Proper and Robust Autoregressive Derivative Adaptive Models

This paper introduces the class of Proper and Robust Autoregressive Derivative Adaptive (PRADA) models, extending score-driven updates beyond the logarithmic scoring rule to all strictly proper and locally proper scoring rules and strictly consistent scoring functions. PRADA updates reduce an expected local divergence measure under misspecification and thereby generalize the information-theoretic foundation of score-driven models beyond the Kullback-Leibler divergence. They are interpreted as the online analogues of M-estimators, and are linked to online Z-estimation through strict identification functions. When derived from scoring functions or identification functions, PRADA updates operate directly on elicitable functionals of the postulated conditional distribution, such as conditional means, quantiles or risk measures, and therefore do not require a parametric model. The results provide general conditions under which updates are guaranteed to reduce their corresponding divergence, stablish robustness through bounded and censored updates, and encompass many existing score-driven inspired models as special cases.