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ESTIMATION OF INTEGRATED VOLATILITY FUNCTIONALS WITH KERNEL SPOT VOLATILITY ESTIMATORS

Published online by Cambridge University Press:  16 October 2025

José E. Figueroa-López ,
Jincheng Pang  and
Bei Wu
José E. Figueroa-López*
Affiliation:
Washington University in St. Louis
Jincheng Pang
Affiliation:
Washington University in St. Louis
Bei Wu
Affiliation:
Upstart
*
Address correspondence to José E. Figueroa-López, Department of Statistics and Data Science, Washington University in St. Louis, St. Louis, MO, USA, e-mail: figueroa-lopez@wustl.edu.
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Abstract

For a multidimensional Itô semimartingale, we consider the problem of estimating integrated volatility functionals. Jacod and Rosenbaum (2013, The Annals of Statistics 41(3), 1462–1484) studied a plug-in type of estimator based on a Riemann sum approximation of the integrated functional and a spot volatility estimator with a forward uniform kernel. Motivated by recent results that show that spot volatility estimators with general two-sided kernels of unbounded support are more accurate, in this article, an estimator using a general kernel spot volatility estimator as the plug-in is considered. A biased central limit theorem for estimating the integrated functional is established with an optimal convergence rate. Central limit theorems for properly de-biased estimators are also obtained both at the optimal convergence regime for the bandwidth and when applying undersmoothing. Our results show that one can significantly reduce the estimator’s bias by adopting a general kernel instead of the standard uniform kernel. Our proposed bias-corrected estimators are found to maintain remarkable robustness against bandwidth selection in a variety of sampling frequencies and functions.

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ARTICLES
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Econometric Theory , First View , pp. 1 - 31
Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

The authors are truly grateful to both Co-Editors (Dr. Dennis Kristensen and Dr. Peter C. B. Phillips), and two anonymous referees for the numerous suggestions that helped to significantly improve the original manuscript. Research supported in part by the NSF grant: DMS-2413557.

References

REFERENCES

Aït-Sahalia, Y., & Jacod, J. (2014). High-frequency financial econometrics . Princeton University Press.Google Scholar
Aït-Sahalia, Y., & Xiu, D. (2019). Principal component analysis of high-frequency data. Journal of the American Statistical Association , 114(525), 287–303.10.1080/01621459.2017.1401542CrossRefGoogle Scholar
Benvenuti, F. (2021). Essays on high-frequency and financial data analysis [Ph.D. thesis]. Aarhus University.Google Scholar
Chen, R. Y. (2019). Inference for volatility functionals of multivariate itô semimartingales observed with jump and noise. Tech. rep., Working Paper. arXiv:1810.04725v2.Google Scholar
Clément, E., Delattre, S., & Gloter, A. (2013). An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility. Stochastic Processes and their Applications , 123(7), 2500–2521.CrossRefGoogle Scholar
Fan, J., & Wang, Y. (2008). Spot volatility estimation for high-frequency data. Statistics and its Interface , 1(2), 279–288.CrossRefGoogle Scholar
Figueroa-López, J. E., & Li, C. (2020). Optimal kernel estimation of spot volatility of stochastic differential equations. Stochastic Processes and their Applications , 130(8), 4693–4720.CrossRefGoogle Scholar
Figueroa-López, J. E., & Wu, B. (2024). Kernel estimation of spot volatility with microstructure noise using pre-averaging. Econometric Theory , 40(3), 558–607.10.1017/S0266466622000470CrossRefGoogle Scholar
Jacod, J., & Protter, P. (2011). Discretization of processes . Springer Science & Business Media.Google Scholar
Jacod, J., & Rosenbaum, M. (2013). Quarticity and other functionals of volatility: Efficient estimation. The Annals of Statistics , 41(3), 1462–1484.CrossRefGoogle Scholar
Jacod, J., & Rosenbaum, M. (2015). Estimation of volatility functionals: The case of a $\sqrt{n}$ window. In Friz, P. K., Gatheral, J., Gulisashvili, A., Jacquier, A., & Teichmann, J. (Eds.), Large deviations and asymptotic methods in finance . Springer International Publishing.Google Scholar
Kristensen, D. (2010). Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory , 26(1), 60–93.10.1017/S0266466609090616CrossRefGoogle Scholar
Li, J., & Liu, Y. (2021). Efficient estimation of integrated volatility functionals under general volatility dynamics. Econometric Theory , 37(4), 664–707.10.1017/S0266466620000274CrossRefGoogle Scholar
Li, J., Liu, Y., & Xiu, D. (2019). Efficient estimation of integrated volatility functionals via multiscale jackknife. The Annals of Statistics , 47(1), 156–176.10.1214/18-AOS1684CrossRefGoogle Scholar
Li, J., & Xiu, D. (2016). Generalized method of integrated moments for high-frequency data. Econometrica , 84(4), 1613–1633.10.3982/ECTA12306CrossRefGoogle Scholar
Mancini, C., Mattiussi, V., & Renò, R. (2015). Spot volatility estimation using delta sequences. Finance & Stochastics , 19(2), 261–293.10.1007/s00780-015-0255-1CrossRefGoogle Scholar
Mykland, P. A., & Zhang, L. (2009). Inference for continuous semimartingales observed at high frequency. Econometrica , 77(5), 1403–1445.Google Scholar
Renault, E., Sarisoy, C., & Werker, B. (2017). Efficient estimation of integrated volatility and related processes. Econometric Theory , 33, 439–478.10.1017/S0266466616000013CrossRefGoogle Scholar
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