Implicit score-driven filters for time-varying parameter models

We propose an observation-driven modeling framework that permits time variation in the model’s parameters using an implicit score-driven (ISD) update. The ISD update maximizes the observation log density with respect to the parameter vector, while penalizing the weighted l2 norm relative to the one-step-ahead prediction. This yields an implicit stochastic-gradient update; taking instead the explicit version produces the popular class of explicit score-driven (ESD) models. Specifically, we show that the ESD update arises as a linearization of the ISD update. By preserving the full density, the ISD update globalizes favorable local properties of the ESD update. Namely, for log-concave observation densities (even when misspecified), ISD models are stable for any learning rate and globally contractive to a pseudo-truth. We demonstrate the usefulness of ISD models in simulations and empirical illustrations for finance and macroeconomics.