On the optimality of score-driven models

Summary Score-driven models have been recently introduced as a general framework to specify time-varying parameters of conditional densities. %The underlying idea is to specify a time-varying parameter as an autoregressive process with innovation given by the score of the associated log-likelihood. The score enjoys stochastic properties that make these models easy to implement and convenient to apply in several contexts, ranging from biostatistics to finance. Score-driven parameter updates have been shown to be optimal in terms of locally reducing a local version of the Kullback–Leibler divergence between the true conditional density and the postulated density of the model. A key limitation of such an optimality property is that it holds only locally both in the parameter space and sample space, yielding to a definition of local Kullback–Leibler divergence that is in fact not a divergence measure. The current paper shows that score-driven updates satisfy stronger optimality properties that are based on a global definition of Kullback–Leibler divergence. In particular, it is shown that score-driven updates reduce the distance between the expected updated parameter and the pseudo-true parameter. Furthermore, depending on the conditional density and the scaling of the score, the optimality result can hold globally over the parameter space, which can be viewed as a generalization of the monotonicity property of the stochastic gradient descent scheme. Several examples illustrate how the results derived in the paper apply to specific models under different easy-to-check assumptions, and provide a formal method to select the link-function and the scaling of the score.