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Asymptotic Inference in Multiple-Threshold Nonlinear Time Series Models

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Author Info

  • Dong Li

    ()
    (Tsinghua University)

  • Shiqing Ling

    (Hong kong University of Science and Technology)

  • Jean-Michel Zakoian

    ()
    (CREST)

Abstract

This paper investigates a class of multiple-threshold models, called Multiple Threshold Double AR (MTDAR) models. A sufficient condition is obtained for the existence and uniqueness of a strictly stationary and ergodic solution to the first-order MTDAR model. We study the Quasi-Maximum Likelihood Estimator (QMLE) of the MTDAR model. The estimated thresholds are shown to be n-consistent, asymptotically independent, and to converge weakly to the smallest minimizer of a two-sided compound Poisson process. The remaining parameters are ?n-consistent and asymptotically multivariate normal. In particular, these results apply to the multiple threshold ARCH model, with or without AR part, and to the multiple threshold AR models with ARCH errors. A score-based test is also presented to determine the number of thresholds in MTDAR models. The limiting distribution is shown to be distribution-free and is easy to implement in practice. Simulation studies are conducted to assess the performance of the QMLE and our score-based test in finite samples. The results are illustrated with an application to the quarterly U.S. real GNP data over the period 1947–2013

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Bibliographic Info

Paper provided by Centre de Recherche en Economie et Statistique in its series Working Papers with number 2013-51.

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Length: 27
Date of creation: Dec 2013
Date of revision:
Handle: RePEc:crs:wpaper:2013-51

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Related research

Keywords: Compound Poisson process; Ergodicity; Quasi-maximum likelihood estimation; Strict stationarity; MTDAR model; Score test;

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References

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  1. Simon M. Potter, 1993. "A Nonlinear Approach to U.S. GNP," UCLA Economics Working Papers 693, UCLA Department of Economics.
  2. Gary Koop & Simon M. Potter, 2004. "Dynamic asymmetries in US unemployment," ESE Discussion Papers 15, Edinburgh School of Economics, University of Edinburgh.
  3. Bruce E. Hansen, 1996. "Sample Splitting and Threshold Estimation," Boston College Working Papers in Economics 319., Boston College Department of Economics, revised 12 May 1998.
  4. Li, Dong & Ling, Shiqing, 2012. "On the least squares estimation of multiple-regime threshold autoregressive models," Journal of Econometrics, Elsevier, vol. 167(1), pages 240-253.
  5. Jesús Gonzalo & Michael Wolf, 2001. "Subsampling inference in threshold autoregressive models," Economics Working Papers 573, Department of Economics and Business, Universitat Pompeu Fabra.
  6. Gourieroux, Christian & Monfort, Alain, 1992. "Qualitative threshold ARCH models," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 159-199.
  7. Mika Meitz & Pentti Saikkonen, 2010. "Parameter estimation in nonlinear AR–GARCH models," Koç University-TUSIAD Economic Research Forum Working Papers 1002, Koc University-TUSIAD Economic Research Forum.
  8. Shiqing Ling, 2004. "Estimation and testing stationarity for double-autoregressive models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 63-78.
  9. Francq, Christian & Roy, Roch & Zakoian, Jean-Michel, 2005. "Diagnostic Checking in ARMA Models With Uncorrelated Errors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 532-544, June.
  10. Li, Dong & Ling, Shiqing & Li, Wai Keung, 2013. "Asymptotic Theory On The Least Squares Estimation Of Threshold Moving-Average Models," Econometric Theory, Cambridge University Press, vol. 29(03), pages 482-516, June.
  11. Li, W K & Ling, Shiqing & McAleer, Michael, 2002. " Recent Theoretical Results for Time Series Models with GARCH Errors," Journal of Economic Surveys, Wiley Blackwell, vol. 16(3), pages 245-69, July.
  12. Rabemananjara, R & Zakoian, J M, 1993. "Threshold Arch Models and Asymmetries in Volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(1), pages 31-49, Jan.-Marc.
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