Estimation of Time Varying Skewness and Kurtosis with an Application to Value at Risk

Abstract

This paper generalizes the Hyperbolic Asymmetric Power ARCH (HYAPARCH) model to allow for time varying skewness and kurtosis in the conditional distribution. This is done by modeling the conditional skewness and degrees of freedom of the skewed Student's t distribution of Lambert and Laurent (2001) as a function of the conditioning information. The proposed specification nests a large number of models in the literature and represents the first attempt to jointly model long memory in volatility and time variation in the third and fourth moments. The finite sample properties of MLE for this class of model are examined. The results indicate that the ARCH class of processes with time varying skewness can be reliably estimated with realistic sample sizes. Simulations and empirical evidence are unable to replicate the findings of Harvey and Siddique (1999), that accounting for time varying skewness reduces the persistence and asymmetry properties of the conditional variance. Simulations also suggest that time varying kurtosis estimation must be viewed with caution, because it can be difficult to identify in the presence of ARCH effects. Application of the HYAPARCH model with time varying skewness and degrees of freedom illustrates the usefulness of the proposed approach. Out of sample forecasts of the value at risk (VaR) however, generally support parsimonious models that assume conditional normality. When forecasting VaR, skewness and leptokurtosis in the unconditional return distribution is generally better captured via an asymmetric conditional variance model with Gaussian innovations.