State-Space Models
In recent years state-space representations and the associated Kalman recursions have had a profound impact on time series analysis and many related areas.
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State-Space Models
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
State-Space Representations The Basic Structural Model State-Space Representation of ARIMA Models The Kalman Recursions Estimation for State-Space Models State-Space Models with Missing Observations The EM Algorithm Generalized State-Space Models
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; 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 9.1. 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 9.2, and a state-space representation is developed for a general ARIMA process in Section 9.3. The Kalman recursions, which play a key role in the analysis of state-space models, are derived in Section 9.4. 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 9.5) and the formulation of state-space models to deal with missing values (Section 9.6). In Section 9.7 we introduce the EM algorithm, an iterative procedure for maximizing the
© Springer International Publishing Switzerland 2016 P.J. Brockwell, R.A. Davis, Introduction to Time Series and Forecasting, Springer Texts in Statistics, DOI 10.1007/978-3-319-29854-2_9
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State-Space Models
likelihood when only a subset of the complete data set is available. The EM algorithm is particularly well suited for estimation problems in the state-space framework. Generalized state-space models are introduced in Section 9.8. These are Bayesian models that can be used to represent time series of many different types, as demonstrated by two applications to time series of count data. Throughout the chapter we shall use the notation {Wt } ∼ WN(0, {Rt }) to indicate that the random vectors Wt have mean 0 and that Rt , if s = t, E Ws Wt = 0, otherwise.
9.1 State-Space Representations A state-space model for a (possibly multivariate) time series {Yt , t = 1, 2, . . .} consists of two equations. The first, known as the observation equation, expresses the w-dimensional observation Yt as a linear function of a v-dimensional state variable Xt plus n
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