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Mortality models incorporating long memory for life table estimation: a comprehensive analysis

Published online by Cambridge University Press:  02 February 2021

Hongxuan Yan ,
Gareth W. Peters  and
Jennifer Chan
Hongxuan Yan*
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China Center for Forecasting Science, Chinese Academy of Sciences, Beijing100190, China
Gareth W. Peters
Affiliation:
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, EdinburghEH14 4AS, UK
Jennifer Chan
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Sydney, 2006, Australia
*
*Corresponding author. E-mail: [email protected]
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Abstract

Mortality projection and forecasting of life expectancy are two important aspects of the study of demography and life insurance modelling. We demonstrate in this work the existence of long memory in mortality data. Furthermore, models incorporating long memory structure provide a new approach to enhance mortality forecasts in terms of accuracy and reliability, which can improve the understanding of mortality. Novel mortality models are developed by extending the Lee–Carter (LC) model for death counts to incorporate a long memory time series structure. To link our extensions to existing actuarial work, we detail the relationship between the classical models of death counts developed under a Generalised Linear Model (GLM) formulation and the extensions we propose that are developed under an extension to the GLM framework known in time series literature as the Generalised Linear Autoregressive Moving Average (GLARMA) regression models. Bayesian inference is applied to estimate the model parameters. The Deviance Information Criterion (DIC) is evaluated to select between different LC model extensions of our proposed models in terms of both in-sample fits and out-of-sample forecasts performance. Furthermore, we compare our new models against existing models structures proposed in the literature when applied to the analysis of death count data sets from 16 countries divided according to genders and age groups. Estimates of mortality rates are applied to calculate life expectancies when constructing life tables. By comparing different life expectancy estimates, results show the LC model without the long memory component may provide underestimates of life expectancy, while the long memory model structure extensions reduce this effect. In summary, it is valuable to investigate how the long memory feature in mortality influences life expectancies in the construction of life tables.

Keywords

Life tableLife expectancyLee–Carter modelFractional integrated modelBayesian inferenceLong memoryGLARMA
Type
Original Research Paper
Information
Annals of Actuarial Science , Volume 15 , Issue 3 , November 2021 , pp. 567 - 604
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Institute and Faculty of Actuaries

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References

Beran, J. (1994). Statistics for Long-Memory Processes, vol. 61. CRC Press, New York.Google Scholar
Brouhns, N., Denuit, M. & Vermunt, J.K. (2002). A poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31(3), 373–393.Google Scholar
Cairns, A.J., Blake, D. & Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. Journal of Risk and Insurance, 73(4), 687–718.CrossRefGoogle Scholar
Cairns, A.J., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A. & Balevich, I. (2009). A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal, 13(1), 1–35.CrossRefGoogle Scholar
Cossette, H., Delwarde, A., Denuit, M., Guillot, F. & Marceau, É. (2007). Pension plan valuation and mortality projection: a case study with mortality data. North American Actuarial Journal, 11(2), 1–34.CrossRefGoogle Scholar
Currie, I. (2006). Smoothing and forecasting mortality rates with p-splines, technical report, The Institute of Actuaries, Heriot Watt University, London, UK.Google Scholar
Currie, I.D. (2013). Fitting models of mortality with generalized linear and non-linear models, technical report, Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, United Kingdom.Google Scholar
Currie, I.D. & Durban, M. (2002). Flexible smoothing with p-splines: a unified approach. Statistical Modelling, 2(4), 333–349.CrossRefGoogle Scholar
Currie, I.D., Durban, M. & Eilers, P.H. (2004). Smoothing and forecasting mortality rates. Statistical Modelling 4(4), 279–298.CrossRefGoogle Scholar
Czado, C., Delwarde, A. & Denuit, M. (2005). Bayesian Poisson log-bilinear mortality projections. Insurance: Mathematics and Economics, 36(3), 260–284.Google Scholar
Davidson, J. & De Jong, R.M. (2000). The functional central limit theorem and weak convergence to stochastic integrals ii: fractionally integrated processes. Econometric Theory, 16(5), 643–666.CrossRefGoogle Scholar
Davis, R.A., Dunsmuir, W.T. & Wang, Y. (1999). Modeling time series of count data. Statistics Textbooks and Monographs, 158(3), 63–114.Google Scholar
De Jong, P. & Heller, G.Z. (2008). Generalized linear models for insurance data, vol. 1. Cambridge University Press, UK.CrossRefGoogle Scholar
Delwarde, A., Denuit, M. & Eilers, P. (2007a). Smoothing the Lee–Carter and poisson log-bilinear models for mortality forecasting: a d log-likelihood approach. Statistical Modelling, 7(1), 29–48.CrossRefGoogle Scholar
Delwarde, A., Denuit, M. & Partrat, C. (2007b). Negative binomial version of the Lee–Carter model for mortality forecasting. Applied Stochastic Models in Business and Industry, 23(5), 385–401.CrossRefGoogle Scholar
Duane, S., Kennedy, A.D., Pendleton, B.J. and Roweth, D. (1987). Hybrid monte carlo. Physics Letters B, 195(2), 216–222.CrossRefGoogle Scholar
Forfar, D., McCutcheon, J. & Wilkie, A. (1988). On graduation by mathematical formula. Journal of the Institute of Actuaries, 115(1), 1–149.CrossRefGoogle Scholar
Fung, M.C., Peters, G.W. & Shevchenko, P.V. (2015). A state-space estimation of the Lee-Carter mortality model and implications for annuity pricing. arXiv preprint .CrossRefGoogle Scholar
Fung, M.C., Peters, G.W. & Shevchenko, P.V. (2017). A unified approach to mortality modelling using state-space framework: characterisation, identification, estimation and forecasting. Annals of Actuarial Science, 11(2), 343–389.CrossRefGoogle Scholar
Fung, M.C., Peters, G.W. & Shevchenko, P.V. (2019). Cohort effects in mortality modelling: a Bayesian state-space approach. Annals of Actuarial Science, 13(1), 109–144.CrossRefGoogle Scholar
Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by browne and draper). Bayesian Analysis, 1(3), 515–534.CrossRefGoogle Scholar
Gelman, A. & Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4), 457–472.CrossRefGoogle Scholar
Geman, S. & Geman, D. (1984). Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1(6), 721–741.CrossRefGoogle Scholar
Giacometti, R., Bertocchi, M., Rachev, S.T. & Fabozzi, F.J. (2012). A comparison of the Lee–Carter model and AR–ARCH model for forecasting mortality rates. Insurance: Mathematics and Economics, 50(1), 85–93.Google Scholar
Girosi, F. & King, G. (2008). Demographic Forecasting, vol. 2. Princeton University Press, UK.CrossRefGoogle Scholar
Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115(1), 513–583.Google Scholar
Granger, C.W. & Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis, 1(1), 15–29.CrossRefGoogle Scholar
Harvey, A.C. (1990). Forecasting, Structural Time Series Models and the Kalman Filter, vol. 1. Cambridge University Press, UK.CrossRefGoogle Scholar
Hosking, J.R. (1981). Fractional differencing. Biometrika, 68(1), 165–176.CrossRefGoogle Scholar
Hunt, A. & Villegas, A.M. (2015). Robustness and convergence in the Lee–Carter model with cohort effects. Insurance: Mathematics and Economics, 64(15), 186–202.Google Scholar
Hurst, H.E. (1951). Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers, 116(1), 770–808.CrossRefGoogle Scholar
Hyndman, R., Koehler, A.B., Ord, J.K. & Snyder, R.D. (2008). Forecasting with Exponential Smoothing: the State Space Approach, vol. 1. Springer-Verlag, Berlin, Heidelberg, New York.CrossRefGoogle Scholar
Hyndman, R.J. & Koehler, A.B. (2006). Another look at measures of forecast accuracy. International Journal of Forecasting, 22(4), 679–688.CrossRefGoogle Scholar
International Monetary Fund, Monetary and Capital Markets, Department (2012). Global financial stability report (2012): The quest for lasting stability. Available online at the address https://www.imf.org/en/Publications/GFSR/Issues/2016/12/31/Global-Financial-Stability-Report-April-2012-The-Quest-for-Lasting-Stability-25343.Google Scholar
Juel, K. (1983). Demographic factors and cancer mortality: a mathematical model for cancer mortality in Denmark 1943–78. International Journal of Epidemiology, 12(4), 419–425.CrossRefGoogle ScholarPubMed
Kitagawa, G. (1987). Non-gaussian state-space modeling of nonstationary time series. Journal of the American Statistical Association, 82(400), 1032–1041.Google Scholar
Koissi, M.-C. & Shapiro, A.F. (2008). The Lee-Carter model under the condition of variables age-specific parameters. In 43rd Actuarial Research Conference, Canada.Google Scholar
Koissi, M.-C., Shapiro, A.F. & Högnôs, G. (2006). Evaluating and extending the Lee–Carter model for mortality forecasting: bootstrap confidence interval. Insurance: Mathematics and Economics 38(1), 1–20.Google Scholar
Lee, R.D. & Carter, L.R. (1992). Modeling and forecasting US mortality. Journal of the American Statistical Association, 87(419), 659–671.Google Scholar
Macdonald, A. (1996). An actuarial survey of statistical models for decrement and transition data-i: multiple state, poisson and binomial models. British Actuarial Journal, 2(1), 129–155.CrossRefGoogle Scholar
Macdonald, A.S., Richards, S.J. & Currie, I.D. (2018). Modelling Mortality with Actuarial Applications, vol. 1. Cambridge University Press, UK.CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. & Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087–1092.CrossRefGoogle Scholar
Neal, R.M. (1994). An improved acceptance procedure for the hybrid Monte Carlo algorithm. Journal of Computational Physics, 111(1), 194–203.CrossRefGoogle Scholar
Ohlsson, E. & Johansson, B. (2010). The basics of pricing with GLMs. In Non-Life Insurance Pricing with Generalized Linear Models, vol. 1. Springer-Verlag, Berlin, Heidelberg, Germany.CrossRefGoogle Scholar
Pedroza, C. (2006). A Bayesian forecasting model: predicting US male mortality. Biostatistics, 7(4), 530–550.CrossRefGoogle Scholar
Plat, R. (2009). On stochastic mortality modeling. Insurance: Mathematics and Economics, 45(3), 393–404.Google Scholar
Rainville, E.D. (1960). Special Functions, vol. 442. Cambridge University Press, New York.Google Scholar
Renshaw, A. & Haberman, S. (2003). Lee–Carter mortality forecasting: a parallel generalized linear modelling approach for England and Wales mortality projections. Journal of the Royal Statistical Society: Series C (Applied Statistics), 52(1), 119–137.Google Scholar
Renshaw, A.E. & Haberman, S. (2006). A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556–570.Google Scholar
Renshaw, A.E., Haberman, S. & Hatzopoulos, P. (1996). The modelling of recent mortality trends in United Kingdom male assured lives. British Actuarial Journal, 2(2), 449–477.CrossRefGoogle Scholar
Särkkä, S. (2013). Bayesian Filtering and Smoothing, vol. 3. Cambridge University Press, UK.CrossRefGoogle Scholar
Shkolnikov, V. (2017). The human mortality database. http://www.mortality.org/.Google Scholar
Sithole, T.Z., Haberman, S. & Verrall, R.J. (2000). An investigation into parametric models for mortality projections, with applications to immediate annuitants and life office pensioners data. Insurance: Mathematics and Economics, 27(3), 285–312.Google Scholar
Spiegelhalter, D.J., Best, N.G., Carlin, B.P. & Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4), 583–639.CrossRefGoogle Scholar
Spiegelhalter, D.J., Best, N.G., Carlin, B.P. & Van Der Linde, A. (2014). The deviance information criterion: 12 years on. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(3), 485–493.CrossRefGoogle Scholar
Stein, E.M. & Weiss, G.L. (1971). Introduction to Fourier Analysis on Euclidean Spaces, vol. 32. Princeton University Press, USA.Google Scholar
Toczydlowska, D., Peters, G.W., Fung, M.C. and Shevchenko, P.V. (2017). Stochastic period and cohort effect state-space mortality models incorporating demographic factors via probabilistic robust principal components. Risks, 5(3), 42.CrossRefGoogle Scholar
Wold, H. (1938). A Study in the Analysis of Stationary Time Series. PhD thesis, Almqvist & Wiksell.Google Scholar
Woodward, W.A., Cheng, Q.C. & Gray, H.L. (1998). A k-factor GARMA long-memory model. Journal of Time Series Analysis, 19(4), 485–504.CrossRefGoogle Scholar
Yan, H., Chan, J.S. & Peters, G.W. (2017). Long memory models for financial time series of counts and evidence of systematic market participant trading behaviour patterns in futures on US treasuries. SSRN: https://ssrn.com/abstract=2962341.CrossRefGoogle Scholar
Yan, H., Peters, G.W. & Chan, J.S. (2018a). Evidence for persistence and long memory features in mortality data. SSRN: http://ssrn.com/abstract=3322611.Google Scholar
Yan, H., Peters, G.W. & Chan, J.S. (2018b). Mortality models incorporating long memory improves life table estimation: a comprehensive analysis (appendix). Available at SSRN: https://ssrn.com/abstract=3149914 or http://dx.doi.org/10.2139/ssrn.3149914.Google Scholar