Moment matrices in conditional heteroskedastic models under elliptical distributions with applications in AR-ARCH models

It is well known that moment matrices play a very important rôle in econometrics and statistics. Liu and Heyde (Stat Pap 49:455–469, 2008) give exact expressions for two-moment matrices, including the Hessian for ARCH models under elliptical distributions. In this paper, we extend the theory by establishing two additional moment matrices for conditional heteroskedastic models under elliptical distributions. The moment matrices established in this paper implement the maximum likelihood estimation by some estimation algorithms like the scoring method. We illustrate the applicability of the additional moment matrices established in this paper by applying them to establish an AR-ARCH model under an elliptical distribution.