Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T21:22:00.340Z Has data issue: false hasContentIssue false
  • English
  • Français

COHERENT FORECASTING OF MORTALITY RATES: A SPARSE VECTOR-AUTOREGRESSION APPROACH

Published online by Cambridge University Press:  23 March 2017

Hong Li  and
Yang Lu
Hong Li
Affiliation:
Aix-Marseille School of Economics, Aix-Marseille University, 2, Rue de la Charite, 13001, Marseille, France
Yang Lu*
Affiliation:
Aix-Marseille School of Economics, Aix-Marseille University, 2, Rue de la Charite, 13001, Marseille, France
*
†Corresponding author, [email protected]. Aix-Marseille School of Economics, Aix-Marseille University, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proposes a spatial-temporal autoregressive model for the mortality surface, where mortality rates of each age depend on the historical values of itself (temporality) and the neighbouring ages (spatiality). The mortality dynamics is formulated as a large, first order vector autoregressive model which encompasses standard factor models such as the Lee and Carter (1992) model. Sparsity and smoothness constraints are then introduced, based on the idea that the nearer the two ages, the more important the dependence between mortalities at these ages. Our model has several novelties. First, it ensures that in the long-run, mortality rates at different ages do not diverge. Second, it provides a natural explanation of the so-called cohort effect without identifiability difficulties. Third, the model is easily extended to the multiple-population case in a coherent way. Finally, the model is associated with a closed form, non-parametric estimation method: the penalized least square, which ensures spatial smoothness of the age-dependent parameters. Using US and UK mortality data, we find that our model produces reasonable projected mortality profile in the long-run, as well as satisfying short-run out-of-sample forecast performance.

Keywords

cohort effectspatial co-integrationpenalized least square
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 2 , May 2017 , pp. 563 - 600
Copyright
Copyright © Astin Bulletin 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

[email protected]. School of Finance, Nankai University, Tongyan Road 38, 300350, Tianjin, P.R.China. Hong Li acknowledges the support of National Natural Science Foundation of China, Grant No. 61673225.

References

Barrieu, P., Bensusan, H., El Karoui, N., Hillairet, C., Loisel, S., Ravanelli, C. and Salhi, Y. (2012) Understanding, Modelling and Managing Longevity Risk: Key Issues and Main Challenges. Scandinavian Actuarial Journal, 2012 (3), 203–231.Google Scholar
Biffis, E. and Millossovich, P. (2006) A Bidimensional Approach to Mortality Risk. Decisions in Economics and Finance, 29 (2), 71–94.Google Scholar
Burman, P., Chow, E. and Nolan, D. (1994) A Cross-Validatory Method for Dependent Data. Biometrika, 81 (2).Google Scholar
Cairns, A. J. G., Blake, D. and Dowd, K. (2006) A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of Risk and Insurance, 73 (4), 687–718.Google Scholar
Cairns, A. J. G., Blake, D. and Dowd, K. (2008) Modelling and Management of Mortality Risk: a Review. Scandinavian Actuarial Journal, 2008 (2-3), 79–113.CrossRefGoogle Scholar
Cairns, A. J. G., Blake, D., Dowd, K., Coughlan, G. and Khalaf-Allah, M. (2011) Bayesian Stochastic Mortality Modelling for Two Populations. ASTIN Bulletin, 41 (1), 29–59.Google Scholar
Cairns, A. J. G., Blake, D., Dowd, K., Coughlan, G. D., Epstein, D., Ong, A. and Balevich, I. (2009) A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States. North American Actuarial Journal, 13 (1), 1–35.Google Scholar
Callot, L., Haldrup, N. and Kallestrup-Lamb, M. (2016) Deterministic and stochastic trends in the lee–carter mortality model. Applied Economics Letters, 23 (7), 486–493.Google Scholar
Chuliá, H., Guillén, M. and Uribe, J. M. (2015) Modeling Longevity Risk With Generalized Dynamic Factor Models and Vine Copulae. ASTIN Bulletin, 46 (01), 1–26.Google Scholar
Debón, A., Montes, F., Mateu, J., Porcu, E. and Bevilacqua, M. (2008) Modelling Residuals Dependence in Dynamic Life Tables: A Geostatistical Approach. Computational Statistics & Data Analysis, 52 (6), 3128–3147.Google Scholar
Delwarde, A., Denuit, M. and Eilers, P. (2007) Smoothing the Lee–Carter and Poisson Log-bilinear Models for Mortality Forecasting: A Penalized Log-likelihood Approach. Statistical Modelling, 7 (1), 29–48.Google Scholar
Dowd, K., Cairns, A. J. G., Blake, D., Coughlan, G. and Khalaf-Allah, M. (2011) A Gravity Model of Mortality Rates for Two Related Populations. North American Actuarial Journal, 15 (2), 334–356.Google Scholar
Engle, R. F. and Granger, C. W. (1987) Co-integration and Error Correction: Representation, Estimation, and Testing. Econometrica, 55 (2), 251–276.Google Scholar
Gaille, S. and Sherris, M. (2011) Modelling Mortality with Common Stochastic Long-Run Trends. The Geneva Papers on Risk and Insurance-Issues and Practice, 36 (4), 595–621.Google Scholar
Granger, C. (1969) Investigating Causal Relations by Econometric Models and Cross-Spectral Methods. Econometrica, 37 (3), 424–38.CrossRefGoogle Scholar
Heckman, J. and Robb, R. (1985) Using Longitudinal Data to Estimate Age, Period and Cohort Effects in Earnings Equations. In Cohort Analysis in Social Research, pages 137–150. Springer.Google Scholar
Hunt, A. and Blake, D. (2015a) Identifiability, Cointegration and the Gravity Model. Pension Institute DP.Google Scholar
Hunt, A. and Blake, D. (2015b) Modelling Longevity Bonds: Analysing the Swiss Re Kortis Bond. Insurance: Mathematics and Economics, 63, 12–29.Google Scholar
Hunt, A. and Villegas, A. M. (2015) Robustness and Convergence in the Lee–Carter Model with Cohort Effects. Insurance: Mathematics and Economics, 64, 186–202.Google Scholar
Hyndman, R. J., Booth, H. and Yasmeen, F. (2013) Coherent Mortality Forecasting: the Product-Ratio Method with Functional Time Series Models. Demography, 50 (1), 261–283.Google Scholar
Kuang, D., Nielsen, B. and Nielsen, J. (2008) Identification of the Age-period-Cohort Model and the Extended Chain-Ladder Model. Biometrika, 95 (4), 979–986.Google Scholar
Lazar, D. and Denuit, M. M. (2009) A Multivariate Time Series Approach to Projected Life Tables. Applied Stochastic Models in Business and Industry, 25 (6), 806–823.CrossRefGoogle Scholar
Lee, R. and Carter, L. (1992) Modeling and Forecasting US Mortality. Journal of the American Statistical Association, 87 (419), 659–671.Google Scholar
Leng, X. and Peng, L. (2016) Inference Pitfalls in Lee-Carter Model for Forecasting Mortality. Insurance: Mathematics and Economics, 70, 58–65.Google Scholar
Li, H., O'Hare, C. and Zhang, X. (2015) A Semiparametric Panel Approach to Mortality Modeling. Insurance: Mathematics and Economics, 61, 264–270.Google Scholar
Li, J. S.-H. and Hardy, M. R. (2011) Measuring Basis Risk in Longevity Hedges. North American Actuarial Journal, 15 (2), 177–200.CrossRefGoogle Scholar
Li, N. and Lee, R. (2005) Coherent Mortality Forecasts for a Group of Populations: An Extension of the Lee-Carter Method. Demography, 42 (3), 575–594.Google Scholar
Li, N., Lee, R. and Gerland, P. (2013) Extending the Lee-Carter Method to Model the Rotation of Age Patterns of Mortality Decline for Long-Term Projections. Demography, 50 (6), 2037–2051.Google Scholar
Litterman, R. B. (1986) Forecasting with Bayesian Vector Autoregressions-Five Years of Experience. Journal of Business & Economic Statistics, 4 (1), 25–38.Google Scholar
Mavros, G., Cairns, A. J. G., Kleinow, T. and Streftaris, G. (2016) Stochastic Mortality Modelling: Key Drivers and Dependent Residuals. Heriot-Watt University DP.Google Scholar
Pace, R. K., Barry, R., Clapp, J. M. and Rodriquez, M. (1998) Spatiotemporal Autoregressive Models of Neighborhood Effects. Journal of Real Estate Finance and Economics, 17 (1), 15–33.CrossRefGoogle Scholar
Plat, R. (2009) Stochastic Portfolio Specific Mortality and the Quantification of Mortality Basis Risk. Insurance: Mathematics and Economics, 45 (1), 123–132.Google Scholar
Racine, J. (1997) Feasible Cross-Validatory Model Selection for General Stationary Processes. Journal of Applied Econometrics, 12 (2), 169–179.3.0.CO;2-P>CrossRefGoogle Scholar
Renshaw, A. and 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
Salhi, Y. and Loisel, S. (2016) Basis risk modelling: a co-integration based approach. forthcoming in Statistics.Google Scholar
Seneta, E. (2006) Non-negative matrices and Markov chains. Springer Science & Business Media.Google Scholar
Tibshirani, R. (1996) Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58 (1), 267–288.Google Scholar
Tsybakov, A. B. and Zaiats, V. (2009) Introduction to Nonparametric Estimation, volume 11. Springer.Google Scholar
Yang, S. S. and Wang, C.-W. (2013) Pricing and Securitization of Multi-Country Longevity Risk with Mortality Dependence. Insurance: Mathematics and Economics, 52 (2), 157–169.Google Scholar
Zellner, A. (1962) An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias. Journal of the American statistical Association, 57 (298), 348–368.Google Scholar
Zhou, R., Wang, Y., Kaufhold, K., Li, J. S.-H. and Tan, K. S. (2014) Modeling Period Effects in Multi-Population Mortality Models: Applications to Solvency II. North American Actuarial Journal, 18 (1), 150–167.Google Scholar