Journal of Population Research

, Volume 35, Issue 3, pp 289–318 | Cite as

Forecasting mortality rates: multivariate or univariate models?

  • Lingbing Feng
  • Yanlin ShiEmail author
Original Research


It is well known that accurate forecasts of mortality rates are essential to various demographic research topics, such as population projections and the pricing of insurance products such as pensions and annuities. In this study, we argue that including the lagged rates of neighbouring ages cannot further improve mortality forecasting after allowing for autocorrelations. This is because the sample cross-correlation function cannot exhibit meaningful and statistically significant correlations. In other words, rates of neighbouring ages are usually not leading indicators in mortality forecasting. Therefore, multivariate stochastic mortality models like the classic Lee–Carter may not necessarily lead to more accurate forecasts, compared with sophisticated univariate models. Using Australian mortality data, simulation and empirical studies employing the Lee–Carter, Functional Data, Vector Autoregression, Autoregression-Autoregressive Conditional Heteroskedasticity and exponential smoothing (ETS) state space models are performed. Results suggest that ETS models consistently outperform the others in terms of forecasting accuracy. This conclusion holds for both female and male mortality data with different empirical features across various forecasting error measurements. Hence, ETS can be a widely useful tool to model and forecast mortality rates in actuarial practice.


Mortality rates Multivariate model Univariate model Exponential smoothing Lee–Carter model 



We are grateful to the Macquarie University and Jiangxi University of Finance and Economics for their support. The authors would also like to thank Rob Hyndman, James Raymer and Hanlin Shang for their helpful comments and suggestions. We are also grateful for the valuable advice provided by the Editor (Santosh Jatrana) and two anonymous referees. The usual disclaimer applies.

Supplementary material

12546_2018_9205_MOESM1_ESM.pdf (499 kb)
Supplementary material 1 (PDF 498 kb)


  1. Bell, W. R. (1997). Comparing and assessing time series methods for forecasting age specific fertility and mortality rates. Journal of Official Statistics, 13, 279.Google Scholar
  2. Booth, H., Hyndman, R., Tickle, L., De Jong, P., et al. (2006). Lee–Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research, 15, 289–310.CrossRefGoogle Scholar
  3. Booth, H., Maindonald, J., & Smith, L. (2002). Applying Lee–Carter under conditions of variable mortality decline. Population Studies, 56(3), 325–336.CrossRefGoogle Scholar
  4. Camarda, C. G. (2012). Mortalitysmooth: An R package for smoothing poisson counts with P-splines. Journal of Statistical Software, 50(1), 1–24.CrossRefGoogle Scholar
  5. Chatfield, C. (1997). Forecasting in the 1990s. Journal of the Royal Statistical Society: Series D (The Statistician), 46(4), 461–473.Google Scholar
  6. Davis, R. A., Zang, P., & Zheng, T. (2016). Sparse vector autoregressive modeling. Journal of Computational and Graphical Statistics, 25(4), 1077–1096.CrossRefGoogle Scholar
  7. Du Preez, J., & Witt, S. F. (2003). Univariate versus multivariate time series forecasting: An application to international tourism demand. International Journal of Forecasting, 19(3), 435–451.CrossRefGoogle Scholar
  8. Gardner, E. S. (1985). Exponential smoothing: The state of the art. Journal of Forecasting, 4(1), 1–28.CrossRefGoogle Scholar
  9. Gardner, E. S., Jr., & McKenzie, E. (1985). Forecasting trends in time series. Management Science, 31(10), 1237–1246.CrossRefGoogle Scholar
  10. Giacometti, R., Bertocchi, M., Rachev, S. T., & Fabozzi, F. J. (2012). A comparison of the Lee–Carter model and AR-ARCH model for forecasting mortality rates. Insurance: Mathematics and Economics, 50(1), 85–93.Google Scholar
  11. Girosi, F., & King, G. (2007). Understanding the Lee-Carter mortality forecasting method 1. Technical report, Rand Corporation, Santa Monica, CA.Google Scholar
  12. Human Mortality Database. (2016). University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany), Accessed 1 Feb 2016.
  13. Hyndman, R. J., & Khandakar, Y. (2008). Automatic time series forecasting: The forecast package for R. Journal of Statistical Software, 27(3), 1–22.CrossRefGoogle Scholar
  14. Hyndman, R. J., & Koehler, A. B. (2006). Another look at measures of forecast accuracy. International Journal of Forecasting, 22(4), 679–688.CrossRefGoogle Scholar
  15. Hyndman, R. J., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2005). Prediction intervals for exponential smoothing using two new classes of state space models. Journal of Forecasting, 24(1), 17–37.CrossRefGoogle Scholar
  16. Hyndman, R., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2008). Forecasting with exponential smoothing: The state space approach. Berlin: Springer.CrossRefGoogle Scholar
  17. Hyndman, R. J., Koehler, A. B., Snyder, R. D., & Grose, S. (2002). A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting, 18(3), 439–454.CrossRefGoogle Scholar
  18. Hyndman, R., & Ullah, S. (2007). Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics & Data Analysis, 51(10), 4942–4956.CrossRefGoogle Scholar
  19. Lee, R. D., & Carter, L. R. (1992). Modeling and forecasting US mortality. Journal of the American statistical association, 87(419), 659–671.Google Scholar
  20. Makridakis, S., & Hibon, M. (2000). The M3-competition: Results, conclusions and implications. International Journal of Forecasting, 16(4), 451–476.CrossRefGoogle Scholar
  21. Ord, J. K., Koehler, A., & Snyder, R. D. (1997). Estimation and prediction for a class of dynamic nonlinear statistical models. Journal of the American Statistical Association, 92(440), 1621–1629.CrossRefGoogle Scholar
  22. Pegels, C. (1969). Exponential forecasting: Some new variations. Management Science, 15(5), 311–315.CrossRefGoogle Scholar
  23. Ramsay, J. O., & Silverman, B. W. (2005). Functional data analysis. Berlin: Springer.CrossRefGoogle Scholar
  24. Renshaw, A. E., & Haberman, S. (2003). Lee–carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics, 33(2), 255–272.Google Scholar
  25. Renshaw, A. E., & Haberman, S. (2006). A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556–570.Google Scholar
  26. Taylor, J. W. (2003). Exponential smoothing with a damped multiplicative trend. International Journal of Forecasting, 19(4), 715–725.CrossRefGoogle Scholar
  27. Trefethen, L. N., & Bau, D. (1997). Numerical linear algebra (Vol. 50). Siam.Google Scholar
  28. Wood, S. N. (2003). Thin plate regression splines. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65(1), 95–114.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.International Institute of Financial StudiesJiangxi University of Finance and EconomicsNanchangChina
  2. 2.Department of Actuarial Studies and Business AnalyticsMacquarie UniversitySydneyAustralia

Personalised recommendations