On the optimality of score-driven models

• 成果类型: Article
• 署名作者: Gorgi, P.; Lauria, C. S. A.; Luati, A.
• 署名单位: Vrije Universiteit Amsterdam; University of Bologna; Imperial College London
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asad067
• 发表日期: 2024
• 页码: 865880
• 关键词: maximum-likelihood-estimation; time-series; AUTOREGRESSIVE MODELS

摘要

Score-driven models have recently been introduced as a general framework to specify time-varying parameters of conditional densities. 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.