Autocorrelation in Stochastic Models of Dutch Mortality Rates


  • Alexander Heinemann



This paper studies different stochastic mortality models with respect to its underlying assumptions. The Lee-Carter model, the rank-p SVD approximation model, the Weighted Least Squares model and the Poisson Bilinear model are discussed and applied to a Dutch data set. In an empirical analysis, it is illustrated how to obtain parameter estimates and how mortality rates can be forecasted. In a second step, the paper examines the model’s underlying assumptions on a residual basis. Several tests are employed testing for normality, homogeneity of variance and autocorrelation. The test results seem to invalidate the models applicability due to failure in the underlying assumptions. In particular the crucial assumption of observational independence does not seem to hold, which may result in prediction intervals that are too narrow. Furthermore, a theoretical explanation for autocorrelation is given and an alternative model (multivariate ARIMA model) is proposed, that does not only rely on a weaker set of assumptions, but also requires less computational effort. Comparing the initial models’ estimates with the estimates of the alternative model, it is concluded that all five stochastic mortality models tend to deliver similar estimates.


Booth, H., Maindonald, J. and Smith, L. (2002). Applying Lee-Carter under Condition of Variable Mortality Decline. Population Studies, 56(3), 325-336.

Box, G. E. P., & Jenkins, G. M. (1976). Time Series Analysis Forecasting and Control. San Francisco: Holden-Day.

Bühlmann, P. (1997). Sieve Bootstrap for Time Series. Bernoulli, 3, 123-148.

Bühlmann, P. (1998). Sieve Bootstrap for Smoothing Nonstationary Time Series. Annals of Statistics, 26, 48-83.

Brillinger (1986). The natural variability of vital rates and associated statistics. Bimetrics, 42(4), 693-734.

Brouhns, N., Denuit, M., Keilegom, I.V.(2005). Bootstrapping the poisson Log-bilinear Regression Approach to the Construction of Projected Lifetables. Scandinavian Actuarial Journal, 3, 212-224.

Brouhns, N., Denuit, M., Vermunt, J.K.(2002). A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31, 373-393.

Caselli, G.,Vallin, J., Wunsch, G.J. (2006). Demography: Analysis and Synthesis. Elsevier, 1, 117- 119.

Dowd et. al. (2010). Evaluating the goodness of fit of stochastic mortality models. Insurance: Mathematics and Economics, 47, 255-265.

Durbin, J., and Watson, G. S. (1950). Testing for Serial Correlation in Least Squares Regression, I. Biometrika 37, 409- 428.

Eckart, C. and Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1(3), 211-218.

Girosi F. and G. King (2007). Understanding the Lee-Carter Mortality Forecasting Method. Working paper, Harvard University, USA.

Hartley, H.O. (1950). The Use of Range in Analysis of Variance. Biometrica Trust, 37(3/4), 271-280.

Human Mortality Database (2013). Netherlands: Complete Data Series. Retrieved from on 10 April, 2013.

Lee, R.D. & Carter, L.R. (1992). Modeling and Forecasting U. S. Mortality. Journal of the American Statistical Association, 87(419), 659-671.

Madsen, R.E., Hansen, L.K., Winther, O. (2004). Singular Value Decomposition and Principal Component Analysis.

Park, H. M. (2008). Univariate Analysis and Normality Test Using SAS, Stata, and SPSS. Indiana University, Indiana. USA.

Pierce, D. A. and Schafer, D. W. (1986). residuals in Generalized Linear Models. Journal of the American Statistical Association, 81(396), 977-986.

Ravishanker, N., Hochberg, Y. , Melnick, E. L. (1987). Approximate Simultaneous Prediction Intervals for Multiple Forecasts Technometrics, 29(3), 371-376.

Renshaw, A. and Haberman, S. (2005). Mortality Reduction Factors Incorporating Cohort Effects. Actuarial Research Paper, Cass Business School, London. UK.

United Nations, Department of Economic and Social Affairs, Population Division (2012). World Mortality Report 2011. United Nations publication.

Wilmoth J.R. (1993). Computational Methods for fitting and Extrapolating the Lee-Carter Model of Mortality Change. Technical Report, University of California, Berkeley. USA.