Asymmetry and Leverage in Conditional Volatility Models

Michael Mcaleer

doi:10.3390/econometrics2030145

Abstract

The three most popular univariate conditional volatility models are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) and Bollerslev (1986), the GJR (or threshold GARCH) model of Glosten, Jagannathan and Runkle (1992), and the exponential GARCH (or EGARCH) model of Nelson (1990, 1991). The underlying stochastic specification to obtain GARCH was demonstrated by Tsay (1987), and that of EGARCH was shown recently in McAleer and Hafner (2014). These models are important in estimating and forecasting volatility, as well as in capturing asymmetry, which is the different effects on conditional volatility of positive and negative effects of equal magnitude, and purportedly in capturing leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility. As there seems to be some confusion in the literature between asymmetry and leverage, as well as which asymmetric models are purported to be able to capture leverage, the purpose of the paper is three-fold, namely, (1) to derive the GJR model from a random coefficient autoregressive process, with appropriate regularity conditions; (2) to show that leverage is not possible in the GJR and EGARCH models; and (3) to present the interpretation of the parameters of the three popular univariate conditional volatility models in a unified manner.

Highlights

IntroductionThe three most popular univariate conditional volatility models are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) [1] and Bollerslev (1986) [2], the GJR (or threshold GARCH) model of Glosten, Jagannathan and Runkle (1992) [3], and the exponential
The three most popular univariate conditional volatility models are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) [1] and Bollerslev (1986) [2], the GJR model of Glosten, Jagannathan and Runkle (1992) [3], and the exponentialGARCH model of Nelson (1990, 1991) [4,5]
These models are important in estimating and forecasting volatility and in capturing asymmetry, which is the different effects on conditional volatility of positive and negative effects of equal magnitude; they are purportedly important in capturing leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility

Summary

Introduction

The three most popular univariate conditional volatility models are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) [1] and Bollerslev (1986) [2], the GJR (or threshold GARCH) model of Glosten, Jagannathan and Runkle (1992) [3], and the exponential. The underlying stochastic specification to obtain GARCH was demonstrated by Tsay (1987) [6], and that of EGARCH was shown recently in McAleer and Hafner (2014) [7] These models are important in estimating and forecasting volatility and in capturing asymmetry, which is the different effects on conditional volatility of positive and negative effects of equal magnitude; they are purportedly important in capturing leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility. The purpose of the paper is three-fold, namely, (1) to derive the GJR model from a random coefficient autoregressive process, with appropriate regularity conditions; (2) to show that leverage is not possible in the GJR and EGARCH models; and (3) to present the interpretation of the parameters of the three popular univariate conditional volatility models in a unified manner.

Random Coefficient Autoregressive Process and GARCH

Random Coefficient Autoregressive Process and GJR

Random Coefficient Complex Nonlinear Moving Average Process and EGARCH

Conclusions