The transformed Gram Charlier distribution: Parametric properties and financial risk applications

Abstract

In this paper we study an extension of the Gram–Charlier (GC) density in Jondeau and Rockinger (2001) which consists of a Gallant and Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated moments, and the closed-form expressions for the expected shortfall (ES) and lower partial moments. We obtain the analytic k th order stationarity conditions for the unconditional moments of the TGARCH model under the transformed GC (TGC) density. In an empirical application to asset return series, we estimate the tail index; backtest the density, VaR and ES; and implement a comparative analysis based on Hansen’s skewed-t distribution. Finally, we present extensions to time-varying conditional skewness and kurtosis, and a new class of mixture densities based on this TGC distribution. • We obtain the parametric properties of the transformed Gram–Charlier density. • We study the unimodality and obtain the true ranges for skewness and kurtosis. • Closed-form expressions for asymmetric-risk measures are obtained. • We study the power-law tail property for asymmetric GARCH models under our density. • The model performs well in both in-sample fitting and out-of-sample backtesting.