We study the asymptotic normality of two feasible estimators of the integrated volatility of volatility based on the Fourier methodology, which does not require the pre-estimation of the spot volatility. We show that the bias-corrected estimator reaches the optimal rate $n^{1/4}$, while the estimator without bias-correction has a slower convergence rate and a smaller asymptotic variance. Additionally, we provide simulation results that support the theoretical asymptotic distribution of the rate-efficient estimator and show the accuracy of the latter in comparison with a rate-optimal estimator based on the pre-estimation of the spot volatility. Finally, using the rate-optimal Fourier estimator, we reconstruct the time series of the daily volatility of volatility of the S\&P500 and EUROSTOXX50 indices over long samples and provide novel insight into the existence of stylized facts about the volatility of volatility dynamics.

翻译：我们根据Fourier方法研究两个可行的综合波动性估计者是否正常,这不需要预先估计点波动性。我们显示,偏差修正估计者达到最佳汇率$n ⁇ 1/4美元,而没有偏差修正的估算者则比较慢的趋同率和较小的零差。此外,我们提供模拟结果,支持节率估测者的理论零差分布,并显示后者与基于预估点波动性的速率-最佳估测者相比的准确性。最后,我们利用最高汇率-最佳四倍估计者,对S ⁇ P500和EUROSTOXX50指数的日波动性进行长期抽样重建时间序列,并对波动性动态的波动性现成事实提供新的洞察力。