Abstract
Mortality forecasts are critically important inputs to the consideration of a range of demographically-related policy challenges facing governments in more developed countries. While methods for jointly forecasting mortality for sub-populations offer the advantage of avoiding undesirable divergence in the forecasts of related populations, little is known about whether they improve forecast accuracy. Using mortality data from ten populations, we evaluate the data fitting and forecast performance of the Poisson common factor model (PCFM) for projecting both sexes’ mortality jointly against the Poisson Lee–Carter model applied separately to each sex. We find that overall the PCFM generates the more desirable results. Firstly, the PCFM ensures that the projected male-to-female ratio of death rates at each age converges to a constant in the long run. Secondly, using out-of-sample analysis, we find that the PCFM provides more accurate projection of the sex ratios of death rates, with the advantage being greater for longer-term forecasts. Thus the PCFM offers a viable and sensible means for coherently forecasting the mortality of both sexes. There are also significant financial implications in allowing for the co-movement of mortality of females and males properly.
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Notes
As in Kleinow (2015), a possible extension to the PCFM structure is to use the same age effect for both sexes, i.e. \(b_{x,1,j} = b_{x,2,j}\). This simplification may further improve the BIC values when the data fitting period is very short and the \(b_{x,i,j}\) patterns are similar between females and males.
In our parameter estimation, the model structure of n = 0 is fitted initially. Then, treating the estimated parameters as known quantities, the additional factors are incorporated into the model structure and these new parameters are computed. There are no convergence problems during the process.
As in Li (2013), we use the same number of additional factors for both sexes in this paper. As a possible extension to the initial PCFM structure, we further test the cases where the number of additional factors of one sex is different to the other. For the ten populations studied in the next section, only two require different numbers of additional factors between the two sexes to achieve the lowest BIC value. Since the trends of the corresponding higher-order additional factors required for these two populations are quite irregular, we choose to simply use the same number of additional factors for both sexes and leave the possible extension for future research.
One may assume that \(\varepsilon_{t}\) and \(\omega_{t,i,j}\) are independent, as in Li and Lee (2005) and Li (2013), or alternatively that they follow a certain correlation structure. This assumption would not affect our analysis, as we focus on the accuracy and reasonableness of the mean forecasts but not the variability.
The MAPE of the fitted log death rates is defined similarly as in (9).
Though the extents are very different, both the PCFM and the Poisson Lee-Carter model produce residuals with some systematic effects along the cohort year. Cohort terms (e.g. Renshaw and Haberman 2006) may be added to improve the model fitting.
The (period) life expectancy value is calculated as \(\sum\nolimits_{t = 0} {{}_{t + 0.5}p_{x} }\), where \({}_{t}p_{x}\) are (period) survival probabilities (of age x for t years) estimated from the observed death rates or projected death rates.
The cohorts considered are not extinct and so the corresponding life expectancy values computed are limited by the data period. For example, the value calculated for an Australian born in 1995 can be interpreted as the number of years expected to be lived within 13 years (but not the entire lifespan).
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Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions, which greatly enhance the presentation of this paper. The authors also gratefully acknowledge financial support from the Institute of Actuaries of Australia via an Australian Actuarial Research Grant. Finally, the authors thank Daniel Ciarliero for his excellent assistance in this research project.
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Li, J., Tickle, L. & Parr, N. A multi-population evaluation of the Poisson common factor model for projecting mortality jointly for both sexes. J Pop Research 33, 333–360 (2016). https://doi.org/10.1007/s12546-016-9173-0
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DOI: https://doi.org/10.1007/s12546-016-9173-0