Abstract
When independence is assumed, forecasts of mortality for subpopulations are almost always divergent in the long term. We propose a method for coherent forecasting of mortality rates for two or more subpopulations, based on functional principal components models of simple and interpretable functions of rates. The product-ratio functional forecasting method models and forecasts the geometric mean of subpopulation rates and the ratio of subpopulation rates to product rates. Coherence is imposed by constraining the forecast ratio function through stationary time series models. The method is applied to sex-specific data for Sweden and state-specific data for Australia. Based on out-of-sample forecasts, the coherent forecasts are at least as accurate in overall terms as comparable independent forecasts, and forecast accuracy is homogenized across subpopulations.
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Notes
If X and Y are two random variables, then Corr(X – Y, X + Y) = Var(X) – Var(Y). So if X and Y have equal variances, then X – Y and X + Y are uncorrelated.
References
Bengtsson, T. (2006). Linear increase in life expectancy: Past and present. In T. Bengtsson (Ed.), Perspectives on mortality forecasting: Vol. III. The linear rise in life expectancy: History and prospects (pp. 83–99). Stockholm: Swedish Social Insurance Agency, Stockholm.
Booth, H., Hyndman, R. J., Tickle, L., & de Jong, P. (2006). Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research, 15(article 9), 289–310. doi:10.4054/DemRes.2006.15.9
Booth, H., Maindonald, J., & Smith, L. (2002). Applying Lee-Carter under conditions of variable mortality decline. Population Studies, 56, 325–336.
Booth, H., & Tickle, L. (2008). Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science, 3(1–2), 3–43.
Box, G. E. P., Jenkins, G. M., & Reinsel, G. C. (2008). Time series analysis: Forecasting and control (4th ed.). Hoboken, NJ: Wiley.
Cairns, A., Blake, D., & Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. The Journal of Risk and Insurance, 73, 687–718.
Cairns, A. J. G., Blake, D., Dowd, K., Coughlan, G. D., Epstein, D., & Khalaf-Allah, M. (2011). Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48, 355–367.
Currie, I. D., Durban, M., & Eilers, P. H. (2004). Smoothing and forecasting mortality rates. Statistical Modelling, 4, 279–298.
Granger, C., & Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis, 1, 15–39.
Greenshtein, E. (2006). Best subset selection, persistence in high-dimensional statistical learning and optimization under l 1 constraint. The Annals of Statistics, 34, 2367–2386.
Greenshtein, E., & Ritov, Y. (2004). Persistence in high-dimensional linear predictor selection and the virtue of overparameterization. Bernoulli, 10, 971–988.
Haslett, J., & Raftery, A. E. (1989). Space-time modelling with long-memory dependence: Assessing Ireland’s wind power resource (with discussion). Applied Statistics, 38, 1–50.
Hosking, J. (1981). Fractional differencing. Biometrika, 68, 165–176.
Human Mortality Database. (2010). University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Retrieved from http://www.mortality.org
Hyndman, R. J. (2008). addb: Australian Demographic Data Bank. R package version 3.222. Retrieved from http://robjhyndman.com/software/addb
Hyndman, R. J. (2010). Demography: Forecasting mortality, fertility, migration and population data. R package version 1.07. With contributions from Heather Booth and Leonie Tickle and John Maindonald. Retrieved from http://robjhyndman.com/software/demography
Hyndman, R. J., & Booth, H. (2008). Stochastic population forecasts using functional data models for mortality, fertility and migration. International Journal of Forecasting, 24, 323–342.
Hyndman, R. J., & Khandakar, Y. (2008). Automatic time series forecasting: The forecast package for R. Journal of Statistical Software, 27(3), 1–22.
Hyndman, R. J., & Shang, H. L. (2009). Forecasting functional time series (with discussion). Journal of the Korean Statistical Society, 38, 199–221.
Hyndman, R. J., & Shang, H. L. (2010). Rainbow plots, bagplots and boxplots for functional data. Journal of Computational and Graphical Statistics, 19, 29–45.
Hyndman, R. J., & Ullah, M. S. (2007). Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics & Data Analysis, 51, 4942–4956.
Kalben, B. B. (2002). Why men die younger: Causes of mortality differences by sex (SOA Monograph M-LI01-1). Schaumburg, IL: Society of Actuaries.
Lee, R. D. (2000). The Lee-Carter method for forecasting mortality, with various extensions and applications. North American Actuarial Journal, 4, 80–92.
Lee, R. D. (2006). Mortality forecasts and linear life expectancy trends. In T. Bengtsson (Ed.), Perspectives on mortality forecasting: Vol. III. The linear rise in life expectancy: History and prospects (pp. 19–39). Stockholm: Swedish National Social Insurance Board.
Lee, R. D., & Carter, L. R. (1992). Modeling and forecasting US mortality. Journal of the American Statistical Association, 87, 659–671.
Lee, R. D., & Miller, T. (2001). Evaluating the performance of the Lee-Carter method for forecasting mortality. Demography, 38, 537–549.
Lee, R., & Nault, F. (1993, August). Modeling and forecasting provincial mortality in Canada. Paper presented at the World Congress of the IUSSP, Montreal, Canada.
Li, N., & Lee, R. (2005). Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography, 42, 575–594.
Oeppen, J. (2008). Coherent forecasting of multiple-decrement life tables: A test using Japanese cause of death data (Technical report). Catalonia, Spain: Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona. Retrieved from http://hdl.handle.net/10256/742
Oeppen, J., & Vaupel, J. W. (2002). Broken limits to life expectancy. Science, 296, 1029–1031.
Ortega, J. A., & Poncela, P. (2005). Joint forecasts of southern European fertility rates with non-stationary dynamic factor models. International Journal of Forecasting, 21, 539–550.
Peiris, M., & Perera, B. (1988). On prediction with fractionally differenced ARIMA models. Journal of Time Series Analysis, 9, 215–220.
Preston, S. H., Heuveline, P., & Guillot, M. (2001). Demography: Measuring and modelling population processes. Oxford, UK: Blackwell.
Reinsel, G. C. (2003). Elements of multivariate time series analysis (2nd ed.). New York: Springer-Verlag.
Renshaw, A., & Haberman, S. (2003). Lee–Carter mortality forecasting: A parallel generalized linear modelling approach for England and Wales mortality projections. Applied Statistics, 52, 119–137.
Shang, H. L., Booth, H., & Hyndman, R. J. (2011). Point and interval forecasts of mortality rates and life expectancy: A comparison of ten principal component methods. Demographic Research, 25(article 5), 173–214. doi:10.4054/DemRes.2011.25.5
Shumway, R. H., & Stoffer, D. S. (2006). Time series analysis and its applications: With R examples (2nd ed.). New York: Springer Science+Business Media, LLC.
White, K. M. (2002). Longevity advances in high-income countries, 1955–96. Population and Development Review, 28, 59–76.
Wilmoth, J. R. (1995). Are mortality projections always more pessimistic when disaggregated by cause of death? Mathematical Population Studies, 5, 293–319.
Wood, S. N. (1994). Monotonic smoothing splines fitted by cross validation. SIAM Journal on Scientific Computing, 15, 1126–1133.
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Hyndman, R.J., Booth, H. & Yasmeen, F. Coherent Mortality Forecasting: The Product-Ratio Method With Functional Time Series Models. Demography 50, 261–283 (2013). https://doi.org/10.1007/s13524-012-0145-5
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DOI: https://doi.org/10.1007/s13524-012-0145-5