In recent years state-space representations and the associated Kalman recursions have had a profound impact on time series analysis and many related areas. The techniques were originally developed in connection with the control of linear systems (for accounts of this subject see Davis and Vinter, 1985, and Hannan and Deistler, 1988). An extremely rich class of models for time series, including and going well beyond the linear ARIMA and classical decomposition models considered so far in this book, can be formulated as special cases of the general state-space model defined below in Section 8.1. In econometrics the structural time series models developed by Harvey (1990) are formulated (like the classical decomposition model) directly in terms of components of interest such as trend, seasonal component, and noise. However, the rigidity of the classical decomposition model is avoided by allowing the trend and seasonal components to evolve randomly rather than deterministically. An introduction to these structural models is given in Section 8.2 and a state-space representation is developed for a general ARIMA process in Section 8.3. The Kalman recursions, which play a key role in the analysis of state-space models, are derived in Section 8.4. These recursions allow a unified approach to prediction and estimation for all processes that can be given a state-space representation. Following the development of the Kalman recursions we discuss estimation with structural models (Section 8.5) and the formulation of state-space models to deal with missing values (Section 8.6).
KeywordsObservation Equation Seasonal Component Basic Structural Model Gaussian Likelihood Error Covariance Matrice
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