Barron-Loss Adaptive Estimation

Classical score-driven models update time-varying parameters using gradients of model-implied log-likelihoods, which can be sensitive to misspecification, outliers, and 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 QSD 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 show that the BLADE update belongs to the class of Proper and Robust Autoregressive Derivative Adaptive (PRADA) models and is therefore expected divergence reducing, with more robust updates achieving an at least as large expected local divergence reduction as less robust ones over explicit intervals of step sizes and, under contamination, of contamination proportions. Monte Carlo experiments confirm these findings and show superior performance relative to GARCH and βt–GARCH models under contamination; an application to Bitcoin log-returns shows BLADE outperforming leading benchmarks out of sample.