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June 2013 Quarticity and other functionals of volatility: Efficient estimation
Jean Jacod, Mathieu Rosenbaum
Ann. Statist. 41(3): 1462-1484 (June 2013). DOI: 10.1214/13-AOS1115

Abstract

We consider a multidimensional Itô semimartingale regularly sampled on $[0,t]$ at high frequency $1/\Delta_{n}$, with $\Delta_{n}$ going to zero. The goal of this paper is to provide an estimator for the integral over $[0,t]$ of a given function of the volatility matrix. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most $\Delta_{n}^{1/4}$, this procedure reaches the parametric rate $\Delta_{n}^{1/2}$, as it is usually the case in integrated functionals estimation. After a suitable bias correction, we obtain an unbiased central limit theorem for our estimator and show that it is asymptotically efficient within some classes of sub models.

Citation

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Jean Jacod. Mathieu Rosenbaum. "Quarticity and other functionals of volatility: Efficient estimation." Ann. Statist. 41 (3) 1462 - 1484, June 2013. https://doi.org/10.1214/13-AOS1115

Information

Published: June 2013
First available in Project Euclid: 1 August 2013

zbMATH: 1292.60033
MathSciNet: MR3113818
Digital Object Identifier: 10.1214/13-AOS1115

Subjects:
Primary: 60F05 , 60G44 , 62F12

Keywords: central limit theorem , efficient estimation , High frequency data , Semimartingale , Volatility estimation

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3 • June 2013
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