Modeling mortality with a Bayesian vector autoregression. (English) Zbl 1452.91244

Summary: Parametric mortality models capture the cross section of mortality rates. These models fit the older ages better, because of the more complex cross section of mortality at younger and middle ages. Dynamic parametric mortality models fit a time series to the parameters, such as a vector autoregression (VAR), in order to capture trends and uncertainty in mortality improvements. We consider the full age range using the Heligman and Pollard model [L. Heligman and J. H. Pollard, “The age pattern of mortality”, J. Inst. Actuar. 107, No. 1, 49–80 (1980; doi:10.1017/S0020268100040257)], a cross-sectional mortality model with parameters that capture specific features of different age ranges. We make the Heligman-Pollard model dynamic using a Bayesian vector autoregressive (BVAR) model for the parameters and compare with more commonly used VAR models. We fit the models using Australian data, a country with similar mortality experience to many developed countries. We show how the BVAR models improve forecast accuracy compared to VAR models and quantify parameter risk which is shown to be significant.


91D20 Mathematical geography and demography
91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI


[1] Avraam, D.; Arnold-Gaille, S.; Jones, B., Time-evolution of age-dependent mortality patterns in mathematical model of heterogeneous human population, Exp. Geront., 60, Supplement C, 18-30 (2014)
[2] Baltagi, B., Econometrics (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1007.62097
[3] Borger, M.; Schupp, J., Modeling trend processes in parametric mortality models, Insurance Math. Econom., 78, 369-380 (2018) · Zbl 1400.91241
[4] Box, G.; Jenkins, G., (Time Series Analysis: Forecasting and Control (Wiley Series in Probability and Statistics): Forecasting and Control. Time Series Analysis: Forecasting and Control (Wiley Series in Probability and Statistics): Forecasting and Control, Wiley Series in Probability and Statistics (1976), Holden-Day San-Francisco)
[5] Brandt, P., MSBVAR: Markov-Switching, Bayesian, Vector Autoregression Models (2011), Retrieved from http://CRAN.R-project.org/package=MSBVAR
[6] Brandt, P.; Freeman, J., Advances in Bayesian time series modeling and the study of politics: Theory testing, forecasting, and policy analysis, Political Anal., 14, 1-36 (2006)
[7] Chan, W. S.; Li, J. S.H.; Li, J., The CBD mortality indexes: Modeling and applications, N. Am. Actuar. J., 18, 1, 38-58 (2014) · Zbl 1412.91037
[8] Congdon, P., Statistical graduation in local demographic analysis and projection, J. R. Stat. Soc. A, 156, 2, 237-270 (1993)
[9] Dellaportas, P.; Smith, A. F.M.; Stavropoulos, P., Bayesian analysis of mortality data, J. R. Stat. Soc. A, 164, 2, 275-291 (2001) · Zbl 1002.91504
[10] Denuit, M.; Frostig, E., Association and heterogeneity of insured lifetimes in the Lee-Carter framework, Scand. Actuar. J., 2007, 1, 1-19 (2007) · Zbl 1141.91025
[11] Denuit, M.; Frostig, E., First-Order Mortality Basis for Life Annuities (2008), Palgrave Macmillan, Nature Publishing Group Metadata Repository [http://www.nature.com/oai/request] (Switzerland) ER
[12] Denuit, M.; Frostig, E., Life insurance mathematics with random life tables, N. Am. Actuar. J., 13, 3, 339-355 (2009)
[13] Doan, T.; Litterman, R.; Sims, C., Forecasting and conditional projection using realistic prior distributions, Econometric Rev., 3, 1, 1-100 (1984) · Zbl 0613.62142
[14] Hamilton, J., Time Series Analysis (1994), Princeton University Press
[15] Hartmann, M., Past and recent attempts to model mortality at all ages, J. Official Stat., 3, 1, 19-36 (1987)
[16] Heligman, L.; Pollard, J. H., The age pattern of mortality, J. Inst. Actuar., 107, 49-80 (1980)
[17] Human Mortality Database. 2010. University of California, Berkeley (USA) Max Planck Institute for Demographic Research (Germany). Retrieved from www.mortality.org or www.humanmortality.de.
[18] Joiner, A., Monetary Policy Effect in an Australian Bayesian VAR ModelWorking Paper (2001), Retrieved from www.rbnz.govt.nz/research/workshops/112040/4apr02joiner.pdf
[19] Kadiyala, K. R.; Karlsson, S., Numerical methods for estimation and inference in Bayesian-models numerical methods for estimation and inference in Bayesian VAR-models, J. Appl. Econometrics, 12, 2, 99-132 (1997)
[20] Lee, R. D., Stochastic demographic forecasting, Int. J. Forecast., 8, 3, 315-327 (1992)
[21] Lee, R. D.; Carter, L. R., Modeling and Forecasting U. S. Mortality, J. Amer. Statist. Assoc., 87, 419, 659-671 (1992) · Zbl 1351.62186
[22] Leng, X.; Peng, L., Inference pitfalls in Lee-Carter model for forecasting mortality, Insurance Math. Econom., 70, Supplement C, 58-65 (2016) · Zbl 1371.91098
[23] Litterman, R. B., Forecasting with Bayesian vector autoregressions: Five years of experience, J. Bus. Econom. Statist., 4, 1, 25-38 (1986)
[24] Lütkepohl, H., 1991. Introduction to Multiple Time Series Analysis. first ed., New York. · Zbl 0729.62085
[25] Lütkepohl, H., 2005. New Introduction to Multiple Time Series Analysis. New York. · Zbl 1072.62075
[26] McNeil, D. R.; Trussell, T. J.; Turner, J. C., Spline interpolation of Demographic data, Demography, 14, 2, 245-252 (1977)
[27] McNown, R.; Rogers, A., Forecasting mortality: A parameterized time series approach, Demography, 26, 4, 645-660 (1989)
[28] Miranda-Agrippino, S.; Ricco, G., Bayesian Vector Autoregressions: Estimation (2019), Oxford University Press, Retrieved from http://oxfordre.com/economics/view/10.1093/acrefore/9780190625979.001.0001/acrefore-9780190625979-e-164
[29] Pedroza, C., A Bayesian forecasting model: predicting U.S. male mortality, Biostat, 7, 4, 530-550 (2006) · Zbl 1170.62397
[30] Peng, R., A method for visualizing multivariate time series data, J. Stat. Softw. Code Snippets, 25, 1, 1-17 (2008), Retrieved from http://www.jstatsoft.org/v25/c01
[31] Reichmuth, W. H.; Sarferaz, S., Bayesian Demographic Modeling and Forecasting: An Application to U.S. Mortality (2008)
[32] Richards, S., A Value-at-Risk framework for longevity trend risk abstract of the London discussion, Br. Actuar. J., 19, 157-167 (2014)
[33] Robertson, J. C.; Tallman, E. W., Vector autoregressions: forecasting and reality, Econ. Rev., Q1, 4-18 (1999), Retrieved from http://ideas.repec.org/a/fip/fedaer/y1999iq1p4-18nv.84no.1.html
[34] Rogers, A., Parameterized multistate population dynamics and projections, J. Amer. Statist. Assoc., 81, 393, 48-61 (1986)
[35] Sharrow, D. J.; Clark, S. J.; Collinson, M. A.; Kahn, K.; Tollman, S. M., The Age-Pattern of Increases in Mortality Affected By HIV: Bayesian Fit of the Heligman-Pollard Model To Data from the Agincourt HDSS Field Site in Rural Northeast South Africa (2010), University of Washington
[36] Sherris, M.; Njenga, C. N., Longevity risk and the econometric analysis of mortality trends and volatility, (SSRN ELibrary (2009)), Retrieved from SSRN: http://ssrn.com/abstract=1458084
[37] Sims, C. A., Macroeconomics and reality, Econometrica, 48, 1, 1-48 (1980)
[38] Sims, C. A.; Zha, T., Bayesian methods for dynamic multivariate models, Internat. Econom. Rev., 39, 4, 949-968 (1998)
[39] Summers, P. M., Forecasting Australia’s economic performance during the Asian crisis, Int. J. Forecast., 17, 3, 499-515 (2001)
[40] Thompson, P. A.; Bell, W. R.; Long, J. F.; Miller, R. B., Multivariate time series projections of parameterized age-specific fertility rates, J. Amer. Statist. Assoc., 84, 407, 689-699 (1989)
[41] Wilmoth, J.; Andreev, K.; Jdanov, D.; Glei, D., Methods protocol for the human mortality database, Human Mortality DatabaseTech. Rep. (2007), Retrieved from www.mortality.org/Public/Docs/MethodsProtocol.pdf
[42] Wong-Fupuy, C.; Haberman, S., Projecting mortality trends: Recent developments in the United Kingdom and the United States, N. Am. Actuar. J., 8, 2, 56-83 (2004) · Zbl 1085.62517
[43] Zivot, E.; Wang, J., Modelling Financial Time Series with S-Plus (2006) · Zbl 1092.91067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.