Necessary and sufficient conditions for the identifiability of observation-driven models

In this contribution we are interested in proving that a given observation-driven model is identifiable. In the case of a GARCH(p, q) model, a simple sufficient condition has been established in Berkes I, Horváth L, Kokoszka P. (2003). Bernoulli 9: 201–227 for showing the consistency of the quasi-maximum likelihood estimator. It turns out that this condition applies for a much larger class of observation-driven models, that we call the class of linearly observation-driven models. This class includes standard integer valued observation-driven time series such as the Poisson autoregression model and its numerous extensions. Our results also apply to vector-valued time series such as the bivariate integer valued GARCH model, to nonlinear models such as the threshold Poisson autoregression or to observation-driven models with exogenous covariates such as the PARX model.