Analysis of Financial Time Series, Third Edition

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

11.27

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

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 ...