Fourth-order moments of augmented arch processes

This paper is concerned with establishing the correspondence between the fourth- order moments structure and the parametric representation for augmented ARCH processes. In an introduction, the possible types of fourth-order moments are classified and some tentative interpretation for each class is given. Then, algebraic formulae are developed that permit the calculation of all fourth- order cross moments that are possibly non-zero within the AARCH class on the basis of known AARCH parameters. These formulae are useful for evaluating properties of estimated AARCH models, if estimation is conducted via maximum likelihood methods or approximations, and also for possible use in method-of-moments estimation.