The Annals of Statistics

Quarticity and other functionals of volatility: Efficient estimation

Jean Jacod and Mathieu Rosenbaum

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 1 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1375362556

Digital Object Identifier
doi:10.1214/13-AOS1115

Mathematical Reviews number (MathSciNet)
MR3113818

Zentralblatt MATH identifier
1292.60033

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

Keywords
Semimartingale high frequency data volatility estimation central limit theorem efficient estimation

Citation

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. https://projecteuclid.org/euclid.aos/1375362556


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