Gram–Charlier densities: Maximum likelihood versus the method of moments

This paper compares two alternative estimation methods for estimating the density underlying financial returns specified in terms of a finite Gram–Charlier (GC) expansion. Maximum likelihood (ML) is the most widely employed method despite the fact that it is only consistent under the Gaussian or the true density, and usually involves convergence problems. Alternatively, the method of moments (MM) is a natural and straightforward procedure, although positivity is only guaranteed in the asymptotic expansion. We show an example for estimating daily returns of the Dow Jones Index with a very long data set, illustrating that both ML and MM yield similar outcomes. Therefore the MM applied to GC densities should be considered as an accurate tool for risk management and forecasting.