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TESTING FOR A UNIT ROOT IN LEE–CARTER MORTALITY MODEL

Published online by Cambridge University Press:  29 August 2017

Xuan Leng  and
Liang Peng
Xuan Leng
Affiliation:
Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands, E-Mail: leng@ese.eur.nl
Liang Peng*
Affiliation:
Department of Risk Management and Insurance, Robinson College of Business, Georgia State University, Atlanta, GA 30303, USA
*
E-Mail: lpeng@gsu.edu
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Abstract

Motivated by a recent discovery that the two-step inference for the Lee–Carter mortality model may be inconsistent when the mortality index does not follow from a nearly integrated AR(1) process, we propose a test for a unit root in a Lee–Carter model with an AR(p) process for the mortality index. Although testing for a unit root has been studied extensively in econometrics, the method and asymptotic results developed in this paper are unconventional. Unlike a blind application of existing R packages for implementing the two-step inference procedure in Lee and Carter (1992) to the U.S. mortality rate data, the proposed test rejects the null hypothesis that the mortality index follows from a unit root AR(1) process, which calls for serious attention on using the future mortality projections based on the Lee–Carter model in policy making, pricing annuities and hedging longevity risk. A simulation study is conducted to examine the finite sample behavior of the proposed test too.

Keywords

AR processLee–Carter modelunit root
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 3 , September 2017 , pp. 715 - 735
Copyright
Copyright © Astin Bulletin 2017 

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References

Bauer, D., Benth, F.E. and Kiesel, R. (2012) Modeling the forward surface of mortality. SIAM Journal on Financial Mathematics, 3, 639–666.Google Scholar
Bisetti, E. and Favero, C.A. (2014) Measuring the impact of longevity risk on pension systems: The case of Italy. North American Actuarial Journal, 18, 87–103.Google Scholar
Brouhns, N., Denuit, M. and Vermunt, J.K. (2002) A Poisson log-linear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31, 373–393.Google Scholar
Cairns, A.J., Blake, D. and Dowd, K. (2008) Modelling and management of mortality risk: A review. Scandinavian Actuarial Journal, 2008, 79–113.Google Scholar
Cairns, A.J., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011) Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48, 355–367.Google Scholar
Chen, H. and Cox, S.H. (2009) Modeling mortality with jumps: Applications to mortality securitization. Journal of Risk and Insurance, 76, 727–751.Google Scholar
Currie, I.D., Durban, M. and Eilers, P.H.C. (2004) Smoothing and forecasting mortality rates. Statistical Modelling, 4, 279–298.Google Scholar
D'Amato, V., Haberman, S., Piscopo, G., Russolillo, M. and Trapani, L. (2014) Detecting common longevity trends by a multiple population approach. North American Actuarial Journal, 18, 139–149.Google Scholar
Frees, E.W., Carriere, J.F. and Valdez, E.A. (1996) Annuity valuation with dependent mortality. Journal of Risk and Insurance, 63, 229–261.Google Scholar
Girosi, F. and King, G. (2007) Understanding the Lee–Carter Mortality Forecasting Method. Working Paper, Harvard University. Available at https://gking.harvard.edu/files/abs/lc-abs.shtml.Google Scholar
Haberman, S. and Renshaw, A.E. (2008) Mortality, longevity and experiments with the Lee–Carter model. Lifetime Data Analysis, 14, 286–315.Google Scholar
Kölbl, F. (2006) Aggregation of AR(2) processes. Technical report at Technische Universität Graz. Available at www.stat.tugraz.at/dthesis/koelbl06.pdf.Google Scholar
Lee, R. and Carter, L. (1992) Modelling and forecasting US mortality. Journal of the American Statistical Association, 87, 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, N. and Lee, R. (2005) Coherent mortality forecasts for a group of populations: An extension of the Lee–Carter method. Demography, 42, 575–594.Google Scholar
Lin, Y., Tan, K.S., Tian, R. and Yu, J. (2014) Downside risk management of a defined benefit plan considering longevity basis risk. North American Actuarial Journal, 18, 68–86.Google Scholar
Njenga, C.N. and Sherris, M. (2011) Longevity risk and the econometric analysis of mortality trends and volatility. Asia-Pacific Journal of Risk and Insurance, 5, 24–76.Google Scholar
Phillips, P.C.B. and Durlauf, S.N. (1986) Multiple time series regression with integrated processes. The Review of Economic Studies, 53, 473–495.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika, 74, 535–547.Google Scholar
Phillips, P.C.B. and Perron, P. (1988) Testing for a unit root in time series regression. Biometrika, 75, 335–346.Google Scholar
Said, S.E. and Dickey, D. (1984) Testing for unit roots in autoregressive moving-average models with unknown order. Biometrika, 71, 599–607.Google Scholar
Xiao, Z. (2014) Unit roots: A selective review of the contributions of Peter C.B. Phillips. Econometric Theory, 30, 775–814.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, 157–169.Google Scholar
Zhang, L., Shen, H. and Huang, J. (2013) Robust regularized singular value decomposition with application to mortality data. Annals of Applied Statistics, 7, 1540–1561.Google Scholar