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

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