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Modeling and Forecasting Mortality With Economic Growth: A Multipopulation Approach

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Demography

Abstract

Research on mortality modeling of multiple populations focuses mainly on extrapolating past mortality trends and summarizing these trends by one or more common latent factors. This article proposes a multipopulation stochastic mortality model that uses the explanatory power of economic growth. In particular, we extend the Li and Lee model (Li and Lee 2005) by including economic growth, represented by the real gross domestic product (GDP) per capita, to capture the common mortality trend for a group of populations with similar socioeconomic conditions. We find that our proposed model provides a better in-sample fit and an out-of-sample forecast performance. Moreover, it generates lower (higher) forecasted period life expectancy for countries with high (low) GDP per capita than the Li and Lee model.

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Notes

  1. For an overview of the related literature, we refer to the section, “Mortality Developments and Economic Growth.”

  2. In Section B of Online Resource 1, we extend the discussion to a more general stochastic mortality model proposed by Hyndman et al. (2013) and find qualitatively similar results. Other multipopulation mortality models, such as those of Cairns et al. (2011b), Dowd et al. (2011), D’Amato et al. (2014), and Salhi and Loisel (2017), can be naturally incorporated in our framework.

  3. The first group includes the same population as the “low-mortality countries” discussed in Li and Lee (2005) except for West Germany. Because mortality data for West Germany are not available before 1956, the inclusion of West Germany would lead to a shorter calibration window for the group. The second group has the same populations as the set of eastern European countries in Li and Lee (2005).

  4. Belarus, Estonia, Latvia, Lithuania, Russia, and Ukraine.

  5. The Maddison Project data are expressed as 1990 international dollars, whereas the ERS data are expressed as 2010 international dollars.

  6. The increasing trend of the first principal component of GDP reflects the common trend in the GDP. In Online Resource 1, we plot the principal component(s) of GDP and the corresponding values of γ x for each group, both multiplied with –1 (Figs. S5 and S6 therein). In that case, we see that the principal component(s) of GDP have a similar trend as the K t in the Li-Lee model.

  7. Our notation corresponds with the definition in Schwarz (1978), and a higher BIC implies a better model fit. The BIC ratios are also used in the literature with a negative sign:

    \( \underset{BIC}{\sim } \) = –2 log \( \widehat{L} \) + m·log M.

    Then, a lower value of the \( \underset{BIC}{\sim } \) ratio indicates a better model fit.

  8. This mortality trend is denoted by κ in Lee and Carter (1992).

  9. The calibration period used in Li and Lee (2005) is from 1952 to 1996.

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Acknowledgments

The authors thank three anonymous referees, the editors, Geng Niu, Michel Vellekoop, seminar participants at the 20th International Congress on Insurance: Mathematics and Economics, Longevity 12, University of Kent, and Universitat de Barcelona for valuable comments.

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Appendix

Appendix

The Rotational Lee-Carter Model

This appendix offers a brief description of the rotational Lee-Carter (R-LC) model, which is a single-population model. For population i, the rotational Lee-Carter model is given by

$$ \log \kern0.2em {m}_{i,x,t}={a}_{i,x}+{b}_{i,x,t}{k}_{i,t}+{\upvarepsilon}_{i,x,t},\kern1.5em {\upvarepsilon}_{i,x,t}\overset{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }N\left(0,{\upsigma}_{\upvarepsilon_{i,x}}^2\right). $$
(23)

For each population i and age x, a i,x is the mortality level—that is, the average of log m i,x,t over time. Moreover, k i,t is the population-specific latent factor, and b i,x,t is the age-specific loading of k i,t . The latent factor k i,t is modeled by a random walk with drift as in Eq. (19). Li et al. (2013) referred to the loadings b i,x,t as the age pattern of mortality-decline rates. They are given by

$$ {b}_{i,x,t}=\left\{\begin{array}{cc}\hfill {b}_{i,x}^O,\hfill & \hfill \mathrm{if}\kern0.5em {l}_{i,t}<{l}_i^O,\hfill \\ {}\hfill \left(1-{w}_i(t)\right){b}_{i,x}^O+{w}_i(t){b}_{i,x}^U,\hfill & \mathrm{if}\kern0.5em {l}_i^O\le {l}_{i,t}<{l}_i^U,\hfill \\ {}\hfill {b}_{i,x}^U,\hfill & \mathrm{if}\kern0.5em {l}_i^U\le {l}_{i,t}.\hfill \end{array}\right. $$
(24)

In Eq. (24), \( {l}_i^O \)and \( {l}_i^U \) are the threshold and the ultimate life expectancy for population i, respectively. Moreover, \( {b}_{i,x}^O \) is the original age pattern of the mortality-decline rates, which are estimated from the original Lee-Carter model, and \( {b}_{i,x}^U \) is the ultimate age pattern of mortality-decline rate, which is flat for most ages and has the same (downward) trend for the old ages as \( {b}_{i,x}^O \). Finally, w i (t) is a time-varying weighting function, which varies from 0 to 1.

The R-LC model is used to project future life expectancies. Initially, when the projected life expectancy in year t is smaller than the threshold life expectancy, w i (t) equals 0, and the mortality-decline rates are set to be the original rates, \( {b}_{i,x}^O \). When the projected life expectancy exceeds the threshold life expectancy, w i (t) increases, and the actual age pattern of mortality-decline rates gradually rotate toward the ultimate rates. Finally, when the projected life expectancy exceeds the ultimate life expectancy, w i (t) equals 1, and the actual age pattern of mortality-decline rate are the same as the ultimate rates. In our estimation, we use the same parameter specifications as used by Li et al. (2013). So, for each population, we let \( {l}_i^O \) = 80 and \( {l}_i^U \) = 102, and \( {b}_{i,x}^U \) is flat for age groups 0 to 70–74. For the construction of the ultimate age pattern of mortality-decline rates and the weighting function, see Li et al. (2013).

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Boonen, T.J., Li, H. Modeling and Forecasting Mortality With Economic Growth: A Multipopulation Approach. Demography 54, 1921–1946 (2017). https://doi.org/10.1007/s13524-017-0610-2

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