Higher-order dependence in the general Power ARCH process and a special case

In this paper we consider a general first-order power ARCH process and, in particular, a special case in which the power parameter approaches zero. These considerations give us the autocorrelation function of the logarithms of the squared observations for first-order exponential and logarithmic GARCH processes. These autocorrelations decay exponentially with the lag and may be used for checking how well an estimated exponential or logarithmic GARCH model characterizes the corresponding autocorrelation structure of the observations. The results of the paper are also useful in illustrating differences in the autocorrelation structures of the classical first-order GARCH and the exponential or logarithmic GARCH models.