  Invertibility conditions for observation-driven time series models often fail
to be guaranteed in empirical applications. As a result, the asymptotic theory
of maximum likelihood and quasi-maximum likelihood estimators may be
compromised. We derive considerably weaker conditions that can be used in
practice to ensure the consistency of the maximum likelihood estimator for a
wide class of observation-driven time series models. Our consistency results
hold for both correctly specified and misspecified models. The practical
relevance of the theory is highlighted in a set of empirical examples. We
further obtain an asymptotic test and confidence bounds for the unfeasible "
true " invertibility region of the parameter space.
