State space models and the Kalman filter

Abstract

The state space form is an enormously powerful tool which opens the way to handling a wide range of time series models. Once a model has been put in state space form, the Kalman filter may be applied and this in turn leads to algorithms for prediction and smoothing. The state space form is described in the first section of this chapter, while the second section develops the Kalman filter. Prediction and smoothing are described in sections 3.5 and 3.6 respectively. The Kalman filter also opens the way to the maximum likelihood estimation of the unknown parameters in a model. This is done via the prediction error decomposition and a full account can be found in section 3.4. The present chapter can be read independently of the rest of the book, and taken as a guide to the uses of the state space models in areas outside engineering. On the other hand, those interested primarily in the practical aspects of structural time series modelling will be reassured to know that they do not have to master all the technical details of the Kalman filter set out here. The most important parts of the chapter with which to become familiar are sections 3.1 and 3.5, the earlier parts of sections 3.2, 3.4 and 3.6, and, for those interested in non-linear models, sub-section 3.7.1. The reader will also benefit by at least skimming through the remaining sections, since there is some reference back to the various algorithms in later chapters and it is useful to have some idea of what these algorithms do and how they fit into the overall picture.