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Double asymptotics for explosive continuous time models

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  • Wang, Xiaohu
  • Yu, Jun

Abstract

This paper establishes a double asymptotic theory for explosive continuous time Lévy-driven processes and the corresponding exact discrete time models. The double asymptotic theory assumes the sample size diverges because the sampling interval (h) shrinks to zero and the time span (N) diverges. Both the simultaneous and sequential double asymptotic distributions are derived. In contrast to the long-time-span asymptotics (N→∞ with fixed h) where no invariance principle applies, the double asymptotic distribution is derived without assuming Gaussian errors, so an invariance principle applies, as the asymptotic theory for the mildly explosive process developed by Phillips and Magdalinos (2007). Like the in-fill asymptotics (h→0 with fixed N) of Perron (1991), the double asymptotic distribution explicitly depends on the initial condition. The convergence rate of the double asymptotics partially bridges that of the long-time-span asymptotics and that of the in-fill asymptotics. Monte Carlo evidence shows that the double asymptotic distribution works well in practically realistic situations and better approximates the finite sample distribution than the asymptotic distribution that is independent of the initial condition. Empirical applications to real Nasdaq prices highlight the difference between the new theory and the theory without taking the initial condition into account.

Suggested Citation

  • Wang, Xiaohu & Yu, Jun, 2016. "Double asymptotics for explosive continuous time models," Journal of Econometrics, Elsevier, vol. 193(1), pages 35-53.
  • Handle: RePEc:eee:econom:v:193:y:2016:i:1:p:35-53
    DOI: 10.1016/j.jeconom.2016.02.014
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    References listed on IDEAS

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    1. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Historical Episodes Of Exuberance And Collapse In The S&P 500," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56, pages 1043-1078, November.
    2. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
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    4. Park, Joon, 2003. "Weak Unit Roots," Working Papers 2003-17, Rice University, Department of Economics.
    5. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2014. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behaviour," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 76(3), pages 315-333, June.
    6. Zhou, Qiankun & Yu, Jun, 2015. "Asymptotic theory for linear diffusions under alternative sampling schemes," Economics Letters, Elsevier, vol. 128(C), pages 1-5.
    7. 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, April.
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    9. Magdalinos, Tassos, 2012. "Mildly explosive autoregression under weak and strong dependence," Journal of Econometrics, Elsevier, vol. 169(2), pages 179-187.
    10. Peter C. B. Phillips & Jun Yu, 2009. "Simulation-Based Estimation of Contingent-Claims Prices," Review of Financial Studies, Society for Financial Studies, vol. 22(9), pages 3669-3705, September.
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    Citations

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    Cited by:

    1. repec:eee:finlet:v:29:y:2019:i:c:p:398-403 is not listed on IDEAS
    2. Sébastien Laurent & Shuping Shi, 2018. "Volatility Estimation and Jump Detection for drift-diffusion Processes," AMSE Working Papers 1843, Aix-Marseille School of Economics, France.
    3. repec:eee:econom:v:209:y:2019:i:2:p:208-237 is not listed on IDEAS
    4. Lui, Yiu Lim & Xiao, Weilin & Yu, Jun, 2018. "The Grid Bootstrap for Continuous Time Models," Economics and Statistics Working Papers 20-2018, Singapore Management University, School of Economics.
    5. Tao, Yubo & Phillips, Peter C.B. & Yu, Jun, 2019. "Random coefficient continuous systems: Testing for extreme sample path behavior," Journal of Econometrics, Elsevier, vol. 209(2), pages 208-237.
    6. repec:eee:ecolet:v:177:y:2019:i:c:p:26-29 is not listed on IDEAS
    7. repec:eee:econom:v:201:y:2017:i:2:p:400-416 is not listed on IDEAS
    8. repec:bla:jtsera:v:39:y:2018:i:6:p:892-908 is not listed on IDEAS
    9. Xiao, Weilin & Yu, Jun, 2019. "Asymptotic theory for rough fractional Vasicek models," Economics Letters, Elsevier, vol. 177(C), pages 26-29.

    More about this item

    Keywords

    Explosive continuous time models; Lévy process; Moderate deviations from unity; Double asymptotics; Invariance principle; Initial condition;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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