Maximum likelihood estimation for generalized autoregressive score models

We study the strong consistency and asymptotic normality of the maximum likelihood estimator for a class of time series models driven by the score function of the predictive likelihood. This class of nonlinear dynamic models includes both new and existing observation driven time series models. Examples include models for generalized autoregressive conditional heteroskedasticity, mixed-measurement dynamic factors, serial dependency in heavy-tailed densities, and other time varying parameter processes. We formulate primitive conditions for global identification, invertibility, strong consistency, and asymptotic normality under correct specification and under mis-specification. We provide various illustrations of how the theory can be applied to specific dynamic models.