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Abstract
The threshold autoregressive (TAR) model is a class of nonlinear time series models that have been widely used in many areas. Due to its nonlinear nature, one major difficulty in fitting a TAR model is the estimation of the thresholds. As a first contribution, this article develops an automatic procedure to estimate the number and values of the thresholds, as well as the corresponding AR order and parameter values in each regime. These parameter estimates are defined as the minimizers of an objective function derived from the minimum description length (MDL) principle. A genetic algorithm (GA) is constructed to efficiently solve the associated minimization problem. The second contribution of this article is the extension of this framework to piecewise TAR modeling; that is, the time series is partitioned into different segments for which each segment can be adequately modeled by a TAR model, while models from adjacent segments are different. For such piecewise TAR modeling, a procedure is developed to estimate the number and locations of the breakpoints, together with all other parameters in each segment. Desirable theoretical results are derived to lend support to the proposed methodology. Simulation experiments and an application to an U.S. GNP data are used to illustrate the empirical performances of the methodology. Supplementary materials for this article are available online.
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Notes on contributors
Chun Yip Yau
Chun Yip Yau, Department of Statistics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong (E-mail: [email protected]). Supported in part by HKSAR-RGC Grants CUHK405012, 405113, and Direct Grant CUHK2060445. Chong Man Tang, Department of Statistics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong (E-mail: [email protected]). Thomas C. M. Lee, Department of Statistics, University of California at Davis, Davis, CA 95616, USA (E-mail: [email protected]). Supported in part by the National Science Foundation under Grants 1007520, 1209226 and 1209232. The authors are most grateful to the reviewers, the associate editor and the editor, Professor Xuming He, for their most constructive and helpful comments, which led to a much improved version of the article.
Chong Man Tang
Chun Yip Yau, Department of Statistics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong (E-mail: [email protected]). Supported in part by HKSAR-RGC Grants CUHK405012, 405113, and Direct Grant CUHK2060445. Chong Man Tang, Department of Statistics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong (E-mail: [email protected]). Thomas C. M. Lee, Department of Statistics, University of California at Davis, Davis, CA 95616, USA (E-mail: [email protected]). Supported in part by the National Science Foundation under Grants 1007520, 1209226 and 1209232. The authors are most grateful to the reviewers, the associate editor and the editor, Professor Xuming He, for their most constructive and helpful comments, which led to a much improved version of the article.
Thomas C. M. Lee
Chun Yip Yau, Department of Statistics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong (E-mail: [email protected]). Supported in part by HKSAR-RGC Grants CUHK405012, 405113, and Direct Grant CUHK2060445. Chong Man Tang, Department of Statistics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong (E-mail: [email protected]). Thomas C. M. Lee, Department of Statistics, University of California at Davis, Davis, CA 95616, USA (E-mail: [email protected]). Supported in part by the National Science Foundation under Grants 1007520, 1209226 and 1209232. The authors are most grateful to the reviewers, the associate editor and the editor, Professor Xuming He, for their most constructive and helpful comments, which led to a much improved version of the article.