13.1 The Linear Model
Definition 13.1.1. A linear state space model consists of two equations: \[\textbf {x}_t = F\,\textbf {x}_{t-1} + \textbf {w}_t \qquad \text {(state equation)},\] \[\textbf {y}_t = H\,\textbf {x}_t + \textbf {v}_t \qquad \text {(observation equation)},\] where \(\textbf {x}_t\) is the unobserved state, \(\textbf {y}_t\) the observation, and \(\textbf {w}_t\), \(\textbf {v}_t\) independent white noise with covariance matrices \(Q\) and \(R\).
Note. The state equation says the system evolves by its own dynamics; the observation equation says we see it imperfectly. Separating the two is the whole idea. A random walk observed with error — \(x_t = x_{t-1}+w_t\), \(y_t = x_t+v_t\) — is the simplest non-trivial case, and it already cannot be written as an ARMA model without first working out what its observable autocorrelation structure is.
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