Unobserved component models, approximate filters and dynamic adaptive mixture models

State estimation in unobserved component models with parameter uncertainty is traditionally performed through approximate filters, where Gaussian distributions with given moments are employed to replace otherwise intractable conditional densities. This paper re-examines signal-plus-noise models where parameter uncertainty is induced by a latent variable that may assume a fixed number of states. First, it is shown that, for these models, the approximate filters commonly adopted in the literature can be obtained as linear combinations of minimum variance linear unbiased estimators. Second, it is observed that they coincide with filters implied by a novel class of dynamic adaptive mixture models, where the parameters of a mixture of distributions evolve over time following a recursion that is based on the score of the one-step-ahead predictive distribution. Focusing on a robust specification, where the mixture components are Student’s t distributions, we prove existence, stationarity, and ergodicity of the data generating process as well as invertibility of the filter, and consistency and asymptotic normality of the maximum likelihood estimator of the static parameters. An application to energy spot prices is discussed, where the novel specification is compared with, and shown to outperform, robust score-driven filters and the related class of mixture autoregressive models.