17.1 The linear model
Definition 17.1.1. A linear state space model consists of two equations: \begin {align*} \mathbf {x}_t &= F\,\mathbf {x}_{t-1} + \mathbf {w}_t \qquad \text {(state equation)} , \\ \mathbf {y}_t &= H\,\mathbf {x}_t + \mathbf {v}_t \qquad \text {(observation equation)} , \end {align*}
where \(\mathbf {x}_t\) is the unobserved state, \(\mathbf {y}_t\) the observation, \(F\) and \(H\) known matrices, and \(\mathbf {w}_t\), \(\mathbf {v}_t\) independent white noise with covariance matrices \(Q\) and \(R\).
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, \[ x_t = x_{t-1} + w_t , \qquad y_t = x_t + v_t , \] is the simplest non-trivial case, the local level model: the level wanders as a random walk, and each observation is the level plus independent noise.
Example 17.1.2. Write the AR(2) process \(X_t = \alpha _1X_{t-1} + \alpha _2X_{t-2} + Z_t\) in state space form.
Solution. Take the state to be the vector of the two most recent values, \(\mathbf {x}_t = (X_t, X_{t-1})^\top \). Then \[ \begin {pmatrix} X_t \\ X_{t-1} \end {pmatrix} = \begin {pmatrix} \alpha _1 & \alpha _2 \\ 1 & 0 \end {pmatrix}\begin {pmatrix} X_{t-1} \\ X_{t-2} \end {pmatrix} + \begin {pmatrix} Z_t \\ 0 \end {pmatrix} , \qquad y_t = \begin {pmatrix} 1 & 0 \end {pmatrix}\mathbf {x}_t . \] The first row of the state equation is the AR(2) itself; the second merely shifts \(X_{t-1}\) down one place. Here \(F = \begin {pmatrix} \alpha _1 & \alpha _2 \\ 1 & 0 \end {pmatrix}\), \(H = (1, 0)\), \(Q = \operatorname {diag}(\sigma _Z^2, 0)\), and \(R = 0\): the observation is exact. Any AR(\(p\)) works the same way, with a \(p\)-vector state, and MA terms can be included by adding past shocks to the state.
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