Asymptotic Properties of GARCH-X Processes

This article considers a GARCH process, generally named as GARCH-X, in which the additional covariate is specified as a positive fractionally integrated process. Recent work on MEM, HEAVY, and Realized GARCH models falls in this category. We investigate the asymptotic properties of this process and show how it explains stylized facts of financial time series such as the long memory property in volatility and leptokurtosis. Not surprisingly, the time series properties of the GARCH-X process with a nonstationary covariate are qualitatively different from those of the GARCH-X process with a stationary covariate. Nevertheless, if the covariate is persistent, the GARCH-X process provides adequate explanations of some stylized facts that the GARCH(1, 1) model cannot capture.