The second moment and the autocovariance function of the squared errors of the GARCH model

Since Bollerslev and Taylor indepedently introduced the GARCH model almost a decade ago many questions have remained unanswered. This paper addresses two of them. First, `What is the autocovariance structure of the squared errors?ʹ and second, `What is the condition on the parameters of the GARCH (p, q) model in order for the fourth moment of the errors to exist?ʹ. In Section 2of this paper we answer the first question and in Section 3we answer the second one. In a recent paper Ding and Granger introduced an extension of the GARCH(1, 1) model which they called the N-component GARCH(1, 1) model and they mentioned that it can be expressed as a GARCH(n, n) model. This GARCH(n, n) representation is presented in Section 3. Finally, in Section 3, we introduce the two component GARCH(n, n) model and we express it as a GARCH(2n, 2n) model.