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
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 ct, and the measurement error et. Equation (11.26) is the state or transition equation that describes a first-order Markov Chain to govern the state transition with innovation ηt. The matrices Tt, Rt, Qt, Zt, and Ht are known and referred to as system matrices. These matrices are often sparse, and they can be functions of some parameters θ, which can be estimated by the maximum-likelihood method.
The state-space model in Eqs. (11.26) and (11.27) can be rewritten in a compact form as
where
and {ut} is a sequence of Gaussian white nosies with mean zero and covariance matrix
The case of diffuse initialization is achieved by using
where ∑٭ and ∑∞ are m × m symmetric positive-definite matrices and λ is a large real number, which can approach infinity. In S-Plus and SsfPack, the notation
is used; see the notation in Table 11.1.
In many applications, the system matrices are time invariant. However, these matrices can be time varying, making the state-space model flexible.