The class of observation-driven models (ODMs) includes the GARCH(1, 1) model as well as integer-valued time series models such as the log-linear Poisson GARCH of order (1, 1) and the NBIN-GARCH(1, 1) models. In this contribution, we treat the case of general-order ODMs in a similar fashion as the extension of the GARCH(1, 1) model to the GARCH(p, q) model. More precisely, we establish the stationarity and the ergodicity as well as the consistency and the asymptotic normality of the maximum likelihood estimator (MLE) for the class of general-order ODMs, under conditions which are easy to verify. We illustrate these results with specific observation-driven time series, namely, the log-linear Poisson GARCH of order (p, q) and the NBIN-GARCH(p, q) models. An empirical study is also provided.