Stock index realized volatility forecasting in the presence of heterogeneous leverage effects and long range dependence in the volatility of realized volatility

Abstract

In this article, we account for the presence of heterogeneous leverage effects and the persistence in the volatility of stock index realized volatility. The Heterogeneous Autoregressive (HAR) Realized Volatility (RV) model is extended in order to account for asymmetric responses to negative and positive shocks occurring at distinct frequencies, as well as, for the long range dependence in the heteroscedastic variance of the residuals. Compared with established HAR and Autoregressive Fractionally Integrated Moving Average (ARFIMA) realized volatility models, the proposed model exhibits superior in sample fitting, as well as, out of sample volatility forecasting performance. The latter is further improved when the Realized Power Variation (RPV) is used as a regressor, while we show that our analysis is also robust against microstructure noise.

Keywords:

• volatility forecasting
• leverage effects
• long memory
• high frequency data

JEL Classification::

• C13
• C22
• C51
• C53

Acknowledgements

The authors thank the two anonymous referees for their helpful comments. The views expressed in this article do not necessarily represent the views of the Bank of Greece.

Notes

¹The sum of squared intraday returns is actually the realized variance. Realized volatility is defined as the square root of realized variance. However, many authors use the term realized volatility interchangeably with the term realized variance.

² ‘Bad news’ in a stock market (i.e. negative returns) tend to increase future volatility more than ‘good news’ (i.e. positive returns). This asymmetry between negative and positive returns is referred to as asymmetric or leverage effect. In theory, the leverage of the company increases as its stock price goes down, i.e. the company uses more debt than owned capital to finance its business activities. This increases the risk of investing in this stock which in turn increases its volatility.

³ The asymmetric behaviour between returns and volatility is well documented in the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) literature. For the use of asymmetric GARCH models in forecasting volatility see Kisinbay (2008).

⁴The Heterogeneous Market Hypothesis (Muller et al., 1993) states that market agents differ with respect to their investment horizon, risk aversion, degree of available information, institutional constraints, transaction costs, etc. This diversity is identified as the root cause of asset volatility, as market agents aim to settle at different asset valuations, according to their individual market view, preferences and expectations.

⁵For practical applications of the HARCH model see McMillan and Speight (2006a, b).

⁶ For the use of Autoregressive (AR) models in realized volatility forecasting see Hooper et al. (2009).

⁷ Note that when , the Realized Power Variation (RPV) is by definition equal to the realized volatility (i.e. ). In this case, the RPV is not robust to jumps and converges to the integrated volatility plus the jump component.

⁸ The daily logarithmic returns are calculated as where is the closing price of day t, (t − 1).

⁹ All estimates were deduced by numerical optimization of the log likelihood function (Maximum Likelihood Estimation, MLE) and they were conducted with the Ox Metrics G@RCH 4.2 package developed by Laurent and Peters (2002).

¹⁰ The transformation in Equation 15 is derived from the realized variance lognormality assumption: A random variable is lognormally distributed if is normally distributed. Then, the expectation of is , with and denoting the mean and the variance of respectively, e.g. see Beltratti and Morana (2005) and Giot and Laurent (2004).