Properties of moments of a family of GARCH processes

This paper considers the moments of a family of first-order GARCH processes. First, a general condition for the existence of any integer moment of the absolute values of the observations is given. Second, a general expression for this moment as a function of lower-order moments is derived. Third, the kurtosis and the autocorrelation function of the squared and absolute-valued observations are derived. The results apply to a number of different GARCH parameterizations. Finally, the existence, or lack thereof, of the theoretical counterpart to the so-called Taylor effect in some members of this GARCH family is discussed. Possibilities of extending the results to higher-order GARCH processes are indicated and potential applications of the statistical theory proposed.