Establishing error bounds for implicit and explicit score-driven filters

We present a novel error bound analysis for explicit score-driven (ESD) filters, better known as generalized autoregressive score (GAS) filters, and their implicit counterparts, referred to as implicit score-driven (ISD) or proximal-parameter (ProPar) filters. This analysis considers potential misspecification with respect to the data generating process (DGP) and the learning rate. Specifically, we establish upper bounds on the asymptotic root mean squared filtering errors (RMSEs) of ISD and ESD iterates. We derive these asymptotic error bounds under more general conditions regarding the DGP compared to conventional error bound analyses. In particular, we find that ISD filtering error iterates remain bounded asymptotically for any learning rate that is a scalar multiple of the identity matrix. In contrast, ESD requires the learning rate to be sufficiently small. These findings are confirmed in a Monte Carlo study examining nine univariate DGPs. Lastly, our theoretical analysis shows that even when the DGP is non-stationary, we can still guarantee asymptotic error bounds when using an identity prediction step in the filter.