Dick van Dijk | Searcharxiv

arXiv2025

Implicit score-driven filters for time-varying parameter models

We propose an observation-driven modeling framework that allows model parameters to vary over time through an implicit score-driven (ISD) update. The ISD update maximizes the logarithmic observation density with respect to the parameter vector while penalizing the weighted L2 norm relative to a one-step-ahead predicted parameter. This yields an implicit stochastic-gradient update. We show that the popular class of explicit score-driven (ESD) models arises when the observation log density is linearly approximated around the prediction. By preserving the full density, the ISD update extends the favorable local properties of the ESD update to a global setting. For log-concave observation densities, whether correctly specified or not, the ISD filter is stable for all learning rates, and its updates are contractive in mean squared error toward the (pseudo-)true parameter at every time step. We demonstrate the usefulness of ISD filters in simulations and empirical applications in finance and macroeconomics.

stat.ME↗

arXiv2025

Gradient-based filtering under misspecification: Stability and error bounds

Can stochastic gradient methods track a moving target? We study the problem of tracking multidimensional time-varying parameters under noisy observations and possible model misspecification. Gradient-based filters update the time-varying parameters using the gradient of a postulated objective function. A natural filtering objective is the logarithm of the postulated observation density, which gives rise to the widely used class of score-driven filters. As in the optimization literature, these filters come in two forms: explicit filters evaluate the gradient at the predicted parameter, whereas implicit filters evaluate it at the updated parameter. For both filter types, we derive novel sufficient conditions for exponential stability of the filtered parameter path, showing that stability can be guaranteed independently of the data-generating process. Under mild additional moment conditions on the data-generating process, we also obtain finite-sample and asymptotic mean squared error bounds relative to the pseudo-true parameter path. For implicit filters, these guarantees hold under weak parameter restrictions. For explicit filters, they additionally require Lipschitz continuity of the score and a sufficiently small learning rate. Simulation studies support our theoretical findings and show that implicit gradient filters outperform explicit ones in both accuracy and stability.

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