On the existence of moments

Abstract

Stock returns are often modeled as having infinite second or fourth moments, with consequences for test statistics which have not yet been fully explored. Conclusions on the existence of moments are usually drawn from a generalized Pareto or simple Pareto tail index estimate. In a recent study McCulloch (1997) demonstrated that this estimator indicates distributions with even finite fourth moments, although the samples were drawn from infinite-variance stable laws, which points out the doubtful role of the tail index estimate as evidence for the finiteness of moments. Based on an fQ-System for continuous unimodal distributions, introduced by Scheffner (1998), we derive an alternative condition for the existence of moments. An estimation algorithm for the fQ-parameters is proposed and an application to the 30 most busy German stocks shows that daily returns can be modeled as being at least approximately fQ-distributed with finite second moments.