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A parsimonious model of subjective life expectancy

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Abstract

On average, “young” people underestimate whereas “old” people overestimate their chances to survive into the future. Such subjective survival beliefs violate the rational expectations paradigm and are also not in line with models of rational Bayesian learning. In order to explain these empirical patterns in a parsimonious manner, we assume that self-reported beliefs express likelihood insensitivity and can, therefore, be modeled as non-additive beliefs. In a next step we introduce a closed form model of Bayesian learning for non-additive beliefs which combines rational learning with psychological attitudes in the interpretation of information. Our model gives a remarkable fit to average subjective survival beliefs reported in the Health and Retirement Study.

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Notes

  1. Our findings confirm results documented in a vast literature on subjective survival probabilities, see below.

  2. For women we do not observe such a clear convergent pattern even for this age group.

  3. That is, the prior and the posterior are conjugate distributions because both belong to the Beta distribution family.

  4. Tonks (1983) introduces a similar model of rational Bayesian learning in which the agent has a normally distributed prior over the mean of a normal distribution and receives normally distributed information.

  5. Besides the formal description of insensitivity to changes in likelihood (Wakker 2004), properties of non-additive beliefs are used in the literature for formal definitions of, e.g., ambiguity and uncertainty attitudes (Schmeidler 1989; Epstein 1999; Ghirardato and Marinacci 2002), as well as pessimism and optimism (Eichberger and Kelsey 1999; Wakker 2001; Chateauneuf et al. 2007). See Wakker (2010) for a textbook treatment.

  6. An axiomatic foundation under the assumption of CEU preferences is provided in Eichberger et al. (2007, 2010).

  7. Sensitivity analysis with respect to this assumption is presented in the online appendix.

  8. Results are available upon request.

  9. A qualitative difference is that inverse u-shaped experience translates into inverse u-shaped likelihood sensitivity.

  10. We owe this interpretation to an anonymous referee.

  11. In particular, it is not the purpose of this paper to analyze how updating of beliefs differs across particular idiosyncratic health shocks or other events that are regarded as relevant for survival belief formation in the literature, such as parental death. As we further discuss in our concluding remarks in Sect. 5, a more in depth analysis of survival belief formation based on idiosyncratic events using our framework is left for future research.

  12. Observe that this weighting scheme implies that our point estimates are identical to a regression based on average survival rates of each group. Parameter estimates from an un-weighted regression are similar and are available upon request.

  13. The value of the two-sided \(t\) test on the difference between the \(R^{2}\)s for men and women is 108.96 with a \(p\) value of 0.0. The values of Jarque-Bera test statistics for normality of the distribution of the bootstrapped \(R^{2}\)s (and their \(p\) values) are at 0.02 (0.99) for men and at 4.49 (0.10) for women so that a standard \(t\) test is applicable.

    Fig. 3
    figure 3

    Actual and predicted survival probabilities for psychologically biased Bayesian learning. Source Own calculations based on HRS, HMD, and SSA data

    Full size image

  14. The \(t\) statistic of the two-sided \(t\) test for equality of the point estimates is \(3.53\).

  15. In this argument, we can ignore for age 80 households the difference between the additive objective measure \(\pi \) and the additive subjective measure \( \tilde{b}\) because the bias factor \(\frac{2\phi ^{m-j}+h}{2+h}\) is close to one for that amount of experience.

    Fig. 5
    figure 5

    Probability weighting functions. Source Own calculations based on HRS, HMD, and SSA data

    Full size image

  16. An alternative interpretation is that focal point answers reflect ambiguity, cf. Hill et al. (2004).

  17. As Mike Hurd pointed out to us, the first selection effect was particularly severe for the early waves of the HRS because people were not followed into nursing homes in the past. Since we use the more recent waves of the HRS where people are in fact followed into nursing homes, selection effects may only play a role for the very old respondents in our sample, if at all.

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Acknowledgments

We thank Axel Börsch-Supan, Andrew Caplin, Alain Chateauneuf, Itzhak Gilboa, Mike Hurd, David Levine, Jürgen Maurer, Dan McFadden, Klaus Ritzberger, Susan Rohwedder, Aylit Romm, Martin Salm, Frank Sloan, Carsten Trenkler, Peter Wakker, David Weir, Joachim Winter, and an anonymous referee as well as several seminar participants at the Workshop on Subjective Probabilities and Expectations, Jackson Hole, Wyoming, 2007, at Universität Mannheim and the 2009 SAET Conference for helpful comments; and Frank Schilbach for his excellent research assistance. Financial support by the Social Security Administration (SSA) under the 2007 Steven H. Sandell Grant Program is gratefully acknowledged. Alex Ludwig further gratefully acknowledges financial support from the German National Research Foundation (DFG) through SFB 504, the State of Baden-Württemberg and the German Insurers Association (GDV). Alex Zimper is grateful for financial support from Economic Research Southern Africa (ERSA).

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  1. CMR, University of Cologne, Albertus-Magnus-Platz, 50923 , Cologne, Germany

    A. Ludwig

  2. Department of Economics, University of Pretoria, Private Bag X20, Hatfield, 0028, South Africa

    A. Zimper

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Ludwig, A., Zimper, A. A parsimonious model of subjective life expectancy. Theory Decis 75, 519–541 (2013). https://doi.org/10.1007/s11238-013-9355-6

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