The approximation bias of Gram-Charlier Expansion in dynamic higher moments modelling

The Gram-Charlier Expansion (GCE) of the Gaussian density under GARCH framework has been widely used to model the conditional dynamic higher moments. Compared with other generalized distributions, GARCH-GCE models describe the dynamic equations of conditional skewness and kurtosis in a more direct way. While GCE function is not always positive, it is often squared and normalized in empirical studies. However, little attention has been paid to the fact that the higher moments of squared-GCE function are different from the original GCE function. This paper derives the correct skewness and kurtosis of the squared-GCE function, and examines the bias. Empirical results suggest a significant bias when distribution parameters are used instead of the correct skewness and kurtosis, which will result in severe underestimate of the VaR.