Asymmetry and leverage in GARCH models: a News Impact Curve perspective

ABSTRACT

Models for conditional heteroskedasticity belonging to the GARCH class are now common tools in many economics and finance applications. Among the many possible competing univariate GARCH models, one of the most interesting groups allows for the presence of the so-called asymmetry or leverage effect. In our view, asymmetry and leverage are two distinct phenomena, both inspired by the seminal work of Black in 1976. We propose definitions of leverage and asymmetry that build on the News Impact Curve, allowing to easily and coherently verify if they are both present. We show that several GARCH models are asymmetric but none is allowing for a proper leverage effect. Finally, we extend the leverage definition to a local leverage effect and show that the AGARCH model is coherent with the presence of local leverage. An empirical analysis completes the paper.

KEYWORDS:

• Conditional volatility models
• GARCH models
• asymmetry
• leverage

JEL CLASSIFICATION:

• C12
• C13
• C22
• C52
• C58

Acknowledgments

Michele Costola acknowledges financial support from the Marie Skłodowska-Curie Actions, European Union, Seventh Framework Program HORIZON 2020 under REA grant agreement n.707070, the research support from the Research Center SAFE, funded by the State of Hessen initiative for research LOEWE.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Engle and Ng (1993) do not explicitly specify the meaning of the letter V in the acronym of the VGARCH model.

² Note of the authors. The citation “Glosten, Jaganathan and Runkle (1989)” refers to the manuscript version of Glosten, Jaganathan and Runkle (1993).

³ Note of the authors. The citation “Engle and Ng (1992)” refers to the manuscript version of Engle and Ng (1993).

⁴ The use of leverage and asymmetry as synonyms is also common in statistical and econometric software. See, among others, Matlab and Eviews.

⁵ Chang and McAleer (2017) associate symmetry to equality of the first order derivative of the conditional variance equation with respect to the shock, evaluated for positive and negative shocks. Therefore, for the EGARCH model, the derivative equals $\frac{\mathrm{\partial }log{h}_t}{\mathrm{\partial }{z}_{t-1}}=\alpha -sign\left(z_{t-1}\right)\kappa$. Consequently, according to Chang and McAleer (2017), symmetry requires $\kappa =0$.

⁶ One can verify that by solving the equality imposing that parameter $\alpha$ and $\gamma$ satisfy the conditions for positivity of conditional volatilities, but $\alpha <0$. For instance by setting $\alpha =-\frac{1}{2\gamma }$.

⁷ As our purpose is purely illustrative on the presence of asymmetry and/or leverage, we do not make model comparisons in terms of forecasts but only with respect to the in-sample fit.

⁸ We do not include the VGARCH model here given its similarity with the AGARCH model in preliminary estimates.:

⁹ The full collection of assets includes $426$ equities. However, we excluded from the analysis of the assets ($31$) with problems of convergence in at least one of the estimated models. Hence, the final number of assets is equal to $395$.

¹⁰ For sake of completeness, we include in Appendix A the percentages of the best overall performing models for all the considered periods. A related study is that of Bampinas, Ladopoulos, and Panagiotidis (2018) which provides descriptive statistics on the conditional variances under alternative distributions estimated using the GARCH, GJR-GARCH and EGARCH models for the constituents of the S&P Composite 1500.

¹¹ The estimates of the parameters are very similar to those obtained by other distributions and are available upon request to the authors.

¹² We standardized the NICs for comparison purposes and hence the NICs of the AGARCH and NGARCH models are equivalents.

Additional information

Funding

This work was supported by the H2020 Marie Skłodowska-Curie Actions [707070].