Quarticity and other functionals of volatility: Efficient estimation

TL;DR

This paper introduces an efficient estimator for integrated functionals of volatility matrices in high-frequency multidimensional stochastic processes, achieving parametric convergence rates and asymptotic efficiency after bias correction.

Contribution

It proposes a simple Riemann sum-based estimator for volatility functionals that attains the parametric rate and is asymptotically efficient after bias correction.

Findings

Estimator reaches the parametric rate of convergence.

Bias correction yields an unbiased central limit theorem.

Estimator is asymptotically efficient within certain models.

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.