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Analysis of Financial Time Series, Third Edition by RUEY S. TSAY

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11.2 Linear State-Space Models

We now consider the general state-space model. Many dynamic time series models in economics and finance can be represented in state-space form. Examples include the ARIMA models, dynamic linear models with unobserved components, time-varying regression models, and stochastic volatility models. A general Gaussian linear state-space model assumes the form

11.26 11.26

11.27 11.27

where st = (st, …, smt)′ is an m-dimensional state vector, yt = (yt, …, ymt)′ is a k-dimensional observation vector, dt and ct are m- and k-dimensional deterministic vectors, Tt and Zt are m × m and k × m coefficient matrices, Rt is an m × n matrix often consisting of a subset of columns of the m × m identity matrix, and {ηt} and et are n- and k-dimensional Gaussian white noise series such that

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where Qt and Ht are positive-definite matrices. We assume that {et} and {ηt} are independent, but this condition can be relaxed if necessary. The initial state s1 is Nμ1I0, 1I0, where μ1I0 and 1I0 are given, and is independent of et and ηt for t > 0.

Equation (11.27) is the measurement or observation equation that relates the vector of observations yt to the state vector st, the explanatory variable ...

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