General-order observation-driven models: ergodicity and consistency of the maximum likelihood estimator

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.