Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-29T00:12:15.568Z Has data issue: false hasContentIssue false

MORTALITY FORECASTING WITH A SPATIALLY PENALIZED SMOOTHED VAR MODEL

Published online by Cambridge University Press:  04 November 2020

Le Chang*
Affiliation:
Research School of Finance, Actuarial Studies, and Statistics Australian National UniversityCanberra, ACT, Australia E-Mail: le.chang@anu.edu.au
Yanlin Shi
Affiliation:
Department of Actuarial Studies and Business Analytics Macquarie University Sydney, NSW, Australia E-Mail: yanlin.shi@mq.edu.au

Abstract

This paper investigates a high-dimensional vector-autoregressive (VAR) model in mortality modeling and forecasting. We propose an extension of the sparse VAR (SVAR) model fitted on the log-mortality improvements, which we name “spatially penalized smoothed VAR” (SSVAR). By adaptively penalizing the coefficients based on the distances between ages, SSVAR not only allows a flexible data-driven sparsity structure of the coefficient matrix but simultaneously ensures interpretable coefficients including cohort effects. Moreover, by incorporating the smoothness penalties, divergence in forecast mortality rates of neighboring ages is largely reduced, compared with the existing SVAR model. A novel estimation approach that uses the accelerated proximal gradient algorithm is proposed to solve SSVAR efficiently. Similarly, we propose estimating the precision matrix of the residuals using a spatially penalized graphical Lasso to further study the dependency structure of the residuals. Using the UK and France population data, we demonstrate that the SSVAR model consistently outperforms the famous Lee–Carter, Hyndman–Ullah, and two VAR-type models in forecasting accuracy. Finally, we discuss the extension of the SSVAR model to multi-population mortality forecasting with an illustrative example that demonstrates its superiority in forecasting over existing approaches.

Type
Research Article
Copyright
© 2020 by Astin Bulletin. All rights reserved

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

References

Barrieu, P., Bensusan, H., El Karoui, N., Hillairet, C., Loisel, S., Ravanelli, C. and Salhi, Y. (2012) Understanding, modelling and managing longevity risk: Key issues and main challenges. Scandinavian Actuarial Journal, 2012(3), 203231.CrossRefGoogle Scholar
Basu, S. and Michailidis, G. (2015) Regularized estimation in sparse high-dimensional time series models. The Annals of Statistics, 43(4), 15351567.CrossRefGoogle Scholar
Beck, A. and Teboulle, M. (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1), 183202.CrossRefGoogle Scholar
Biffis, E. and Millossovich, P. (2006) A bidimensional approach to mortality risk. Decisions in Economics and Finance, 29(2), 7194.CrossRefGoogle Scholar
Booth, H., Hyndman, R., Tickle, L. and De Jong, P. (2006) Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research, 15, 289310.CrossRefGoogle Scholar
Cairns, A.J., Blake, D., Dowd, K., Coughlan, G.D. and Khalaf-Allah, M. (2011) Bayesian stochastic mortality modelling for two populations. ASTIN Bulletin, 41, 2959. ISSN 1783-1350.Google Scholar
Cressie, N. and Wikle, C.K. (2015) Statistics for Spatio-Temporal Data. New Jersey: John Wiley & Sons.Google Scholar
Daubechies, I., Defrise, M. and De Mol, C. (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 57(11), 14131457.CrossRefGoogle Scholar
Davis, R.A., Zang, P. and Zheng, T. (2016) Sparse vector autoregressive modeling. Journal of Computational and Graphical Statistics, 25(4), 10771096.CrossRefGoogle Scholar
Debón, A., Montes, F., Mateu, J., Porcu, E. and Bevilacqua, M. (2008) Modelling residuals dependence in dynamic life tables: A geostatistical approach. Computational Statistics & Data Analysis, 52(6), 31283147.CrossRefGoogle Scholar
Diebold, F.X. and Mariano, R.S. (2002) Comparing predictive accuracy. Journal of Business & Economic Statistics, 20(1), 134144.Google Scholar
Donoho, D.L. and Johnstone, J.M. (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425455.CrossRefGoogle Scholar
Fan, J., Lv, J. and Qi, L. (2011) Sparse high-dimensional models in economics. Annual Review of Economics, 3(1), 291317.Google ScholarPubMed
Friedman, J., Hastie, T. and Tibshirani, R. (2008) Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3), 432441.CrossRefGoogle ScholarPubMed
Guibert, Q., Lopez, O. and Piette, P. (2019) Forecasting mortality rate improvements with a high-dimensional var. Insurance: Mathematics and Economics, 88, 255272.Google Scholar
Mortality Database, Human. (2019) University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). URL http://www.mortality.org.Google Scholar
Hunt, A. and Blake, D. (2014) A general procedure for constructing mortality models. North American Actuarial Journal, 18(1), 116138.CrossRefGoogle Scholar
Hyndman, R.J., Booth, H. and Yasmeen, F. (2013) Coherent mortality forecasting: the product-ratio method with functional time series models. Demography, 50(1), 261283.Google ScholarPubMed
Hyndman, R.J. and Ullah, M.S. (2007) Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics & Data Analysis, 51(10), 49424956.CrossRefGoogle Scholar
Lazar, D. and Denuit, M.M. (2009) A multivariate time series approach to projected life tables. Applied Stochastic Models in Business and Industry, 25(6), 806823.CrossRefGoogle Scholar
Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting US mortality. Journal of the American Statistical Association, 87(419), 659–671.Google Scholar
Li, H. and Lu, Y. (2017) Coherent forecasting of mortality rates: A sparse vector-autoregression approach. ASTIN Bulletin: The Journal of the IAA, 47(2), 563600.CrossRefGoogle Scholar
Li, H., Lu, Y. and Lyu, P. (2018) Coherent mortality forecasting for less developed countries. SSRN Scholarly Paper ID 3209392. URL http://dx.doi.org/10.2139/ssrn.3209392.CrossRefGoogle Scholar
Li, J.S.-H. and Hardy, M.R. (2011) Measuring basis risk in longevity hedges. North American Actuarial Journal, 15(2), 177200.CrossRefGoogle Scholar
Li, N. and Lee, R. (2005) Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography, 42(3), 575594.CrossRefGoogle ScholarPubMed
Renshaw, A.E. and Haberman, S. (2006) A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556570.Google Scholar
Salhi, Y. and Loisel, S. (2017) Basis risk modelling: A cointegration-based approach. Statistics, 51(1), 205221.CrossRefGoogle Scholar
Song, S. and Bickel, P.J. (2011) Large vector auto regressions. arXiv preprint arXiv:1106.3915.Google Scholar
Villegas, A.M., Haberman, S., Kaishev, V.K. and Millossovich, P. (2017) A comparative study of two-population models for the assessment of basis risk in longevity hedges. ASTIN Bulletin: The Journal of the IAA, 47(3), 631679.CrossRefGoogle Scholar
Yang, S.S. and Wang, C.-W. (2013) Pricing and securitization of multi-country longevity risk with mortality dependence. Insurance: Mathematics and Economics, 52(2), 157169.Google Scholar
Zhou, R., Wang, Y., Kaufhold, K., Li, J.S.-H. and Tan, K.S. (2014) Modeling period effects in multi-population mortality models: Applications to solvency ii. North American Actuarial Journal, 18(1), 150167.CrossRefGoogle Scholar
Zou, H. and Hastie, T. (2005) Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (statistical methodology), 67(2), 301320.CrossRefGoogle Scholar