  This paper is concerned with some properties of the generalized GARCH models,
obtained by extending GARCH models with exogenous variables, the so-called
GARCH extended (GARCHX) models. For these, we establish sufficient conditions
for some properties such as stationarity, existence of moments, ergodicity,
geometric ergodicity, consistence and asymptotic normality of likelihood
estimators of the model parameters. For some of these properties we show that
the conditions that we propose are also necessary. We further provide examples
and applications to illustrate and highlight the importance of our findings.
