Modelling and forecasting mortality distributions in England and Wales using the Lee–Carter model

Abstract

Lee and Carter proposed in 1992 a non-linear model m _xt  = exp (a _x  + b _x k _t  + ϵ _xt ) for fitting and forecasting age-specific mortality rates at age x and time t. For the model parameter estimation, they employed the singular value decomposition method to find a least squares solution. However, the singular value decomposition algorithm does not provide the standard errors of estimated parameters, making it impossible to assess the accuracy of model parameters. This article describes the Lee–Carter model and the technical procedures to fit and extrapolate this model. To estimate the precision of the parameter estimates of the Lee–Carter model, we propose a binomial framework, whose parameter point estimates can be obtained by the maximum likelihood approach and interval estimates by a bootstrap approach. This model is used to fit mortality data in England and Wales from 1951 to 1990 and to forecast mortality change from 1991 to 2020. The Lee–Carter model fits these mortality data very well with R ² being 0.9980. The estimated overall age pattern of mortality a_x is very robust whereas there is considerable uncertainty in b_x (changes in the age pattern over time) and k_t (overall change in mortality). The fitted log age-specific mortality rates have been declining linearly from 1951 to 1990 at different paces and the projected rates will continue to decline in such a way in the 30 years prediction period.

Keywords:

• Lee–Carter model
• single value decomposition
• binomial distribution
• bootstrap
• mortality forecasting

Acknowledgement

We wish to thank Professor Michael Murphy for his helpful comments on an earlier draft of this paper.