Volatility of Volatility : Estimation and Tests Based on Noisy High Frequency Data with Jumps

We establish a feasible central limit theorem with convergence rate $n^{1/8}$ for the estimation of the {integrated volatility of volatility} (VoV) based on noisy high-frequency data with jumps. This is the first inference theory ever built for VoV estimation under such a general setup. The central limit theorem is applied to provide interval estimates of the VoV and conduct hypothesis tests. Furthermore, when one is interested in the null hypothesis that the VoV is zero, we show that a more powerful test can be established based on a VoV estimator with a convergence rate $n^{1/5}$ under the null. Empirical results on the S\&P 500 and individual stocks show strong evidence of non-zero VoV