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Forecasting mortality rate improvements with a high-dimensional VAR. (English) Zbl 1425.91223

Summary: Forecasting mortality rates is a problem which involves the analysis of high-dimensional time series. Most of usual mortality models propose to decompose the mortality rates into several latent factors to reduce this complexity. These approaches, in particular those using cohort factors, have a good fit, but they are less reliable for forecasting purposes. One of the major challenges is to determine the spatial-temporal dependence structure between mortality rates given a relatively moderate sample size. This paper proposes a large vector autoregressive (VAR) model fitted on the differences in the log-mortality rates, ensuring the existence of long-run relationships between mortality rate improvements. Our contribution is threefold. First, sparsity, when fitting the model, is ensured by using high-dimensional variable selection techniques without imposing arbitrary constraints on the dependence structure. The main interest is that the structure of the model is directly driven by the data, in contrast to the main factor-based mortality forecasting models. Hence, this approach is more versatile and would provide good forecasting performance for any considered population. Additionally, our estimation allows a one-step procedure, as we do not need to estimate hyper-parameters. The variance-covariance matrix of residuals is then estimated through a parametric form. Secondly, our approach can be used to detect nonintuitive age dependence in the data, beyond the cohort and the period effects which are implicitly captured by our model. Third, our approach can be extended to model the several populations in long run perspectives, without raising issue in the estimation process. Finally, in an out-of-sample forecasting study for mortality rates, we obtain rather good performances and more relevant forecasts compared to classical mortality models using the French, US and UK data. We also show that our results enlighten the so-called cohort and period effects for these populations.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62M20 Inference from stochastic processes and prediction
91D20 Mathematical geography and demography
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[1] Barrieu, P.; Bensusan, H.; El Karoui, N.; Hillairet, C.; Loisel, S.; Ravanelli, C.; Salhi, Y., Understanding, modelling and managing longevity risk: key issues and main challenges, Scand. Actuar. J., 2012, 3, 203-231, (2012) · Zbl 1277.91073
[2] Basu, S.; Michailidis, G., Regularized estimation in sparse high-dimensional time series models, Ann. Statist., 43, 4, 1535-1567, (2015) · Zbl 1317.62067
[3] Bickel, P. J.; Levina, E., Covariance regularization by thresholding, Ann. Statist., 6, 2577-2604, (2008) · Zbl 1196.62062
[4] Bien, J.; Tibshirani, R. J., Sparse estimation of a covariance matrix, Biometrika, 98, 4, 807-820, (2011) · Zbl 1228.62063
[5] Bohk-Ewald, C.; Rau, R., Probabilistic mortality forecasting with varying age-specific survival improvements, Genus, 73, 1, 1, (2017)
[6] Booth, H.; Maindonald, J.; Smith, L., Applying Lee-Carter under conditions of variable mortality decline, Popul. Stud., 56, 3, 325-336, (2002)
[7] Booth, H.; Tickle, L., Mortality modelling and forecasting: a review of methods, Ann. Actuar. Sci., 3, 1-2, 3-43, (2008)
[8] Börger, M.; Fleischer, D.; Kuksin, N., Modeling the mortality trend under modern solvency regimes, Astin Bull., 44, 01, 1-38, (2014)
[9] Boumezoued, A., 2016. Improving HMD mortality estimates with HFD fertility data, HAL preprint: hal-01270565.
[10] Brouhns, N.; Denuit, M.; Vermunt, J. K., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance Math. Econom., 31, 3, 373-393, (2002) · Zbl 1074.62524
[11] Cairns, A. J.G.; Blake, D.; Dowd, K., A two-factor model for stochastic mortality with parameter uncertainty: Theory and Calibration, J. Risk Insur., 73, 4, 687-718, (2006)
[12] Cairns, A. J.G.; Blake, D.; Dowd, K., Modelling and management of mortality risk: a review, Scand. Actuar. J., 2008, 2-3, 79-113, (2008) · Zbl 1224.91048
[13] Cairns, A. J.; Blake, D.; Dowd, K.; Coughlan, G. D.; Epstein, D.; Ong, A.; Balevich, I., A quantitative comparison of stochastic mortality models using data from England and Wales and the united states, N. Am. Actuar. J., 13, 1, 1-35, (2009)
[14] Cairns, A. J.; Blake, D.; Dowd, K.; Coughlan, G. D.; Khalaf-Allah, M., Bayesian stochastic mortality modelling for two populations, Astin Bull., 41, 01, 29-59, (2011)
[15] Cairns, A. J.G.; Blake, D.; Dowd, K.; Kessler, A. R., Phantoms never die: living with unreliable population data, J. R. Statist. Soc.: Ser. A, 179, 4, 975-1005, (2016)
[16] Cairns, A.J.G., Kallestrup-Lamb, M., Rosenskjold, C.P., Blake, D., Dowd, K., et al., 2016. Modelling Socio-Economic Dierences in the Mortality of Danish Males Using a New Auence Index. Tech. rep. Working paper, Heriot-Watt University. Department of Economics and Business Economics, Aarhus University.
[17] Chai, C. M.H.; Tak Siu, K.; Zhou, X., A double-exponential GARCH model for stochastic mortality, Eur. Actuar. J., 3, 2, 385-406, (2013)
[18] Chatterjee, A.; Lahiri, S. N., Bootstrapping lasso estimators, J. Amer. Statist. Assoc., 106, 494, 608-625, (2011) · Zbl 1232.62088
[19] Chen, H.; MacMinn, R.; Sun, T., Multi-population mortality models: A factor copula approach, (Special Issue: Longevity Nine - the Ninth International Longevity Risk and Capital Markets Solutions Conference, Vol. 63, (2015)), 135-146, Insurance: Mathematics and Economics · Zbl 1348.91131
[20] Christiansen, M. C.; Spodarev, E.; Unseld, V., Differences in European mortality rates: A geometric approach on the age – period plane, Astin Bull., 45, 03, 477-502, (2015) · Zbl 1390.91173
[21] Currie, I. D.; Durban, M.; Eilers, P. H.C., Smoothing and forecasting mortality rates, Statist. Model., 4, 4, 279-298, (2004), (00309) · Zbl 1061.62171
[22] Dokumentov, A., Hyndman, R.J., 2018. smoothAPC: Smoothing of Two-Dimensional Demographic Data, Optionally Taking into Account Period and Cohort Effects. R package version 0.3.
[23] Dokumentov, A.; Hyndman, R. J.; Leonie, T., Bivariate smoothing of mortality surfaces with cohort and period ridges, Stat, 7, 1, Article e199 pp., (2018)
[24] Doukhan, P.; Pommeret, D.; Rynkiewicz, J.; Salhi, Y., A class of random field memory models for mortality forecasting, Insurance Math. Econom., 77, 97-110, (2017) · Zbl 1422.62309
[25] Dowd, K.; Cairns, A. J.G.; Blake, D.; Coughlan, G. D.; Marwa, K.-A., A gravity model of mortality rates for two related populations, N. Am. Actuar. J., 15, 2, 334-356, (2011) · Zbl 1228.91032
[26] Enchev, V.; Kleinow, T.; Cairns, A. J.G., Multi-population mortality models: fitting, forecasting and comparisons, Scand. Actuar. J., 2017, 4, 319-342, (2016) · Zbl 1401.62206
[27] Fan, J.; Lv, J.; Qi, L., Sparse high dimensional models in economics, Annu. Rev. Econ., 3, 291-317, (2011)
[28] Friedman, J.; Hastie, T.; Tibshirani, R., Regularization paths for generalized linear models via coordinate descent, J. Stat. Softw., 33, 1, 1-22, (2010)
[29] Furman, Y., VAR Estimation with the Adaptive Elastic Net. SSRN Scholarly Paper ID 2456510, (2014), Social Science Research Network
[30] Gefang, D., Bayesian doubly adaptive elastic-net Lasso for VAR shrinkage, Int. J. Forecast., 30, 1, 1-11, (2014)
[31] Granger, C. W.J., Investigating Causal relations by econometric models and cross-spectral methods, Econometrica, 37, 3, 424-438, (1969) · Zbl 1366.91115
[32] Haberman, S.; Renshaw, A., Parametric mortality improvement rate modelling and projecting, Insurance Math. Econom., 50, 3, 309-333, (2012) · Zbl 1237.91129
[33] Hahn, L. J., A Bayesian Multi-Population Mortality Projection Model, (2014), Universität Ulm, (Master thesis)
[34] Hoerl, A. E.; Kennard, R. W., Ridge regression: Biased estimation for nonorthogonal problems, Technometrics, 12, 1, 55-67, (1970) · Zbl 0202.17205
[35] Human Mortality Database, 2019. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (data downloaded on 28.01.19).
[36] Hunt, A.; Blake, D., A general procedure for constructing mortality models, N. Am. Actuar. J., 18, 1, 116-138, (2014) · Zbl 1412.91045
[37] Hunt, A.; Villegas, A. M., Robustness and convergence in the Lee-Carter model with cohort effects, Insurance Math. Econom., 64, 186-202, (2015) · Zbl 1348.62241
[38] Huynen, M. M.; Martens, P.; Schram, D.; Weijenberg, M. P.; Kunst, A. E., The impact of heat waves and cold spells on mortality rates in the dutch population, Environ. Health Perspect., 109, 5, 463-470, (2001)
[39] Hyndman, R.J., 2019. demography: Forecasting Mortality, Migration, Fertility and Population Data. (R package version 1.21).
[40] Hyndman, R. J.; Ullah, S., Robust forecasting of mortality and fertility rates: A functional data approach, Comput. Statist. Data Anal., 51, 10, 4942-4956, (2007) · Zbl 1162.62434
[41] Izraelewicz, E., L’effet moisson - l’impact des catastrophes vie sur la mortalité à long terme - Exemple de la canicule de l’été 2003, Bull. Français d’Actuar., 12, 24, 113-159, (2012)
[42] Jarner, S. F.; Kryger, E. M., Modelling adult mortality in small populations: The saint model, Astin Bull., 41, 02, 377-418, (2011) · Zbl 1239.91128
[43] Lee, R. D.; Carter, L. R., Modeling and forecasting U.S. mortality, J. Amer. Statist. Assoc., 87, 419, 659-671, (1992) · Zbl 1351.62186
[44] Li, N.; Lee, R., Coherent mortality forecasts for a group of populations: An extension of the lee-Carter method, Demography, 42, 3, 575-594, (2005)
[45] Li, N.; Lee, R.; Gerland, P., Extending the lee-Carter method to model the rotation of age patterns of mortality-decline for long-term projection, Demography, 50, 6, 2037-2051, (2013)
[46] Li, H.; Lu, Y., Coherent forecasting of mortality rates: A sparse vector-autoregression approach, ASTIN Bull.: J. IAA, 47, 2, 563-600, (2017) · Zbl 1390.62215
[47] Li, H.; O’hare, C.; Vahid, F., Two-dimensional kernel smoothing of mortality surface: An evaluation of cohort strength, J. Forecast., 35, 6, 553-563, (2016) · Zbl 1376.62065
[48] Opgen-Rhein, R.; Strimmer, K., From correlation to causation networks: a simple approximate learning algorithm and its application to high-dimensional plant gene expression data, BMC Syst. Biol., 1, 1, 37, (2007)
[49] Perron, P., Trends and random walks in macroeconomic time series: Further evidence from a new approach, J. Econ. Dyn. Control, 12, 2-3, 297-332, (1988) · Zbl 0659.62128
[50] Plat, R., On stochastic mortality modeling, Insurance Math. Econom., 45, 3, 393-404, (2009) · Zbl 1231.91227
[51] R Core Team, R., R: A Language and Environment for Statistical Computing, (2019), R Foundation for Statistical Computing
[52] Renshaw, A. E.; Haberman, S., A cohort-based extension to the lee-Carter model for mortality reduction factors, Insurance Math. Econom., 38, 3, 556-570, (2006) · Zbl 1168.91418
[53] Renshaw, A. E.; Haberman, S., On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling, Insurance Math. Econom., 42, 2, 797-816, (2008) · Zbl 1152.91598
[54] Said, S. E.; Dickey, D. A., Testing for unit roots in autoregressive-moving average models of unknown order, Biometrika, 71, 3, 599-607, (1984) · Zbl 0564.62075
[55] Salhi, Y.; Loisel, S., Basis risk modelling: a cointegration-based approach, Statistics, 51, 1, 205-221, (2017) · Zbl 1369.62285
[56] Schäfer, J.; Strimmer, K., A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics, Statist. Appl. Genet. Mol. Biol., 4, 1, (2005)
[57] Song, S., Bickel, P.J., 2011. Large vector auto regressions. arXiv preprint arXiv:1106.3915.
[58] Spodarev, E., Shmileva, E., Roth, S., 2013. Extrapolation of stationary random fields, arXiv preprint arXiv:1306.6205. · Zbl 1327.62483
[59] Tibshirani, R., Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B Stat. Methodol., 58, 1, 267-288, (1996) · Zbl 0850.62538
[60] Toulemon, L.; Barbieri, M., The mortality impact of the august 2003 heat wave in France: Investigating the ‘harvesting’ effect and other long-term consequences, Popul. Stud., 62, 1, 39-53, (2008)
[61] Vazzoler, S., Frattarolo, L., Billio, M., 2016. sparsevar: A Package for Sparse VAR/VECM Estimation. Tech. rep. R package version 0.0.10.
[62] Villegas, A.M., Kaishev, V., Millossovich, P., 2017. StMoMo: An R Package for Stochastic Mortality Modelling. R package version 0.4.1.
[63] Willets, R., The cohort effect: insights and explanations, Br. Actuar. J., 10, 4, 833-877, (2004)
[64] Wilms, I.; Croux, C., Forecasting using sparse cointegration, Int. J. Forecast., 32, 4, 1256-1267, (2016)
[65] Zou, H.; Hastie, T., Regularization and variable selection via the elastic net, J. R. Stat. Soc. Ser. B Stat. Methodol., 67, 2, 301-320, (2005) · Zbl 1069.62054
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