  The class of observation-driven models (ODMs) includes many models of
non-linear time series which, in a fashion similar to, yet different from,
hidden Markov models (HMMs), involve hidden variables. Interestingly, in
contrast to most HMMs, ODMs enjoy likelihoods that can be computed exactly with
computational complexity of the same order as the number of observations,
making maximum likelihood estimation the privileged approach for statistical
inference for these models. A celebrated example of general order ODMs is the
GARCH$(p,q)$ model, for which ergodicity and inference has been studied
extensively. However little is known on more general models, in particular
integer-valued ones, such as the log-linear Poisson GARCH or the NBIN-GARCH of
order $(p,q)$ about which most of the existing results seem restricted to the
case $p=q=1$. Here we fill this gap and derive ergodicity conditions for
general ODMs. The consistency and the asymptotic normality of the maximum
likelihood estimator (MLE) can then be derived using the method already
developed for first order ODMs.
