The Estimation of Leverage effect with High Frequency Data

The leverage effect has become an extensively studied phenomenon which describes the (usually) negative relation between stock returns and their volatility. Although this characteristic of stock returns is well acknowledged, most studies of the phenomenon are based on cross-sectional calibration with parametric models. On the statistical side, most previous works are conducted over daily or longer return horizons, and few of them have carefully studied its estimation, especially with high frequency data. However, estimation of the leverage effect is important because sensible inference is possible only when the leverage effect is estimated reliably. In this paper, we provide nonparametric estimation for a class of stochastic measures of leverage effect. In order to construct estimators with good statistical properties, we introduce a new stochastic leverage effect parameter. The estimators and their statistical properties are provided in cases both with and without microstructure noise, under the stochastic volatility model. In asymptotics, the consistency and limiting distribution of the estimators are derived and corroborated by simulation results. For consistency, a previously unknown bias correction factor is added to the estimators. Applications of the estimators are also explored. This estimator provides the opportunity to study high frequency regression, which leads to the prediction of volatility using not only previous volatility but also the leverage effect. The estimator also reveals a theoretical connection between skewness and the leverage effect, which further leads to the prediction of skewness. Furthermore, adopting the ideas similar to these, it is easy to extend these methods to other important aspects of stock returns, such as volatility of volatility.