Annals of Statistics

Quarticity and other functionals of volatility: Efficient estimation

Jean Jacod and Mathieu Rosenbaum

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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.

Article information

Ann. Statist., Volume 41, Number 3 (2013), 1462-1484.

First available in Project Euclid: 1 August 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G44: Martingales with continuous parameter 62F12: Asymptotic properties of estimators

Semimartingale high frequency data volatility estimation central limit theorem efficient estimation


Jacod, Jean; Rosenbaum, Mathieu. Quarticity and other functionals of volatility: Efficient estimation. Ann. Statist. 41 (2013), no. 3, 1462--1484. doi:10.1214/13-AOS1115.

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