Chapter 7 Univariate Models

These models can be written in the form: \[\begin{equation} \begin{gathered} x_{t} = b_t x_{t-1} + u_t + \beta_t c_t + w_{t}, \text{ } w_t \sim \,\text{N}(0,q_t) \\ y_{t} = z_t x_{t} + a_t + \psi_t d_t + v_{t}, \text{ } v_t \sim \,\text{N}(0,r_t) \end{gathered} \tag{7.1} \end{equation}\] where \(c_t\) and \(d_t\) are inputs (no missing values) and \(y_t\) are observations (missing values allowed). \(x_t\) are the states. Everything else is an estimated parameter. Parameters can be time-varying. If time-varying, then specified as a 3D array with time in the 3rd dimension.