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On the jump activity index for semimartingales

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  • Jing, Bing-Yi
  • Kong, Xin-Bing
  • Liu, Zhi
  • Mykland, Per

Abstract

Empirical evidence of asset price discontinuities or “jumps” in financial markets has been well documented in the literature. Recently, Aït-Sahalia and Jacod (2009b) defined a general “jump activity index” to describe the degree of jump activities for asset price semimartingales, and provided a consistent estimator when the underlying process contains both a continuous and a jump component. However, only large increments were used in their estimator so that the effective sample size is very small even for large sample sizes. In this paper, we explore ways to improve the Aït-Sahalia and Jacod estimator by making use of all increments, large and small. The improvement is verified through simulations. A real example is also given.

Suggested Citation

  • Jing, Bing-Yi & Kong, Xin-Bing & Liu, Zhi & Mykland, Per, 2012. "On the jump activity index for semimartingales," Journal of Econometrics, Elsevier, vol. 166(2), pages 213-223.
    • Handle: RePEc:eee:econom:v:166:y:2012:i:2:p:213-223
    DOI: 10.1016/j.jeconom.2011.09.036
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    References listed on IDEAS

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    1. Ole E. Barndorff-Nielsen & Neil Shephard, 2006. "Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation," Journal of Financial Econometrics, Oxford University Press, vol. 4(1), pages 1-30.
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    6. Sun, Yucheng & Xu, Wen & Zhang, Chuanhai, 2023. "Identifying latent factors based on high-frequency data," Journal of Econometrics, Elsevier, vol. 233(1), pages 251-270.
    7. Leong, Minhao & Kwok, Simon, 2023. "The pricing of jump and diffusive risks in the cross-section of cryptocurrency returns," Journal of Empirical Finance, Elsevier, vol. 74(C).
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    11. Qiang Liu & Zhi Liu & Chuanhai Zhang, 2020. "Heteroscedasticity test of high-frequency data with jumps and microstructure noise," Papers 2010.07659, arXiv.org.
    12. Todorov, Viktor, 2019. "Nonparametric inference for the spectral measure of a bivariate pure-jump semimartingale," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 419-451.
    13. Hounyo, Ulrich & Varneskov, Rasmus T., 2017. "A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation," Journal of Econometrics, Elsevier, vol. 198(1), pages 10-28.
    14. Jos'e E. Figueroa-L'opez & Cheng Li & Jeffrey Nisen, 2018. "Optimal Iterative Threshold-Kernel Estimation of Jump Diffusion Processes," Papers 1811.07499, arXiv.org, revised Apr 2020.
    15. Deniz Erdemlioglu & Christopher J. Neely & Xiye Yang, 2023. "Systemic Tail Risk: High-Frequency Measurement, Evidence and Implications," Working Papers 2023-016, Federal Reserve Bank of St. Louis.
    16. Boswijk, H. Peter & Laeven, Roger J.A. & Yang, Xiye, 2018. "Testing for self-excitation in jumps," Journal of Econometrics, Elsevier, vol. 203(2), pages 256-266.
    17. Figueroa-López, José E. & Nisen, Jeffrey, 2013. "Optimally thresholded realized power variations for Lévy jump diffusion models," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2648-2677.
    18. Kwok, Simon, 2020. "Nonparametric Inference of Jump Autocorrelation," Working Papers 2020-09, University of Sydney, School of Economics, revised Jan 2021.
    19. Adam D. Bull, 2014. "Near-optimal estimation of jump activity in semimartingales," Papers 1409.8150, arXiv.org, revised Jan 2016.
    20. Fabian Mies & Ansgar Steland, 2019. "Nonparametric Gaussian inference for stable processes," Statistical Inference for Stochastic Processes, Springer, vol. 22(3), pages 525-555, October.
    21. Zhi Liu, 2017. "Jump-robust estimation of volatility with simultaneous presence of microstructure noise and multiple observations," Finance and Stochastics, Springer, vol. 21(2), pages 427-469, April.
    22. Ulrich Hounyo & Rasmus T. Varneskov, 2015. "A Local Stable Bootstrap for Power Variations of Pure-Jump Semimartingales and Activity Index Estimation," CREATES Research Papers 2015-26, Department of Economics and Business Economics, Aarhus University.
    23. Xin-Bing Kong, 2017. "On the number of common factors with high-frequency data," Biometrika, Biometrika Trust, vol. 104(2), pages 397-410.
    24. Torben G. Andersen & Nicola Fusari & Viktor Todorov & Rasmus T. Varneskov, 2018. "Option Panels in Pure-Jump Settings," CREATES Research Papers 2018-04, Department of Economics and Business Economics, Aarhus University.
    25. Mykland, Per A. & Zhang, Lan, 2016. "Between data cleaning and inference: Pre-averaging and robust estimators of the efficient price," Journal of Econometrics, Elsevier, vol. 194(2), pages 242-262.

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