealth, by Bayesian estimation, reporting posterior distributions of $(\beta,\gamma,\psi)$ for each country; no particular estimator, likelihood basis, or sampler is prescribed, provided the quantities used in item (2) are computed on a stated common basis; (2) compare this specification against the CRRA restriction $\psi=1/\gamma$ by formal Bayesian model comparison (marginal data densities/normalization constants), stating the basis on which the two marginal likelihoods are made comparable and how sensitive the comparison is to the likelihood specification (SNP bases and lag selection, where these enter the implementation) and to prior choices; (3) determine whether the source study's cross-country inferences — near risk neutrality for Sweden, wide zero-including posteriors for Norway, significant $\gamma\approx6.18$ for the UK — are artifacts of the CRRA restriction or robust features of the data, and, for its trivially small model-implied equity premia (below 1%), determine whether these are attributable to that restriction, to the way the premium is computed in the source study, or robust; and (4) assess whether preference enrichment or alternative data treatments — for example the dividend/index construction about which the source itself cautions that the calculated index accuracy "may be periodized somewhat inaccurately giving small index return mistakes", different consumption aggregates, or higher frequency — can move the model-implied premia toward the observed historical premia of these countries.

The model-implied equity premium is used here in the source study's sense: the expected difference between the log return on equity and the log risk-free return, $E_{t}(\log(1+r^{e}_{t+1})-\log(1+r^{rf}_{t+1}))$, derived from the specification's stochastic discount factor. Under the source's lognormal CRRA derivation this difference reduces to $(2\beta-1)\cdot(\omega_{2}/2)$, where $\omega_{2}$ is the consumption-growth variance (0.0558% in the source), so that reported expression depends on $\beta$ and $\omega_{2}$ and not directly on $\gamma$; the premium half of item (3) is therefore posed as an attribution to be established rather than as a settled consequence of the restriction. Any change of convention (arithmetic rather than log premia, index rather than risk-free benchmark, sample rather than model-implied) is to be stated explicitly.

The two specifications compared in item (2) are related in two different senses, and the answer should say which one it uses: at the level of preferences, CRRA is the nested restriction $\psi=1/\gamma$ of the recursive class, so the comparison is a Bayes factor for that restriction and is sensitive to the prior placed on $\psi$; at the level of the source's nonparametric-Bayes implementation, the two estimated likelihood objects may sit on different moment manifolds and SNP bases, in which case they are non-nested as likelihood objects and their normalization constants are comparable only on a stated common basis (common data, likelihood dimension, and prior treatment).

Items (1), (2), and (4) jointly serve the single objective of item (3): establishing which preference specification these three countries' consumption and asset data support and whether separating risk aversion from intertemporal substitution changes what the data say about European consumers' risk preferences and the equity premium. All three countries are in scope, and item (3) is assessed separately for each country; findings that remain inconclusive or prior-dependent, or that differ between the risk-aversion/EIS inferences and the implied premia, are reported as such.

References

1. Solibakke, P. B. (2024). Constant relative risk aversion utility and consumption CAPM: discount factors and risk aversions for Norway, Sweden, and the UK. Cogent Economics & Finance 12(1), 2299609. DOI: 10.1080/23322039.2023.2299609. URL: https://doi.org/10.1080/23322039.2023.2299609
2. Zappa, E., Holmes-Cerfon, M., & Goodman, J. (2018). Monte Carlo on manifolds: Sampling densities and integrating functions. Communications on Pure and Applied Mathematics 71(12), 2609–2647. DOI: 10.1002/cpa.21783. URL: https://doi.org/10.1002/cpa.21783
3. Gallant, A. R. (2022). Nonparametric Bayes subject to overidentified moment conditions. Journal of Econometrics 228(1), 27–38. DOI: 10.1016/j.jeconom.2021.02.005. URL: https://doi.org/10.1016/j.jeconom.2021.02.005
4. Hansen, L. P., & Singleton, K. J. (1982). Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50(5), 1269–1286. DOI: 10.2307/1911873. URL: https://doi.org/10.2307/1911873
5. Mehra, R., & Prescott, E. C. (1985). The equity premium: A puzzle. Journal of Monetary Economics 15(2), 145–161. DOI: 10.1016/0304-3932(85)90061-3. URL: https://doi.org/10.1016/0304-3932(85)90061-3
6. Epstein, L. G., & Zin, S. E. (1989). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57(4), 937–969. DOI: 10.2307/1913778. URL: https://doi.org/10.2307/1913778
7. Epstein, L. G., & Zin, S. E. (1991). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. Journal of Political Economy 99(2), 263–286. DOI: 10.1086/261750. URL: https://doi.org/10.1086/261750
8. Issler, J. V., & Piqueira, N. S. (2000). Estimating relative risk aversion, the discount rate, and the intertemporal elasticity of substitution in consumption for Brazil using three types of utility function. Brazilian Review of Econometrics 20(2), 201–239. DOI: 10.12660/bre.v20n22000.2758. URL: https://doi.org/10.12660/bre.v20n22000.2758
9. Bansal, R., & Yaron, A. (2004). Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59(4), 1481–1509. DOI: 10.1111/j.1540-6261.2004.00670.x. URL: https://doi.org/10.1111/j.1540-6261.2004.00670.x
10. Chen, X., Favilukis, J., & Ludvigson, S. C. (2013). An estimation of economic models with recursive preferences. Quantitative Economics 4(1), 39–83. DOI: 10.3982/qe97. URL: https://doi.org/10.3982/qe97
11. Dimson, E., Marsh, P., & Staunton, M. (2003). Global evidence on the equity risk premium. Journal of Applied Corporate Finance 15(4), 27–38. DOI: 10.1111/j.1745-6622.2003.tb00524.x. URL: https://doi.org/10.1111/j.1745-6622.2003.tb00524.x
12. Dimson, E., Marsh, P., & Staunton, M. (2008). The worldwide equity premium: A smaller puzzle. In Handbook of the Equity Risk Premium (Chapter 11, pp. 467–514). Elsevier. DOI: 10.1016/b978-044450899-7.50023-3. URL: https://doi.org/10.1016/b978-044450899-7.50023-3