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Double Asymptotics for Explosive Continuous Time Models

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  • Xiaohu Wang

    () (School of Economics and Sim Kee Boon Institute for Financial Economics, Singapore Management University)

  • Jun Yu

    () (Sim Kee Boon Institute for Financial Economics, School of Economics and Lee Kong Chian School of Business)

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Abstract

This paper develops a double asymptotic limit theory for the persistent parameter (k) in explosive continuous time models driven by Lévy processes with a large number of time span (N) and a small number of sampling interval (h). The simultaneous double asymptotic theory is derived using a technique in the same spirit as in Phillips and Magdalinos (2007) for the mildly explosive discrete time model. Both the intercept term and the initial condition appear in the limiting distribution. In the special case of explosive continuous time models driven by the Brownian motion, we develop the limit theory that allows for the joint limits where N ! 1 and h ! 0 simultaneously, the sequential limits where N ! 1 is followed by h ! 0, and the sequential limits where h ! 0 is followed by N ! 1. All three asymptotic distributions are the same.

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Bibliographic Info

Paper provided by Singapore Management University, School of Economics in its series Working Papers with number 16-2012.

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Length: 27 pages
Date of creation: Jan 2012
Date of revision:
Publication status: Published in SMU Economics and Statistics Working Paper Series
Handle: RePEc:siu:wpaper:16-2012

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Related research

Keywords: Explosive; Continuous Time; Lévy Process; Invariance Principle; Double Asymptotics;

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  1. Tang, Cheng Yong & Chen, Song Xi, 2009. "Parameter estimation and bias correction for diffusion processes," Journal of Econometrics, Elsevier, vol. 149(1), pages 65-81, April.
  2. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-778, 04.
  3. Aue, Alexander & Horv th, Lajos, 2007. "A Limit Theorem For Mildly Explosive Autoregression With Stable Errors," Econometric Theory, Cambridge University Press, vol. 23(02), pages 201-220, April.
  4. Peter C. B. Phillips & Jun Yu, 2009. "Dating the Timeline of Financial Bubbles During the Subprime Crisis," Working Papers 18-2009, Singapore Management University, School of Economics.
  5. Peter C.B. PHILIPS & Yangru WU & Jun YU, 2009. "Explosive Behavior in the 1990s Nasdaq: When Did Exuberance Escalate Asset Values?," Working Papers 19-2009, Singapore Management University, School of Economics.
  6. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
  7. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
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