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

• Published in: Insurance, mathematics & economics Vol. 51; no. 3; pp. 531 - 537
• Main Authors: Del Brio, Esther B., Perote, Javier
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-11-2012; Elsevier Sequoia S.A
• Subjects: Capital returns; Convergence; Estimating techniques; Estimation; Financial returns density; Generalized method of moments; Mathematics; Maximum likelihood; Maximum likelihood method; Method of moments; Probability distribution; Rates of return; Risk management; Semi-nonparametric method; Studies; Value at risk; Value at risk; C13; C14; C58; Financial returns density; Semi-nonparametric method; Maximum likelihood; G17; Method of moments
• ISSN: 0167-6687; 1873-5959
• DOI: 10.1016/j.insmatheco.2012.07.005

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.
► Gram–Charlier expansion accurately approximates the density of financial returns. ► Fitted Gram–Charlier densities provide more accurate risk measures. ► Consistency of ML estimates is not guaranteed for non-Normal densities. ► ML produces convergence problems for truncated Gram–Charlier densities. ► MM is a consistent and accurate method for fitting Gram–Charlier densities.