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A GROUP REGULARISATION APPROACH FOR CONSTRUCTING GENERALISED AGE-PERIOD-COHORT MORTALITY PROJECTION MODELS

Published online by Cambridge University Press:  09 November 2021

Dilan SriDaran ,
Michael Sherris ,
Andrés M. Villegas  and
Jonathan Ziveyi Open the ORCID record for Jonathan Ziveyi [Opens in a new window]
Dilan SriDaran
Affiliation:
School of Risk and Actuarial Studies and ARC Centre of Excellence in Population Ageing Research (CEPAR) UNSW Sydney Sydney, NSW 2052, Australia E-Mail: [email protected]
Michael Sherris
Affiliation:
School of Risk and Actuarial Studies and ARC Centre of Excellence in Population Ageing Research (CEPAR) UNSW Sydney Sydney, NSW 2052, Australia E-Mail: [email protected]
Andrés M. Villegas*
Affiliation:
School of Risk and Actuarial Studies and ARC Centre of Excellence in Population Ageing Research (CEPAR) UNSW Sydney Sydney, NSW 2052, Australia
Jonathan Ziveyi
Affiliation:
School of Risk and Actuarial Studies and ARC Centre of Excellence in Population Ageing Research (CEPAR) UNSW Sydney Sydney, NSW 2052, Australia E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

Given the rapid reductions in human mortality observed over recent decades and the uncertainty associated with their future evolution, there have been a large number of mortality projection models proposed by actuaries and demographers in recent years. Many of these, however, suffer from being overly complex, thereby producing spurious forecasts, particularly over long horizons and for small, noisy data sets. In this paper, we exploit statistical learning tools, namely group regularisation and cross-validation, to provide a robust framework to construct discrete-time mortality models by automatically selecting the most appropriate functions to best describe and forecast particular data sets. Most importantly, this approach produces bespoke models using a trade-off between complexity (to draw as much insight as possible from limited data sets) and parsimony (to prevent over-fitting to noise), with this trade-off designed to have specific regard to the forecasting horizon of interest. This is illustrated using both empirical data from the Human Mortality Database and simulated data, using code that has been made available within a user-friendly open-source R package StMoMo.

Keywords

Mortality projectionregularisationcross-validationage-period-cohort model
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 52 , Issue 1 , January 2022 , pp. 247 - 289
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Aro, H. and Pennanen, T. (2011) A user-friendly approach to stochastic mortality modelling. European Actuarial Journal, 1(2), 151–167.CrossRefGoogle Scholar
Atance, D. and DebÓn, A. (2020) A comparison of forecasting mortality models using resampling methods Mathematics, 8(9), 1550. https://doi.org/10.3390/math8091550.CrossRefGoogle Scholar
Barigou, K., Loisel, S. and Salhi, Y. (2021) Parsimonious predictive mortality modeling by regularization and cross-validation with and without Covid-type effect. Risks, 9(1), 5.CrossRefGoogle Scholar
Bergmeir, C. and Bentez, J.M. (2012) On the use of cross-validation for time series predictor evaluation. Information Sciences, 191, 192–213. https://doi.org/10.1016/j.ins.2011.12.028.CrossRefGoogle Scholar
Bergmeir, C., Costantini, M. and Bentez, J.M. (2014) On the usefulness of cross-validation for directional forecast evaluation. Computational Statistics and Data Analysis, 76, 132–143. https://doi.org/10.1016/j.csda.2014.02.001.CrossRefGoogle Scholar
Bergmeir, C., Hyndman, R. and Koo, B. (2018) A note on the validity of cross-validation for evaluating autoregressive time series prediction. Computational Statistics and Data Analysis, 120, 70–83. https://doi.org/10.1016/j.csda.2017.11.003.CrossRefGoogle Scholar
Biffis, E. (2005) Affine processes for dynamic mortality and actuarial valuations. Insurance: Mathematics and Economics, 37(3), 443–468. https://doi.org/10.1016/j.insmatheco.2005.05.003.Google Scholar
Booth, H., Hyndman, R.J., Tickle, L. and de Jong, P. (2006) Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions. Demography, 15, 289–310. https://doi.org/10.4054/DemRes.2006.15.9.CrossRefGoogle Scholar
Booth, H. and Tickle, L. (2008) Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science, 31(1-2), 3–43. https://doi.org/10.1017/S1748499500000440.CrossRefGoogle Scholar
Breheny, P. and Huang, J. (2015) Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors. Statistics and Computing, 25(2), 173–87. https://doi.org/10.1007/s11222-013-9424-2.CrossRefGoogle ScholarPubMed
Brouhns, N., Denuit, M. and Vermunt, J. (2002a) A Poisson log-linear regression approach to the construction of projected life tables. Insurance: Mathematics and Economics, 31(3), 373–393.Google Scholar
Brouhns, N., Denuit, M. and Vermunt, J.K. (2002b) Measuring the longevity risk in mortality projections. Bulletin of the Swiss Association of Actuaries, 2(1), 105–130.Google Scholar
Cairns, A., Blake, D. and Dowd, K. (2006) A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. The Journal of Risk and Insurance, 73(4), 687–718.CrossRefGoogle Scholar
Cairns, A., Blake, D., Dowd, K., Coughlan, G., Epstein, D. and Khalaf-Allah, M. (2011) Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48(3), 355–367. https://doi.org/10.1016/j.insmatheco.2010.12.005.Google Scholar
Cairns, A., Blake, D., Dowd, K., Coughlan, G., Epstein, D., Ong, A. and Balevich, I. (2010) 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
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2008) Mortality density forecasts: An analysis of six stochastic mortality models. Pensions Institute Discussion Paper PI-0801. http://papers.ssrn.com/sol3/papers.cfm?abstract%7B/_%7Did=1340353.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011) Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48(3), 355–367.Google Scholar
Currie, I. (2006) Smoothing and forecasting mortality rates with p-splines. https://www.macs.-hw.ac.uk/Iain.Google Scholar
Currie, I. (2016) On fitting generalized linear and non-linear models of mortality. Scandinavian Actuarial Journal, (4), 356–383. https://doi.org/10.1080/03461238.2014.928230.CrossRefGoogle Scholar
Currie, I. (2020) Constraints, the identifiability problem and the forecasting of mortality. Annals of Actuarial Science, 14(2), 537–566. https://doi.org/10.1017/S1748499520000020.CrossRefGoogle Scholar
Currie, I., Durban, M. and Eilers, P. (2004) Smoothing and forecasting mortality rates. Statistical Modelling, 4(4), 279–298.CrossRefGoogle Scholar
Delwarde, A., Denuit, M. and Eilers, P. (2007) Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting: A penalized log-likelihood approach. Statistical Modelling, 7(1), 29–48. https://doi.org/10.1177/1471082X0600700103.CrossRefGoogle Scholar
Devriendt, S., Antonio, K., Reynkens, T. and Verbelen, R. (2020) Sparse regression with multi-type regularized feature modeling. Insurance: Mathematics and Economics, 96, 248–261.Google Scholar
Dowd, K., Cairns, A., Blake, D., Coughlan, G., Epstein, D. and Khalaf-Allah, M. (2009) Backtesting stochastic mortality models: An ex post evaluation of multiperiod-ahead density forecasts. North American Actuarial Journal, 14(3), 281–98.CrossRefGoogle Scholar
Dowd, K., Cairns, A.J.G. and Blake, D. (2020) CBDX: A workhorse mortality model from the Cairns–Blake–Dowd family. Annals of Actuarial Science, 14(2), 445–460. https://doi.org/10.1017/s1748499520000159.CrossRefGoogle Scholar
Fan, J. and Li, R. (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96(456), 1348–1360. https://doi.org/10.1198/016214501753382273.CrossRefGoogle Scholar
Gneiting, T. and Raftery, A. (2007) Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477), 359–78.CrossRefGoogle Scholar
Green, K. and Scott Armstrong, J. (2015) Simple versus complex forecasting: The evidence. Journal of Business Research, 68(8), 1678–1685. https://doi.org/10.1016/j.jbusres.2015.03.026.CrossRefGoogle Scholar
Guibert, Q., Lopez, O. and Piette, P. (2019) Forecasting mortality rate improvements with a high-dimensional VAR. Insurance: Mathematics and Economics, 88, 255–72. https://doi.org/10.1016/j.insmatheco.2019.07.004.Google Scholar
Haberman, S. and Renshaw, A. (2011) A comparative study of parametric mortality projection models. Insurance: Mathematics and Economics, 48(1), 35–55. https://doi.org/10.1016/j.insmatheco.2010.09.003.Google Scholar
Hainaut, D. and Denuit, M. (2020) Wavelet-based feature extraction for mortality projection. ASTIN Bulletin, 50(3), 675–707. https://doi.org/10.1017/asb.2020.18.CrossRefGoogle Scholar
Hastie, T., Tibshirani, R. and Friedman, J. (2001) The Elements of Statistical Learning. https://doi.org/10.1007/b94608.CrossRefGoogle Scholar
Holford, T.R. (1983) The estimation of age, period and cohort effects for vital rates. Biometrics, 331–324. https://doi.org/10.2307/2531004.Google ScholarPubMed
Human Mortality Database. (2020) University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). www.mortality.org.Google Scholar
Hunt, A. and Blake, D. (2014) A General procedure for constructing mortality models. North American Actuarial Journal, 18(1), 116–38. https://doi.org/10.1080/10920277.2013.852963.CrossRefGoogle Scholar
Hunt, A. and Blake, D. (2020a) Identifiability in age/period/cohort mortality models. Annals of Actuarial Science, 14(2), 550–536. https://doi.org/10.1017/S1748499520000123.Google Scholar
Hunt, A. and Blake, D. (2020b) Identifiability in age/period mortality models. Annals of Actuarial Science, 14(2), 461–499. https://doi.org/10.1017/s1748499520000111.CrossRefGoogle Scholar
Hunt, A. and Blake, D. (2021a) A Bayesian approach to modeling and projecting cohort effects. North American Actuarial Journal, 25(sup1), S235–S254. https://doi.org/10.1080/10920277.2019.1649157.CrossRefGoogle Scholar
Hunt, A. and Blake, D. (2021b) On the structure and classification of mortality models. North American Actuarial Journal, 25(sup1), S215–S234. https://doi.org/10.1080/10920277.2019.1649156.CrossRefGoogle Scholar
Hunt, A. and Villegas, A.M. (2015) Robustness and convergence in the Lee-Carter model with cohorts. Insurance: Mathematics and Economics, 64, 186–202.Google Scholar
Hyndman, R., Booth, H., Tickle, L. and Maindonald, J. (2017) Package ‘demography’. http://cran.r-project.org/package=demogra.Google Scholar
James, G., Witen, D., Hastie, T. and Tibshirani, R. (2013) An Introduction to Statistical Learning with Applications in R. New York: Springer.CrossRefGoogle Scholar
Karanikolos, M., Leon, D.A., Smith, P.C. and McKee, M. (2012) Minding the gap: Changes in life expectancy in the Baltic States compared with Finland. Journal of Epidemiology and Community Health, 66(11), 1043–1049. https://doi.org/10.1136/jech-2011-200879.CrossRefGoogle ScholarPubMed
Lee, R. and Carter, L. (1992) Modeling and forecasting U.S. Mortality. Journal of the American Statistical Association, 87(419), 673–674.Google Scholar
Li, H. and O’Hare, C. (2017) Semi-parametric extensions of the Cairns–Blake–Dowd model: A one-dimensional kernel smoothing approach. Insurance: Mathematics and Economics, 77, 166–176.CrossRefGoogle Scholar
Li, H. and Shi, Y. (2021) Mortality forecasting with an age-coherent sparse VAR model. Risks, 9(35). https://doi.org/10.1017/asb.2020.39.CrossRefGoogle Scholar
Li, J.S.H., Zhou, R., Liu, Y., Graziani, G., Dale Hall, R., Haid, J., Peterson, A. and Pinzur, L. (2020) Drivers of mortality dynamics: Identifying age/period/cohort components of historical U.S. mortality improvements. North American Actuarial Journal, 24(2), 228–50. https://doi.org/10.1080/10920277.2020.1716808.CrossRefGoogle Scholar
Li, J.S.H., Zhou, R. and Hardy, M. (2015) A step-by-step guide to building two-population stochastic mortality models. Insurance: Mathematics and Economics, 63, 121–34. https://doi.org/10.1016/j.insmatheco.2015.03.021.Google Scholar
Macdonald, A.S., Richards, S.J. and Currie, I.D. (2018) Modelling Mortality with Actuarial Applications. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Njenga, C.N. and Sherris, M. (2011) Longevity risk and the econometric analysis of mortality trends and volatility. Asia-Pacific Journal of Risk and Insurance, 5(2). https://doi.org/10.2202/2153-3792.1115.CrossRefGoogle Scholar
Plat, R. (2009) On stochastic mortality modeling. Insurance: Mathematics and Economics, 45(3), 393–404. https://doi.org/10.1016/j.insmatheco.2009.08.006.Google Scholar
Racine, J. (2000) Consistent cross-validatory model-selection for dependent data: hv-block cross-validation. Journal of Econometrics, 99(1), 39–61. https://doi.org/10.1016/S0304-4076(00)00030-0.CrossRefGoogle Scholar
Renshaw, A. and Haberman, S. (2003a) Lee-Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics, 33(2), 255–72.Google Scholar
Renshaw, A. and Haberman, S. (2003b) On the forecasting of mortality reduction factors. Insurance: Mathematics and Economics, 32(3), 379–401. https://doi.org/10.1016/S0167-6687(03)00118-5.Google Scholar
Renshaw, A. and Haberman, S. (2006) A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556–570. https://doi.org/10.1016/j.insmatheco.2005.12.001.Google Scholar
Tashman, L.J. (2000) Out-of-sample tests of forecasting accuracy: An analysis and review. International Journal of Forecasting, 16(4), 437–50. https://doi.org/10.1016/S0169-2070(00)00065–0.CrossRefGoogle Scholar
Tibshirani, R. (1996) Regression selection and shrinkage via the lasso. Journal of the Royal Statistical Society, 58(1), 267–288. https://doi.org/10.2307/2346178.Google Scholar
Vandekerckhove, J., Matzke, D. and Wagenmakers, E.-J. (2015). Model comparison and the principle of parsimony. In The Oxford Handbook of Computational and Mathematical Psychology. https://doi.org/10.1093/oxfordhb/9780199957996.013.14.Google Scholar
Venter, G. and Sahin, S. (2018) Parsimonious parameterization of age-period-cohort models by Bayesian shrinkage. ASTIN Bulletin, 48(1), 89–110. https://doi.org/10.1017/asb.2017.21.CrossRefGoogle Scholar
Villegas, A., Kaishev, V. and Millossovich, P. (2018) StMoMo: An R package for stochastic mortality modeling. Journal of Statistical Software 84(3). https://doi.org/10.18637/jss.v084.i03.CrossRefGoogle Scholar
Yuan, M. and Lin, Y. (2006) Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society. Series B: Statistical Methodology, 68(1), 49–67. https://doi.org/10.1111/j.1467-9868.2005.00532.x.CrossRefGoogle Scholar
Zhang, C.H. (2010) Nearly unbiased variable selection under minimax concave penalty. Annals of Statistics, 38(2), 894–942. https://doi.org/10.1214/09-AOS729.CrossRefGoogle Scholar
Zou, H. and Hastie, T. (2005) Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society, 67(2), 301–20. https://doi.org/10.1111/j.1467-9868.2005.00503.x.CrossRefGoogle Scholar