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Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales

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  • Diop, Assane
  • Jacod, Jean
  • Todorov, Viktor

Abstract

We derive Central Limit Theorems for the convergence of approximate quadratic variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case of an Itô semimartingale having a non-vanishing continuous martingale part. Here we focus on the case where the continuous martingale part vanishes and find faster rates of convergence, as well as very different limiting processes.

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  • Diop, Assane & Jacod, Jean & Todorov, Viktor, 2013. "Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 839-886.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:3:p:839-886
    DOI: 10.1016/j.spa.2012.11.003
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    References listed on IDEAS

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    1. Woerner Jeannette H. C., 2003. "Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models," Statistics & Risk Modeling, De Gruyter, vol. 21(1/2003), pages 47-68, January.
    2. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
    3. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2003. "Modeling and Forecasting Realized Volatility," Econometrica, Econometric Society, vol. 71(2), pages 579-625, March.
    4. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
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    Cited by:

    1. Segal, Gill & Shaliastovich, Ivan & Yaron, Amir, 2015. "Good and bad uncertainty: Macroeconomic and financial market implications," Journal of Financial Economics, Elsevier, vol. 117(2), pages 369-397.
    2. Bibinger, Markus & Winkelmann, Lars, 2015. "Econometrics of co-jumps in high-frequency data with noise," Journal of Econometrics, Elsevier, vol. 184(2), pages 361-378.
    3. Todorov, Viktor, 2013. "Power variation from second order differences for pure jump semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2829-2850.
    4. Kim Christensen & Mikkel Slot Nielsen & Mark Podolskij, 2021. "High-dimensional estimation of quadratic variation based on penalized realized variance," Papers 2103.03237, arXiv.org.
    5. Heiny, Johannes & Podolskij, Mark, 2021. "On estimation of quadratic variation for multivariate pure jump semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 234-254.
    6. Kim Christensen & Mikkel Slot Nielsen & Mark Podolskij, 2023. "High-dimensional estimation of quadratic variation based on penalized realized variance," Statistical Inference for Stochastic Processes, Springer, vol. 26(2), pages 331-359, July.

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