Online Learning of Scale Parameters in Score-Driven Filters

Score-driven filters update a time-varying parameter by multiplying a scaled log-likelihood score by a scale parameter that controls the magnitude of the update. We name this scale parameter gain, consider it 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; a scalar gain selects distance along a line, whereas a diagonal gain selects coordinatewise transmission rates and may change direction. Gain selection becomes a conditional one-step predictive decision problem with a Kullback-Leibler objective. Our central observation is that the negative product-of-scores feedback employed in accelerated score-driven recursions can be read as the stochastic gradient of this predictive loss, offering a new variational perspective. Adaptive gain learning can therefore be viewed as an online prediction problem, where the current score provides the context for predicting the next gain. 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 an unbounded exponential link, with the strongest improvements observed in multi-crisis markets.