Modeling the Interactions between Volatility and Returns using EGARCH‐M

Abstract

An EGARCH-M model, in which the logarithm of scale is driven by the score of the conditional distribution, is shown to be theoretically tractable as well as practically useful. A two-component extension makes it possible to distinguish between the short-and long-run effects of returns on volatility, and the resulting short-and long-run volatility components are then allowed to have different effects on returns, with the long-run component yielding the equity risk premium. The EGARCH formulation allows for more flexibility in the asymmetry of the volatility response (leverage) than standard GARCH models and suggests that, for weekly observations on two major stock market indices, the short-term response is close to being anti-symmetric. component, as found by Engle and Lee (1999) and others. Indeed, short-run volatility may even decrease after a good day, because it calms the market. Standard GARCH models are unable to identify this effect.Adrian and Rosenberg (2008) proposed a two-component exponential GARCH-M (EGARCH-M) model. They showed that it captures the asymmetric response of short-and long-term volatility to returns, and that the long-run component is positively correlated with the equity risk premium. Here we demonstrate that letting the dynamics be driven by the score of the conditional distribution yields a model that is theoretically tractable as well as practically useful. Models constructed using the conditional score were introduced into the literature by Creal et al. (2011Creal et al. ( , 2013, where they are called Generalized Autoregressive Score (GAS) models, and Harvey (2013), where they are called Dynamic Conditional Score (DCS) models. The classic EGARCH specification in Nelson (1991), which is the one used by Adrian and Rosenberg, is sensitive to outliers and it has the unfortunate theoretical property that unconditional moments do not exist when the conditional distribution is Student's t. The DCS model resolves these problems and, in doing so, yields a specification which is open to the development of a full asymptotic theory for the distribution of the maximum likelihood estimator. This contrasts with standard formulations of ARCH-M models, where the asymptotic theory appears to be intractable.Section 2 sets out the DCS formulation of the basic EGARCH-M model with a conditional Student's t-distribution and outlines the associated statistical theory, including an easily implemented condition for invertibility. This is followed by subsections on the asymmetric response of volatility to returns (leverage), two components, skewness and splines. In Section 3 various models are fitted to weekly data on NASDAQ and NIKKEI returns. The risk-free rate of return is then introduced as an explanatory variable, as in Scruggs (1998), and the model is fitted to weekly excess returns on S&P500 over a 60-year period. Section 4 reports the results of a small forecasting study to provide reassurance on the predictive performance of DCS models. Section 5 concludes. Appendices A to C are provided online as Ap...

Score-driven models for realized volatility

“…Taking logarithms and …tting a corresponding EGB2 gives the same estimates and score residuals. The seasonal pattern is similar to that re- vealed by the preliminary modelling and the leverage e¤ect is stronger in the short-term component, a …nding that is not unusual for returns or range; see Harvey and Lange (2018) and Harvey (2013, pp 176-81).…”

A coupled component DCS-EGARCH model for intraday and overnight volatility

“…Applying Theorem 3.1 in Bougerol (1993), if for some integer p 1, E log(sup Harvey and Lange (2018) for the univariate model. Otherwise, our condition is in general more restrictive than theirs.…”

Score-Driven Time Series Models

“…The equity risk premium is then captured by the long-run component, with an equilibrium level of + 1 exp ! : Harvey and Lange (2018) demonstrate that a two-component score-driven model with symmetric long-run volatility, that is 1 = 0; coupled with antisymmetric short-run volatility, that is 2 = 0; provides a good …t and yields a plausible interpretation of market behaviour. This accords with the conclusion of Adrian and Rosenberg (2008, p 3015), in that the short-run component appears to capture shocks to market skewness, whereas the long-run component is related to business cycle risk.…”