Empirical‐Process Limit Theory and Filter Approximation Bounds for Score‐Driven Time Series Models

This article examines the filtering and approximation-theoretic properties of score-driven time series models. Under specific Lipschitz-type and tail conditions, new results are derived, leading to maximal and deviation inequalities for the filtering approximation error using empirical process theory. This approach allows the study of the asymptotic behavior of the empirical distribution function and empirical process of the approximated noise, extending the results of Francq and Zakoïan (2022) for generalized autoregressive conditional heteroskedasticity models. For general score-driven models, however, it is proven that the asymptotic distribution of the empirical process of the approximated noise is model-dependent and influenced by the estimation of model parameters. This contrasts with well-known results for linear and some nonlinear time series models, and is mainly due to the fact that the finite-dimensional static model parameters affect both the noise density and the score-driven process. The goodness-of-fit problem is then considered, and an application of these results is demonstrated with the Beta-GARCH(1,1) model, a popular score-driven time series model.