The time variation in second and higher-order moments is an important phenomenon for assessing (tail) risk, constructing hedge strategies, and pricing assets. Exponentially weighted moving average (EWMA) methods have proved to be useful tools for capturing such time variation in a parsimonious and effective way. Here, we develop a new empirical methodology that extends and improves upon the standard EWMA approach. Our framework uses the higher-moment properties of the forecasting distribution to drive the dynamics of volatilities and other timevarying parameters. This ensures that the new method is robust to outliers if a non-normal forecasting distribution is used, as is typically the case when forecasting financialasset returns. The new method is easy to implement and remains similar in spirit to the highly familiar EWMA approach of RiskMetricsTM.

The score-driven EWMA (SD-EWMA) model that we propose builds on a new observation-driven methodology, namely the generalized autoregressive score (GAS) dynamics; see Creal, Koopman, and Lucas (2011, 2013) and Harvey (2013). In particular, we consider an integrated version of the score-driven dynamics. The analogy is simple: just as the standard EWMA approach is a special case of the IGARCH(1,1) model of Bollerslev (1986) and Engle (1982), the proposed SD-EWMA approach is a special case of the IGAS(1,1) model of Creal et al. (2013). Its key feature is the fact that the time-varying parameter dynamics are driven by the score of the forecasting distribution. Empirical evidence of the usefulness of score-driven dynamics is provided by Creal, Schwaab, Koopman, and Lucas (2014), Harvey and Luati (2014), and Lucas, Schwaab, and Zhang (2014), for example, while Blasques, Koopman, and Lucas (2015) demonstrate the information-theoretic optimality properties of score-driven updates.

The intuition for using the score is straightforward. As an example, consider forecasting a time-varying variance of a fat-tailed distribution. If one uses the standard EWMA approach, a large absolute return has a major impact on the next period’s estimated variance, due to the use of squared returns in the variance updating equation. Given the integrated nature of the EWMA dynamics, this impact affects a large number of the subsequent volatility estimates. If one accounts for the fat-tailedness of the return distribution by using a score-driven propagation mechanism for the variances, the impact of incidental tail observations is reduced substantially. This mitigation or robustifying mechanism is particularly important in our current context with integrated (infinite memory) dynamics.