Consistency, distributional convergence, and optimality of time-varying parameters in score-driven models

Observation-driven models for time series have a long history in statistics and econometrics, but are typically studied under the assumption of correct dynamic specification. We develop an in-fill asymptotic framework to study the limiting behavior of the estimated (i.e., filtered) time-varying parameter paths obtained with such models in (severely) mis-specified settings. We show that despite such mis-specification, the filtered paths, particularly those from the class of score-driven models of Creal et al. (2011, 2013) and Harvey (2013), still converge in probability to the Kullback-Leibler optimal time-varying parameter paths, even in severely mis-specified settings. We obtain distributional convergence results for the filtering errors and formulate the observation-driven filter that minimizes the asymptotic filter error variance. Such an optimal filter again has score-driven features. The results substantially generalize earlier findings, which we demonstrate by applying the new theory to time-varying tail shape models, dynamic copulas, and time-varying regression models. We further highlight the practical relevance of the asymptotic results by using them to construct pointwise intervals that quantify the uncertainty of filtered parameter paths based on observation-driven filters and apply these to the volatility path of intraday Pfizer log-returns.