Asymptotic results for the Fourier estimator of the integrated quarticity

In this paper we prove a central limit theorem for the Fourier quarticity estimator. We obtain a new consistency result and we show that the estimator reaches the parametric rate 1/2. The optimal variance is obtained, with a suitable choice of the number of frequencies employed to compute the Fourier coefficients of the volatility, while the limiting distribution has a bias. As a by-product, thanks to the Fourier methodology, we obtain consistent estimators of any even power of the volatility function and an estimator of the spot quarticity. We assess the finite sample performance of the Fourier quarticity estimator in a numerically exercise with different market micro-structure frictions.