Diagnostic Checking of Unobserved-Components Time Series Models

Abstract

Abstract Diagnostic checking of the specification of a time series model is normally carried out using the innovations—that is, the one-step-ahead prediction errors. In an unobserved-components model, other residuals are available. These auxiliary residuals are estimators of the disturbances associated with the unobserved components. The auxiliary residuals are functions of the innovations, but they present the information in a different way. This can lead to the discovery of features of a fitted model that are not apparent from the innovations themselves. Unfortunately, the auxiliary residuals suffer from the disadvantage that they are serially correlated, even in a correctly specified model with known parameters. The purpose of this article is to show how the properties of auxiliary residuals may be obtained, how they are related to each other and to the innovations, and how they can be used to construct test statistics. The methods extend straightforwardly to models containing observed explanatory variables.