Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting: a penalized log-likelihood approach

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 (alpha(x) + beta K-x(t)). The log of the time series of age-specific death rates is thus expressed as the sum of an age-specific component alpha(x), that is independent of time and another component that is the product of a timevarying parameter Kt reflecting the general level of mortality, and an age-specific component beta(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 Kt's are then modeled and projected as a stochastic time series using standard Box Jenkins methods. However, the estimated fix'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, beta(x)'s in the Lee-Carter and Poisson log-bilinear models for mortality projection. To this end, a penalized least-squares/maximum likelihood analysis is performed. The optimal value of the smoothing parameter is selected with the help of cross, validation.