Dynamic Models for Volatility and Heavy Tails

Abstract

The volatility of financial returns changes over time and, for the last thirty years, Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models have provided the principal means of analyzing, modeling and monitoring such changes. Taking into account that financial returns typically exhibit heavy tails - that is, extreme values can occur from time to time - Andrew Harvey's new book shows how a small but radical change in the way GARCH models are formulated leads to a resolution of many of the theoretical problems inherent in the statistical theory. The approach can also be applied to other aspects of volatility. The more general class of Dynamic Conditional Score models extends to robust modeling of outliers in the levels of time series and to the treatment of time-varying relationships. The statistical theory draws on basic principles of maximum likelihood estimation and, by doing so, leads to an elegant and unified treatment of nonlinear time-series modeling.

Score Driven Exponentially Weighted Moving Average and Value-at-Risk Forecasting

“…As discussed in Creal et al [2013] and Harvey [2013], the weight factor in front of y 2 t in equation (6) has a robustifying effect on the volatility dynamics. If y t lies in the tails of the conditional distribution at time t, the volatility is increased, but not by the full y 2 t .…”

“…To account for the shape of the conditional forecasting distribution in constructing an EWMA scheme, we use the generalized autoregressive score (GAS) framework of Creal et al [2011of Creal et al [ , 2013; see also Harvey [2013]. show that updating the time-varying parameters by the score of the forecasting distribution always locally improves the Kullback-Leibler divergence between the model and the true, unknown data generating process.…”

“…As 6 it is well established that financial returns are typically fat-tailed, it makes much more sense to use an SD-EWMA scheme based upon a fat-tailed distribution. In this paper we follow Creal et al [2011Creal et al [ , 2013 and Harvey [2013] and use the Student's t (and later also the skewed Student's t) distribution with ν degrees of freedom,…”

“…One way forward is to use the skewed Student's t distribution with associated score and information matrix expressions as derived in for example Gómez et al [2007] and discussed in the score-driven setting by Harvey [2013]. The density of the skewed Student's t distribution is given by…”

“…The score-driven EWMA (SD-EWMA) model we propose builds on a new observation driven methodology, namely the generalized autoregressive score (GAS) dynamics; see Creal et al [2011Creal et al [ , 2013 and Harvey [2013]. In particular, we consider an integrated version of the score-driven dynamics.…”