The Estimation of Leverage Effect With High-Frequency Data

Abstract

The leverage effect has become an extensively studied phenomenon that 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 article, we provide nonparametric estimation for a class of stochastic measures of leverage effect. 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 the estimation of the leverage effect, it is easy to extend the methods to study other important aspects of stock returns, such as volatility of volatility.

KEY WORDS:

• Consistency
• Discrete observation
• Efficiency
• Itô process
• Microstructure noise
• Realized volatility
• Skewness
• Stable convergence

Notes

The relative change between time lag and lagged leverage effect should be maintained, even if the consistency may be an concern, since the consistency can be achieved by a bias correction multiplier from our study shown later.

To get from local boundedness to results that cover the whole time interval, use arguments as in chap. 2.4.5 (p. 160-161) of Mykland and Zhang (2012). |σ_t| is locally bounded from above by continuity. The assumptions guarantee that the equivalent martingale measure for X exists locally. This is used in the proofs; see the beginning of Section A.1.

One can also consider a kernel estimator of the spot volatility in (5), by applying the methods in Kristensen (2010), with some adaptation. A detailed study is beyond the scope of this article.

Suppose that all relevant processes (X_t, σ_t, etc.) are adapted to the filtration . Let Z_n be a sequence of -measurable random variables. We say that Z_n converges stably in law to Z as n → ∞ if Z is measurable with respect to an extension of so that for all and for all bounded continuous g, EI_Ag(Z_n) → EI_Ag(Z) as n → ∞. The same definition applies to triangular arrays.

See Footnote 4.

Here and in the continuation of Remark 1, we assume that the denominator in (9(9) ) is nonzero.

See Footnote 4.

Such as Zhang, Mykland, and Aït-Sahalia (2005), Zhang (2006), Barndorff-Nielsen et al. (2008a), Reiss (2010), and Xiu (2010), as well as the preaveraging papers cited in the text.

See Footnote 4.

See Footnote 4.

The total numbers of days removed are 21 for 2008, 27 for 2009, 60 for 2010, and 43 for 2011.

The main motivation for us to include “leverage effect scaled returns” is the earlier empirical findings on asymmetric impact of positive or negative returns on the volatility process. To capture this asymmetric impact in the prediction model, we include the extra term. We realize that Jacod (1994, 1996), Barndorff-Nielsen, and Shephard (2005), and others, showed that the correlation (leverage effect) has no impact on the asymptotic distribution. These observations seem to suggest that asymmetries do not matter for forecasting, but that is not so. The concept of the news impact curve (Engle and Ng 1993) was originally formulated within the context of daily ARCH-type models. In the models of Engle and Ng (1993), the returns are included in the volatility prediction models by differently scaling the positive or negative returns. Here we applied the intra-day data over small time interval to estimate integrated volatility, but predict the next volatility over a much longer time interval (two-day ahead). Therefore, the prediction falls into a comparative low frequency setting. The leverage effect turns out to have a significant impact on the volatility prediction which is supported by our empirical finding. This inclusion of leverage effect scaled returns in the volatility prediction model is further supported by the findings in Chen and Ghysels (2011), where they are dealing with two different frequencies and reaching a similar conclusion.