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.