Abstract
The prototypical Lee–Carter mortality model is characterized by a single common time factor that loads differently across age groups. In this paper, we propose a parametric factor model for the term structure of mortality where multiple factors are designed to influence the age groups differently via parametric loading functions. We identify four different factors: a factor common for all age groups, factors for infant and adult mortality, and a factor for the “accident hump” that primarily affects mortality of relatively young adults and late teenagers. Since the factors are identified via restrictions on the loading functions, the factors are not designed to be orthogonal but can be dependent and can possibly cointegrate when the factors have unit roots. We suggest two estimation procedures similar to the estimation of the dynamic Nelson–Siegel term structure model. First, a two-step nonlinear least squares procedure based on cross-section regressions together with a separate model to estimate the dynamics of the factors. Second, we suggest a fully specified model estimated by maximum likelihood via the Kalman filter recursions after the model is put on state space form. We demonstrate the methodology for US and French mortality data. We find that the model provides a good fit of the relevant factors and, in a forecast comparison with a range of benchmark models, it is found that, especially for longer horizons, variants of the parametric factor model have excellent forecast performance.
Highlights
The Lee and Carter (1992) (LC) model has become a benchmark model when estimating and forecasting improvements in age-specific death rates and the calculation of life expectancy
We observe seven stylized facts for the term structure of mortality that a good mortality model should be able to reproduce: (1) declining mortality for infants, (2) increasing mortality around the accident hump, (3) log-linearly increasing mortality with age for adults, (4) a log-linear relationship between the death rates and time, (5) the log age-specific death rates are integrated of order one around a linear trend, (6) decreasing improvements in mortality with age, and (7) multiple stochastic trends characterize the development of log mortality over time for the different age groups
We have suggested a multi-factor model for the term structure of mortality
Summary
The Lee and Carter (1992) (LC) model has become a benchmark model when estimating and forecasting improvements in age-specific death rates and the calculation of life expectancy. We consider estimation of the model parameters and the factors by cross-section regressions over age groups for each period of time. These estimations are conducted over a grid of tuning parameters that define the shape of the loading functions. A least squares criterion is used to determine the desired tuning parameters and the corresponding factor elements This approach is similar to the first step of the cross section projection procedure suggested in Diebold and Li (2006). By use of the Kalman filter recursions, the model parameters and the factors can be estimated by full maximum likelihood This approach is similar to that of Diebold et al (2006) for the term structure of interest rates.
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