Implicit score-driven filters for time-varying parameter models

2025-12-02

AI Summary

This paper addresses the challenge of modeling time-varying parameters in financial and macroeconomic time series. We propose an implicit score-driven filtering framework that jointly maximizes the observed log-density and penalizes parameter deviations from predictions, enabling robust time-varying estimation in nonlinear dynamic systems. Our key contribution is the first introduction of an implicit update mechanism: leveraging optimization theory for log-concave observation densities, it preserves full posterior density information while ensuring filter stability and mean-squared error contraction—eliminating the sensitivity to learning-rate tuning inherent in explicit methods. The algorithm integrates implicit stochastic gradient updates, weighted ℓ₂ regularization, and one-step-ahead prediction. Empirical results demonstrate substantial improvements in estimation accuracy and stability across diverse financial and macroeconomic datasets, with strong robustness under model misspecification.

Technology Category

Machine Learning: OptimizationReasoning under Uncertainty: Stochastic OptimizationIntelligent Robots: State Estimation

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose an observation-driven modeling framework that permits time variation in the model parameters using 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 if the observation log density is linearly approximated around the prediction. By preserving the full density, the ISD update globalizes favorable local properties of the ESD update. Namely, for log-concave observation densities, whether correctly specified or not, the ISD filter is stable for all learning rates, while 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 illustrations in finance and macroeconomics.

Problem

• Develops implicit score-driven filters for time-varying parameter models
• Enhances stability and contraction properties for log-concave observation densities
• Applies framework to financial and macroeconomic empirical illustrations

Innovation

• Implicit score-driven update for time-varying parameters
• Maximizes log density with L2 penalty on parameters
• Ensures stability for all learning rates in log-concave densities