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This article considers a GARCH process, generally named as GARCHX,
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.
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