Rate-Efficient Asymptotic Normality for the Fourier Estimator of the Leverage Process

We prove a Central Limit Theorem for two estimators of the leverage process based on the Fourier method of [Malliavin and Mancino, 2009], showing that they reach the optimal rate 1/4 and a smaller variance with respect to different estimators based on a pre-estimation of the instantaneous volatility. The obtained limiting distributions of the estimators are confirmed by simulation results. Further, we exploit the availability of efficient leverage estimates to show, using S&P500 prices, that adding an extra term which accounts for the leverage effect to the Heterogeneous Auto-Regressive volatility model by [Corsi, 2009], increases the explanatory power of the latter.