Negative binomial version of the Lee–Carter model for mortality forecasting

Abstract

SUMMARYMortality 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 + x t ). 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 a time-varying parameter t reflecting the general level of mortality, and an age-specific component x that represents how rapidly or slowly mortality at each age varies when the general level of mortality changes. The parameters are usually estimated via singular value decomposition or via maximum likelihood in a binomial or Poisson regression model. This paper demonstrates that it is possible to take into account the overdispersion present in the mortality data by estimating the parameter in a negative binomial regression model.

Prospective mortality tables: Taking heterogeneity into account

“…A Poisson log-likelihood approach has been developed in Brouhns et al (2002b), Brouhns et al (2002a) and Renshaw and Haberman (2003) to remedy to some of the drawbacks of the Lee-Carter approach, such as for instance the assumed homoskedasticity of the errors. Cosette et al (2007) use a binomial maximum likelihood, and a negative binomial version of the Lee-Carter model has been developed by Delwarde et al (2007) to take into account the over-dispersion phenomenon.…”

Bayesian mortality forecasting with overdispersion

“…In particular, we found in our preliminary study that the computational gain from marginalising µ xt substantially outweighs the burden of dealing with the more complicated negative binomial likelihood (by comparing the effective number of samples generated per unit time). Note that this model has already been considered by Delwarde et al (2007), but within the classical framework.…”

“…Their approach also suffers from the issue that the relationship between the expectation, variance and probability function of death data under the model are internally inconsistent (see Li et al (2009)). Delwarde et al (2007) then proposed a direct extension of the Poisson LC model to form the negative binomial LC model (again, they did not consider the construction of prediction intervals). In addition, Li et al (2009) attempted to account for mortality variations by introducing an age-specific latent variable that accounts for heterogeneity of individuals, which upon marginalisation, leads to the negative binomial LC model as well.…”

Lee-Carter state space modeling: Application to the Malaysia mortality data

“…It becomes the dominant model widely used by actuaries, demographers and many other practitioners to model and forecast age-specific mortality. The approach has undergone various extensions and modifications exemplified in the works of Brouhns et al [6], Cairns et al [7], De Jong and Tickle [8], Delwarde et al [9], Hyndman and Shahid Ullah [10] and Mitchell et al [11], in which attentions were given to modification in the estimation of the first equation of the model by retaining the time series method in the second equation. Pedroza [12] highlighted the fact that the prediction interval based on the original LC model only accounts for the error of time series model while ignoring the errors in estimating the parameters and the variance of the error term in the modeling of the first equation.…”