Autoregressive Conditional Duration and FIGARCH Models: Origins of Long Memory

Abstract

Although properties of ARCH(∞) model are well investigated, existence of long memory FIGARCH and IARCH solution was not established in the literature. These two popular ARCH type models which are widely used in applied literature, were causing theoretical controversy because of suspicion that other solutions besides the trivial zero one, do not exist. Since ARCH models with non-zero intercept have a unique stationary solution and exclude long memory, existence of finite variance FIGARCH and IARCH models and, thus, possibility of long memory in ARCH setting was doubtful. The present paper solves this controversy by showing that FIGARCH and IARCH equations have a non-trivial covariance stationary solution, and that such solution exhibits long memory. Existence and uniqueness of stationary Integrated AR(∞) processes is also discussed, and long memory as their inherited feature is established. Summarizing, we show that covariance stationary IARCH, FIEGARCH and IAR(∞) processes exist, their class is wide, and they all have long memory. (joint work with D Surgailis and A Skarnulis)