Quarticity and other functionals of volatility: Efficient estimation

Abstract

We consider a multidimensional Ito 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.

Published in at http://dx.doi.org/10.1214/13-AOS1115 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• A remark on the rates of convergence for integrated volatility estimation in the presence of jumps

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