Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T15:09:10.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Identifiability in age/period/cohort mortality models

Published online by Cambridge University Press:  04 June 2020

Andrew Hunt Open the ORCID record for Andrew Hunt [Opens in a new window]  and
David Blake Open the ORCID record for David Blake [Opens in a new window]
Andrew Hunt*
Affiliation:
Cass Business School, City University London, London, UK
David Blake
Affiliation:
Pensions Institute, Cass Business School, City University London, London, UK
*
* Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The addition of a set of cohort parameters to a mortality model can generate complex identifiability issues due to the collinearity between the dimensions of age, period and cohort. These issues can lead to robustness problems and difficulties making projections of future mortality rates. Since many modern mortality models incorporate cohort parameters, we believe that a comprehensive analysis of the identifiability issues in age/period/cohort mortality models is needed. In this paper, we discuss the origin of identifiability issues in general models before applying these insights to simple but commonly used mortality models. We then discuss how to project mortality models so that our forecasts of the future are independent of any arbitrary choices we make when fitting a model to data in order to identify the historical parameters.

Keywords

Mortality modellingAge/period/cohort modelsIdentification issuesProjectionC15C51C53
Type
Paper
Information
Annals of Actuarial Science , Volume 14 , Issue 2 , September 2020 , pp. 500 - 536
Copyright
© Institute and Faculty of Actuaries 2020

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.)

References

Booth, H., Maindonald, J. & Smith, L. (2002). Applying Lee-Carter under conditions of variable mortality decline. Population Studies, 56(3), 325–336. Available online at the address http://www.ncbi.nlm.nih.gov/pubmed/12553330CrossRefGoogle ScholarPubMed
Börger, M., Fleischer, D. & Kuksin, N. (2013). Modeling the mortality trend under modern solvency regimes. ASTIN Bulletin, 44(1), 1–38.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D. & Dowd, K. (2006). Pricing death: frameworks for the valuation and securitization of mortality risk. ASTIN Bulletin, 36(1), 79–120.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. & Khalaf-Allah, M. (2011). Mortality density forecasts: an analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48(3), 355–367. Available online at the address http://linkinghub.elsevier.com/retrieve/pii/S0167668710001484Google Scholar
Cairns, A.J.G., 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
Cairns, A.J.G., Blake, D., Dowd, K. & Kessler, A. (2015). Phantoms never die: living with unreliable population data. Journal of the Royal Statistical Society: Series A (Statistics in Society), Forthcoming.Google Scholar
Carstensen, B. (2007). Age-period-cohort models for the Lexis diagram. Statistics in Medicine, 26, 3018–3045.CrossRefGoogle ScholarPubMed
Clayton, D. & Schifflers, E. (1987). Models for temporal variation in cancer rates. II: age-period-cohort models. Statistics in Medicine, 6(4), 469–481. Available online at the address http://www.ncbi.nlm.nih.gov/pubmed/3629048CrossRefGoogle ScholarPubMed
Fienberg, S.E. & Mason, W.M. (1979). Identification and estimation of age-period-cohort models in the analysis of discrete archival data. Sociological Methodology, 10, 1–67.CrossRefGoogle Scholar
Glenn, N. (1976). Cohort analysts’ futile quest: statistical attempts to separate age, period and cohort effects. American Sociological Review, 41(5), 900–904.CrossRefGoogle Scholar
Haberman, S. & Renshaw, A. (2009). On age-period-cohort parametric mortality rate projections. Insurance: Mathematics and Economics, 45(2), 255–270.Google Scholar
Haberman, S. & Renshaw, A. (2011). A comparative study of parametric mortality projection models. Insurance: Mathematics and Economics, 48(1), 35–55. Available online at the address http://linkinghub.elsevier.com/retrieve/pii/S0167668710001022Google Scholar
Hobcraft, J., Menken, J. & Preston, S.H. (1982). Age, period and cohort effects in demography: a review. Population Index, 48(1), 4–43.CrossRefGoogle ScholarPubMed
Holford, T.R. (1983). The estimation of age, period and cohort effects for vital rates. Biometrics, 39(2), 311–324. Available online at the address http://www.ncbi.nlm.nih.gov/pubmed/6626659CrossRefGoogle ScholarPubMed
Human Mortality Database (2014). Human mortality database, technical report, University of California, Berkeley and Max Planck Institute for Demographic Research. Available online at the address www.mortality.orgGoogle Scholar
Hunt, A. & Blake, D. (2014). A general procedure for constructing mortality models. North American Actuarial Journal, 18(1), 116–138. Available online at the address http://www.pensions-institute.org/workingpapers/wp1301.pdfCrossRefGoogle Scholar
Hunt, A. & Blake, D. (2015). Modelling longevity bonds: analysing the Swiss Re Kortis Bond. Insurance: Mathematics and Economics, 63, 12–29.Google Scholar
Hunt, A. & Blake, D. (2018). Identifiability, cointegration and the gravity model. Insurance: Mathematics and Economics, 78, 306–368.Google Scholar
Hunt, A. & Blake, D. (2020a). A Bayesian approach to modelling and projecting cohort effects. North American Actuarial Journal (Forthcoming).CrossRefGoogle Scholar
Hunt, A. & Blake, D. (2020b). Identifiability in age/period mortality models. Annals of Actuarial Science (Forthcoming).CrossRefGoogle Scholar
Hunt, A. & Blake, D. (2020c). On the structure and classification of mortality models. North American Actuarial Journal (Forthcoming).CrossRefGoogle Scholar
Hunt, A. & Villegas, A.M. (2015). Robustness and convergence in the Lee-Carter model with cohort effects. Insurance: Mathematics and Economics, 64, 186–202.Google Scholar
Kuang, D., Nielsen, B. & Nielsen, J.P. (2008a). Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika 95(4), 987–991.CrossRefGoogle Scholar
Kuang, D., Nielsen, B. & Nielsen, J.P. (2008b). Identification of the age-period-cohort model and the extended chain-ladder model. Biometrika, 95(4), 979–986.CrossRefGoogle Scholar
Lee, R.D. & Carter, L.R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87(419), 659–671.Google Scholar
Murphy, M. (2009). The “golden generations” in historical context. British Actuarial Journal, 15(S1), 151–184. Available online at the address http://www.journals.cambridge.org/abstract_S1357321700005559CrossRefGoogle Scholar
Murphy, M. (2010). Re-examining the dominance of birth cohort effects on mortality. Population and Development Review, 36(2), 365–390. Available online at the address http://www.ncbi.nlm.nih.gov/pubmed/20734557CrossRefGoogle Scholar
Nielsen, B. & Nielsen, J.P. (2014). Identification and forecasting in mortality models. The Scientific World Journal, Article ID 347043.CrossRefGoogle Scholar
O’Brien, R.M. (2000). Age period cohort characteristic models. Social Science Research, 29(1), 123–139. Available online at the address http://linkinghub.elsevier.com/retrieve/pii/S0049089X99906567CrossRefGoogle Scholar
O’Brien, R.M. (2011). Constrained estimators and age-period-cohort models. Sociological Methods & Research, 40(3), 419–452. Available online at the address http://ssagepub.com/cgi/doi/10.1177/0049124111415367CrossRefGoogle Scholar
O’Hare, C. & Li, Y. (2012). Explaining young mortality. Insurance: Mathematics and Economics, 50(1), 12–25.Google Scholar
Osmond, C. (1985). Using age, period and cohort models to estimate future mortality rates. International Journal of Epidemiology, 14(1), 124–129. Available online at the address http://www.ncbi.nlm.nih.gov/pubmed/3988427CrossRefGoogle ScholarPubMed
Plat, R. (2009). On stochastic mortality modeling. Insurance: Mathematics and Economics, 45(3), 393–404. Available online at the address http://www.sciencedirect.com/science/article/pii/S0167668709000973Google 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
Richards, S.J. (2008). Detecting year-of-birth mortality patterns with limited data. Journal of the Royal Statistical Society: Series A (Statistics in Society), 171(1), 279–298.Google Scholar
Rodgers, W. (1982). Estimable functions of age, period, and cohort effects. American Sociological Review, 47(6), 774–787.CrossRefGoogle Scholar
Tuljapurkar, S., Li, N., & Boe, C. (2000). A universal pattern of mortality decline in the G7 countries. Nature, 405(6788), 789–792.CrossRefGoogle ScholarPubMed
Willets, R. (1999). Mortality in the Next Millennium. Staple Inn Actuarial Society, London, UK.Google Scholar
Willets, R. (2004). The cohort effect: insights and explanations. British Actuarial Journal, 10(4), 833–877.CrossRefGoogle Scholar
Wilmoth, J.R. (1990). Variation in vital rates by age, period and cohort. Sociological Methodology, 20, 295–335.CrossRefGoogle Scholar
Yang, Y., Fu, W.J. & Land, K.C. (2004). A methodological comparison of age-period-cohort models: the intrinsic estimator and conventional generalized linear models. Sociological Methodology, 34, 75–110.CrossRefGoogle Scholar