Positive Definiteness of Multivariate Densities Based on Hermite Polynomials

This paper develops both univariate and multivariate distributions based on Gram-Charlier and Edgeworth expansions, attempting to ensure non negativity by exploiting the orthogonal properties of the Hermite polynomials. The article motivates the problems underlying some specifications (in particular those involving other conditional moments beyond the variance) and provides empirical examples comparing the performance of these positive definite densities to the univariate and multivariate versions of the so-called Edgeworth-Sargan distribution when fitting stock market indices. The fitted densities perform similarly and thus the use of the positive versions depends on other econometric considerations rather than accuracy.