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Smoothing the Lee–Carter and Poisson log-bilinear models for mortality forecasting

A penalized log-likelihood approach

Antoine Delwarde, Institut des Sciences Actuarielles, Université Catholique de Louvain Belgium.

Michel Denuit

Michel Denuit, Institut de Statistique Universite Catholique de Louvain,Voie du Roman Pays, 20 B-1348 Louvain-la-Neuve Belgium. E-mail: denuit{at}stat.ucl.ac.be

Paul Eilers

Paul Eilers, Medical Statistics, Leiden University Medical Centre, The Netherlands.

Mortality improvements pose a challenge for the planning of public retirement systems as well as for the private life annuities business. For public policy, as well as for the management of financial institutions, it is important to forecast future mortality rates. Standard models for mortality forecasting assume that the force of mortality at age x in calendar year t is of the form exp(_x + ß_xt ). The log of the time series of age-specific death rates is thus expressed as the sum of an age-specific component _x that is independent of time and another component that is the product of time-varying parameter _t reflecting the general level of mortality, and an age-specific component ß_x that represents how rapidly or slowly mortality at each age varies when the general level of mortality changes. This model is fitted to historical data. The resulting estimated _t 's are then modeled and projected as stochastic time series using standard Box–Jenkins methods. However, the estimated ß_x's exhibit an irregular pattern in most cases, and this produces irregular projected life tables. This article demonstrates that it is possible to smooth the estimated ß_x's in the Lee–Carter and Poisson log-bilinear models for mortality projection. To this end, penalized least-squares/maximum likelihood analysis is performed. The optimal value of the smoothing parameter is selected with the help of cross validation.

Key Words: cross validation • life table • mortality projection • roughness penalty • smoothing

Statistical Modelling, Vol. 7, No. 1, 29-48 (2007)
DOI: 10.1177/1471082X0600700103


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