## Necessary and sufficient conditions for the identifiability of observation-driven models.(English)Zbl 07364926

Summary: 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.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60J05 Discrete-time Markov processes on general state spaces 62F12 Asymptotic properties of parametric estimators 62M05 Markov processes: estimation; hidden Markov models
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