Online Learning of Scale Parameters in Score-Driven Filters

2026-08-10

AI Summary

This work addresses the challenge of manually tuning the scale parameter in score-driven filters by proposing an online adaptive gain learning method. Treating the gain as a decision variable, the approach optimizes it in real time by minimizing the Kullback–Leibler divergence of the one-step-ahead predictive density. A bounded gain learning framework is developed based on mirror descent, incorporating a persistence-induced Bregman regularization mechanism and accompanied by a dynamic regret bound. The method integrates key techniques from online convex optimization, including stochastic gradients, projection, and discounted updates. Empirical evaluations on both synthetic data and equity volatility forecasting tasks demonstrate that the proposed bounded mirror gain strategy either outperforms or matches fixed-gain alternatives, effectively curbing extreme fluctuations—particularly excelling in volatile, crisis-prone market environments.

Abstract

Score-driven filters multiply a scaled log-likelihood score by a gain that controls the update magnitude. We treat this gain as a decision variable and study its online learning. Conditional on the current state, observation, score, and scaling rule, each admissible gain induces a reachable next state and a one-step-ahead predictive density: scalar gains govern distance along a line, while diagonal gains govern coordinatewise transmission. Gain selection is therefore a conditional predictive decision problem with a Kullback-Leibler objective. For a scalar unscaled gain, the negative raw product of consecutive scores is the stochastic gradient of this loss; positive aGAS scaling only rescales the effective step. Monotone differentiable gain links induce mirror-descent geometries on bounded gain domains, while persistence yields a Bregman pull towards a reference gain. Under convexity, compactness, and regularity conditions, we establish dynamic-regret bounds for projected and discounted mirror updates relative to time-varying, current-information comparators. Simulations illustrate the roles of scaling, link geometry, persistence, and coordinatewise transmission rates. An out-of-sample panel of equity-index volatilities shows that the bounded mirror gain generally matches or outperforms a constant gain while avoiding the extreme spikes of a nominally unbounded exponential link, with the strongest improvements observed in multi-crisis markets.

Problem

• online learning
• score-driven filters
• scale parameters
• gain selection
• predictive density

Innovation

• score-driven filters
• online learning
• mirror descent
• dynamic regret
• gain adaptation