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Double Asymptotics for Explosive Continuous Time Models

Author

Listed:
  • 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)

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.

Suggested Citation

  • Xiaohu Wang & Jun Yu, 2012. "Double Asymptotics for Explosive Continuous Time Models," Working Papers 16-2012, Singapore Management University, School of Economics.
    • Handle: RePEc:siu:wpaper:16-2012
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Park, Joon, 2003. "Weak Unit Roots," Working Papers 2003-17, Rice University, Department of Economics.
    4. 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.
    5. Zhou, Qiankun & Yu, Jun, 2015. "Asymptotic theory for linear diffusions under alternative sampling schemes," Economics Letters, Elsevier, vol. 128(C), pages 1-5.
    6. Yu, Jun, 2014. "Econometric Analysis Of Continuous Time Models: A Survey Of Peter Phillips’S Work And Some New Results," Econometric Theory, Cambridge University Press, vol. 30(4), pages 737-774, August.
    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.
    8. Aue, Alexander & Horváth, Lajos, 2007. "A Limit Theorem For Mildly Explosive Autoregression With Stable Errors," Econometric Theory, Cambridge University Press, vol. 23(2), pages 201-220, April.
    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," The Review of Financial Studies, Society for Financial Studies, vol. 22(9), pages 3669-3705, September.
    11. Peter C. B. Phillips & Jun Yu, 2011. "Dating the timeline of financial bubbles during the subprime crisis," Quantitative Economics, Econometric Society, vol. 2(3), pages 455-491, November.
    12. Peter C. B. Phillips & Yangru Wu & Jun Yu, 2011. "EXPLOSIVE BEHAVIOR IN THE 1990s NASDAQ: WHEN DID EXUBERANCE ESCALATE ASSET VALUES?," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 52(1), pages 201-226, February.
    13. Qiankun Zhou & Jun Yu, 2010. "Asymptotic Distributions of the Least Squares Estimator for Diffusion Processes," Working Papers 20-2010, Singapore Management University, School of Economics.
    14. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    15. 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.
    16. Peter C. B. Phillips, 2005. "Jackknifing Bond Option Prices," The Review of Financial Studies, Society for Financial Studies, vol. 18(2), pages 707-742.
    17. Wang, Xiaohu & Yu, Jun, 2015. "Limit theory for an explosive autoregressive process," Economics Letters, Elsevier, vol. 126(C), pages 176-180.
    18. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    19. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    20. Hu, Yaozhong & Long, Hongwei, 2009. "Least squares estimator for Ornstein-Uhlenbeck processes driven by [alpha]-stable motions," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2465-2480, August.
    21. Perron, Pierre, 1991. "A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case without an Intercept," Econometrica, Econometric Society, vol. 59(1), pages 211-236, January.
    22. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Limit Theory Of Real‐Time Detectors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56, pages 1079-1134, November.
    23. 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.
    24. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    25. Phillips, Peter C.B. & Magdalinos, Tassos, 2009. "Unit Root And Cointegrating Limit Theory When Initialization Is In The Infinite Past," Econometric Theory, Cambridge University Press, vol. 25(6), pages 1682-1715, December.
    26. Peter C. B. Phillips & Shu-Ping Shi & Jun Yu, 2011. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behavior," Working Papers 15-2011, Singapore Management University, School of Economics.
    27. Shimizu, Yasutaka, 2009. "Notes on drift estimation for certain non-recurrent diffusion processes from sampled data," Statistics & Probability Letters, Elsevier, vol. 79(20), pages 2200-2207, October.
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    Citations

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    Cited by:

    1. Chen, Ye & Phillips, Peter C.B. & Yu, Jun, 2017. "Inference in continuous systems with mildly explosive regressors," Journal of Econometrics, Elsevier, vol. 201(2), pages 400-416.
    2. Laurent, Sébastien & Shi, Shuping, 2020. "Volatility estimation and jump detection for drift–diffusion processes," Journal of Econometrics, Elsevier, vol. 217(2), pages 259-290.
    3. Katsuto Tanaka & Weilin Xiao & Jun Yu, 2020. "Maximum Likelihood Estimation for the Fractional Vasicek Model," Econometrics, MDPI, vol. 8(3), pages 1-28, August.
    4. Cagli, Efe Caglar, 2019. "Explosive behavior in the prices of Bitcoin and altcoins," Finance Research Letters, Elsevier, vol. 29(C), pages 398-403.
    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. Yiu Lim Lui & Weilin Xiao & Jun Yu, 2022. "The Grid Bootstrap for Continuous Time Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(3), pages 1390-1402, June.
    7. Stelios Arvanitis & Tassos Magdalinos, 2018. "Mildly Explosive Autoregression Under Stationary Conditional Heteroskedasticity," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 892-908, November.
    8. Wang, Xiaohu & Xiao, Weilin & Yu, Jun, 2023. "Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process," Journal of Econometrics, Elsevier, vol. 232(2), pages 389-415.
    9. Xiao, Weilin & Yu, Jun, 2019. "Asymptotic theory for rough fractional Vasicek models," Economics Letters, Elsevier, vol. 177(C), pages 26-29.
    10. Junichi Hirukawa & Sangyeol Lee, 2021. "Asymptotic properties of mildly explosive processes with locally stationary disturbance," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(4), pages 511-534, May.
    11. Tao, Yubo & Phillips, Peter C.B. & Yu, Jun, 2017. "Random Coefficient Continuous Systems: Testing for Extreme Sample Path Behaviour," Economics and Statistics Working Papers 18-2017, Singapore Management University, School of Economics.
    12. Chambers, MJ, 2016. "The Effects of Sampling Frequency on Detrending Methods for Unit Root Tests," Economics Discussion Papers 16062, University of Essex, Department of Economics.

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    More about this item

    Keywords

    Explosive; Continuous Time; Lévy Process; Invariance Principle; Double Asymptotics;
    All these keywords.

    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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