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Threshold Autoregressive Nearest-Neighbour Models for Claims Reserving

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  • Siu, Tak Kuen

Abstract

Motivated by claims reserving in run-off triangles, a class of threshold autoregressive nearest-neighbour (TAR-NN) models extending a major class of parametric nonlinear time series models, namely threshold autoregressive (TAR) models, is introduced. The proposed class of models also introduces a flexible regime-switching mechanism to nearest-neighbour models. Attention is given to a sub-class of TAR-NN models, namely self-exciting threshold autoregressive nearest-neighbour models (SETAR-NN), for uses in claims reserving. The (strict) stationarity and geometric ergodicity of the SETAR-NN model, and more generally, a two-dimensional nonlinear autoregressive random field, are discussed. The conditional least-square (CLS) method is used to estimate the SETAR-NN model and some of its nested models. Simulation studies on the parameter estimates from the CLS method are conducted. Using real insurance claims data and stochastic simulations, the applications of the SETAR-NN model and the nested models for projecting future claims liabilities are discussed. Comparisons of those models with the Bootstrap-Chain-Ladder (BCL) model for claims reserving are provided.

Suggested Citation

  • Siu, Tak Kuen, 2025. "Threshold Autoregressive Nearest-Neighbour Models for Claims Reserving," Econometrics and Statistics, Elsevier, vol. 33(C), pages 180-208.
    • Handle: RePEc:eee:ecosta:v:33:y:2025:i:c:p:180-208
    DOI: 10.1016/j.ecosta.2022.03.006
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    as
    1. C. S. Wong & W. K. Li, 2000. "On a mixture autoregressive model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 95-115.
    2. Piet de Jong, 2006. "Forecasting Runoff Triangles," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 28-38.
    3. Tata Subba Rao & Granville Tunnicliffe Wilson & Shiu Fung Wong & Howell Tong & Tak Kuen Siu & Zudi Lu, 2017. "A New Multivariate Nonlinear Time Series Model for Portfolio Risk Measurement: The Threshold Copula-Based TAR Approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(2), pages 243-265, March.
    4. Mack, Thomas, 1993. "Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates," ASTIN Bulletin, Cambridge University Press, vol. 23(2), pages 213-225, November.
    5. Wai-Sum Chan & Albert Wong & Howell Tong, 2004. "Some Nonlinear Threshold Autoregressive Time Series Models for Actuarial Use," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(4), pages 37-61.
    6. Peters, Gareth W. & Wüthrich, Mario V. & Shevchenko, Pavel V., 2010. "Chain ladder method: Bayesian bootstrap versus classical bootstrap," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 36-51, August.
    7. Hansen Bruce E., 1997. "Inference in TAR Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 2(1), pages 1-16, April.
    8. Pešta, Michal & Okhrin, Ostap, 2014. "Conditional least squares and copulae in claims reserving for a single line of business," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 28-37.
    9. S.T. Boris Choy & Jennifer S.K. Chan & Udi E. Makov, 2016. "Robust Bayesian analysis of loss reserving data using scale mixtures distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(3), pages 396-411, March.
    10. Gisler, Alois & Wüthrich, Mario V., 2008. "Credibility for the Chain Ladder Reserving Method," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 565-600, November.
    11. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, March.
    12. Taylor, G. C. & Ashe, F. R., 1983. "Second moments of estimates of outstanding claims," Journal of Econometrics, Elsevier, vol. 23(1), pages 37-61, September.
    13. Gareth W. Peters & Mario V. Wuthrich & Pavel V. Shevchenko, 2010. "Chain ladder method: Bayesian bootstrap versus classical bootstrap," Papers 1004.2548, arXiv.org.
    14. England, P.D. & Verrall, R.J., 2002. "Stochastic Claims Reserving in General Insurance," British Actuarial Journal, Cambridge University Press, vol. 8(3), pages 443-518, August.
    15. Tweedie, Richard L., 1975. "Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space," Stochastic Processes and their Applications, Elsevier, vol. 3(4), pages 385-403, October.
    16. Verrall, R.J., 1990. "Bayes and Empirical Bayes Estimation for the Chain Ladder Model," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 217-243, November.
    17. Haastrup, Svend & Arjas, Elja, 1996. "Claims Reserving in Continuous Time; A Nonparametric Bayesian Approach," ASTIN Bulletin, Cambridge University Press, vol. 26(2), pages 139-164, November.
    18. England, Peter & Verrall, Richard, 1999. "Analytic and bootstrap estimates of prediction errors in claims reserving," Insurance: Mathematics and Economics, Elsevier, vol. 25(3), pages 281-293, December.
    19. Mack, Thomas, 1991. "A Simple Parametric Model for Rating Automobile Insurance or Estimating IBNR Claims Reserves," ASTIN Bulletin, Cambridge University Press, vol. 21(1), pages 93-109, April.
    20. Mack, Thomas, 1994. "Which stochastic model is underlying the chain ladder method?," Insurance: Mathematics and Economics, Elsevier, vol. 15(2-3), pages 133-138, December.
    21. Ma, Tao & Zhou, Zhou & Abdulhai, Baher, 2015. "Nonlinear multivariate time–space threshold vector error correction model for short term traffic state prediction," Transportation Research Part B: Methodological, Elsevier, vol. 76(C), pages 27-47.
    22. Verrall, Richard, 1996. "Claims reserving and generalised additive models," Insurance: Mathematics and Economics, Elsevier, vol. 19(1), pages 31-43, December.
    23. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    24. Stock, James H, 1987. "Asymptotic Properties of Least Squares Estimators of Cointegrating Vectors," Econometrica, Econometric Society, vol. 55(5), pages 1035-1056, September.
    25. R. J. Verrall, 2004. "A Bayesian Generalized Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(3), pages 67-89.
    26. Verrall, R. J., 2000. "An investigation into stochastic claims reserving models and the chain-ladder technique," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 91-99, February.
    27. Wilkie, A.D., 1995. "More on a Stochastic Asset Model for Actuarial Use," British Actuarial Journal, Cambridge University Press, vol. 1(5), pages 777-964, December.
    28. Verrall, R.J. & England, P.D., 2005. "Incorporating expert opinion into a stochastic model for the chain-ladder technique," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 355-370, October.
    29. Miranda, María Dolores Martínez & Nielsen, Jens Perch & Verrall, Richard, 2012. "Double Chain Ladder," ASTIN Bulletin, Cambridge University Press, vol. 42(1), pages 59-76, May.
    30. Whitten, S.P. & Thomas, R.G., 1999. "A Non-Linear Stochastic Asset Model for Actuarial Use," British Actuarial Journal, Cambridge University Press, vol. 5(5), pages 919-953, December.
    31. Huang, Jinlong & Qiu, Chunjuan & Wu, Xianyi & Zhou, Xian, 2015. "An individual loss reserving model with independent reporting and settlement," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 232-245.
    32. Dong, A.X.D. & Chan, J.S.K., 2013. "Bayesian analysis of loss reserving using dynamic models with generalized beta distribution," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 355-365.
    33. Tong, Howell, 2015. "Threshold models in time series analysis—Some reflections," Journal of Econometrics, Elsevier, vol. 189(2), pages 485-491.
    34. Bühlmann, Hans & De Felice, Massimo & Gisler, Alois & Moriconi, Franco & Wüthrich, Mario V., 2009. "Recursive Credibility Formula for Chain Ladder Factors and the Claims Development Result," ASTIN Bulletin, Cambridge University Press, vol. 39(1), pages 275-306, May.
    35. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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