Barron-Loss adaptive estimation

The score-driven framework relies on pre-specified scoring rules tied to assumed conditional densities, making it vulnerable to misspecification under outliers or structural breaks. We embed the flexible Barron loss within the quasi score-driven (QSD) framework, allowing the degree of robustness to be learned from the data. The resulting Barron-Loss Adaptive Estimation (BLADE) filter generates a strictly stationary, ergodic, and invertible sequence of time-varying parameters under mild regularity conditions. Within an extended quasi score-driven estimation framework, obtained by generalizing the required moment condition, the associated estimator is shown to be consistent and asymptotically normal. The Barron loss is strictly consistent for a family of functionals indexed by the shape parameter γ, enabling smooth adaptation between classical and robust targets. We establish that the BLADE update belongs to the class of Proper and Robust Autoregressive Derivative Adaptive (PRADA) models and is therefore expected divergence reducing. We further establish that more robust updates achieve an at least as large expected local divergence reduction as less robust ones over explicit intervals, of step sizes in a non-contaminated setting and of contamination proportions under a contaminated updating draw, providing a formal robustness guarantee against corrupted observations. Monte Carlo experiments under non-contaminated and contaminated estimation environments confirm these theoretical findings and demonstrate superior performance relative to GARCH and βt-GARCH models when outliers are present.