Robust Observation-Driven Models Using Proximal-Parameter Updates

Abstract
We propose a novel observation-driven modeling framework that allows for timevariation in the model’s parameters using a proximal-parameter (ProPar) update.The ProPar update is the solution to an optimization problem that maximizes thelogarithmic observation density with respect to the parameter, while penalizing thesquared distance of the parameter from its one-step-ahead prediction. The asso-ciated first-order condition has the form of an implicit stochastic-gradient update;replacing this implicit update with its explicit counterpart yields the popular classof score-driven models. Key advantages of the ProPar setup are stronger invertibil-ity properties (especially under model misspecification) as well as extended (globalrather than local) optimality properties. For the class of postulated observation den-sities whose logarithm is concave, ProPar’s robustness is evident from its (i) mutedresponse to large shocks in endogenous and exogenous variables, (ii) stability un-der poorly specified learning rates, and (iii) global contractivity towards a pseudo-truth—in all cases, even under model misspecification. We illustrate the generalapplicability and the practical usefulness of the ProPar framework for time-varyingregressions, volatility, and quantiles. Joint paper with Rutger-Jan Lange and Bram van Os.

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