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Optimal iterative threshold-kernel estimation of jump diffusion processes

Author

Listed:
  • José E. Figueroa-López

    (Washington University in St. Louis)

  • Cheng Li

    (Citadel Securities)

  • Jeffrey Nisen

    (Barclays)

Abstract

In this paper, we propose a new threshold-kernel jump-detection method for jump-diffusion processes, which iteratively applies thresholding and kernel methods in an approximately optimal way to achieve improved finite-sample performance. As in Figueroa-López and Nisen (Stoch Process Appl 123(7):2648–2677, 2013), we use the expected number of jump misclassifications as the objective function to optimally select the threshold parameter of the jump detection scheme. We prove that the objective function is quasi-convex and obtain a new second-order infill approximation of the optimal threshold in closed form. The approximate optimal threshold depends not only on the spot volatility $$\sigma _t$$ σ t , but also the jump intensity and the value of the jump density at the origin. Estimation methods for these quantities are then developed, where the spot volatility is estimated by a kernel estimator with thresholding and the value of the jump density at the origin is estimated by a density kernel estimator applied to those increments deemed to contain jumps by the chosen thresholding criterion. Due to the interdependency between the model parameters and the approximate optimal estimators built to estimate them, a type of iterative fixed-point algorithm is developed to implement them. Simulation studies for a prototypical stochastic volatility model show that it is not only feasible to implement the higher-order local optimal threshold scheme but also that this is superior to those based only on the first order approximation and/or on average values of the parameters over the estimation time period.

Suggested Citation

  • José E. Figueroa-López & Cheng Li & Jeffrey Nisen, 2020. "Optimal iterative threshold-kernel estimation of jump diffusion processes," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 517-552, October.
    • Handle: RePEc:spr:sistpr:v:23:y:2020:i:3:d:10.1007_s11203-020-09211-7
    DOI: 10.1007/s11203-020-09211-7
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    References listed on IDEAS

    as
    1. Corsi, Fulvio & Pirino, Davide & Renò, Roberto, 2010. "Threshold bipower variation and the impact of jumps on volatility forecasting," Journal of Econometrics, Elsevier, vol. 159(2), pages 276-288, December.
    2. Yacine Aït-Sahalia & Jean Jacod, 2014. "High-Frequency Financial Econometrics," Economics Books, Princeton University Press, edition 1, number 10261.
    3. 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.
    4. Jacod, Jean, 2008. "Asymptotic properties of realized power variations and related functionals of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 517-559, April.
    5. Cecilia Mancini, 2009. "Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 270-296, June.
    6. Foster, Dean P & Nelson, Daniel B, 1996. "Continuous Record Asymptotics for Rolling Sample Variance Estimators," Econometrica, Econometric Society, vol. 64(1), pages 139-174, January.
    7. Figueroa-López, José E. & Mancini, Cecilia, 2019. "Optimum thresholding using mean and conditional mean squared error," Journal of Econometrics, Elsevier, vol. 208(1), pages 179-210.
    8. 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.
    9. José E. Figueroa-López & Jeffrey Nisen, 2019. "Second-order properties of thresholded realized power variations of FJA additive processes," Statistical Inference for Stochastic Processes, Springer, vol. 22(3), pages 431-474, October.
    10. repec:hal:journl:peer-00741630 is not listed on IDEAS
    11. Cecilia Mancini & Vanessa Mattiussi & Roberto Renò, 2015. "Spot volatility estimation using delta sequences," Finance and Stochastics, Springer, vol. 19(2), pages 261-293, April.
    12. Figueroa-López, José E. & Li, Cheng, 2020. "Optimal kernel estimation of spot volatility of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4693-4720.
    13. Kristensen, Dennis, 2010. "Nonparametric Filtering Of The Realized Spot Volatility: A Kernel-Based Approach," Econometric Theory, Cambridge University Press, vol. 26(1), pages 60-93, February.
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    Cited by:

    1. Milan Kumar Das & Anindya Goswami & Sharan Rajani, 2023. "Inference of Binary Regime Models with Jump Discontinuities," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 49-86, May.

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