Stability and performance guarantees for misspecified multivariate score-driven filters

We address the problem of tracking multivariate unobserved time-varying parameters under potential model misspecification. Specifically, we examine implicit and explicit score-driven (ISD and ESD) filters, which update parameter predictions using the gradient of the postulated logarithmic observation density (commonly referred to as the score). For both filter types, we derive novel sufficient conditions that ensure the invertibility of the filtered parameter path and the existence of a finite mean squared error (MSE) bound relative to the pseudo-true parameter path. Our (non-)asymptotic MSE bounds rely on mild moment conditions on the data-generating process, while our invertibility result is agnostic about the true process. For the ISD filter, concavity of the postulated log density combined with simple parameter restrictions is sufficient (though not necessary) to guarantee stability. In contrast, the ESD filter additionally requires the score to be Lipschitz continuous. We validate our theoretical findings and highlight the superior stability and performance of ISD over ESD filters through extensive simulation studies. Finally, we demonstrate the practical relevance of our approach through an empirical application to U.S. Treasury-bill rates.