Stability and performance guarantees for misspecified multivariate score-driven filters

Can stochastic gradient methods track a moving target? We address the problem of tracking multivariate time-varying parameters under noisy observations and potential model misspecification. Specifically, we examine implicit and explicit score-driven (ISD and ESD) filters, which update parameter predictions using the gradient of the logarithmic postulated observation density (commonly referred to as the score). For both filter types, we derive novel sufficient conditions that ensure the exponential stability 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 stability results are agnostic about the true process. For the ISD filter, concavity of the postulated log density combined with simple parameter restrictions is sufficient to guarantee stability. In contrast, the ESD filter additionally requires the score to be Lipschitz continuous and the learning rate to be sufficiently small. We validate our theoretical findings through simulation studies, showing that ISD filters outperform ESD filters in terms of accuracy and stability.