9.2 DLM in state-space form

A DLM is a state-space model and can be written in MARSS form:

\[\begin{equation} \begin{gathered} \tag{9.4} y_t = \mathbf{F}^{\top}_t \boldsymbol{\theta}_t + e_t \\ \boldsymbol{\theta}_t = \mathbf{G} \boldsymbol{\theta}_{t-1} + \mathbf{w}_t \\ \Downarrow \\ y_t = \mathbf{Z}_t \mathbf{x}_t + v_t \\ \mathbf{x}_t = \mathbf{B} \mathbf{x}_{t-1} + \mathbf{w}_t \end{gathered} \end{equation}\]

Note that DLMs include predictor variables (covariates) in the observation equation much differently than other forms of MARSS models. In a DLM, \(\mathbf{Z}\) is a matrix of predictor variables and \(\mathbf{x}_t\) are the time-evolving regression parameters.

\[\begin{equation} y_t = \boxed{\mathbf{Z}_t \mathbf{x}_t} + v_t. \end{equation}\]

In many other MARSS models, \(\mathbf{d}_t\) is a time-varying column vector of covariates and \(\mathbf{D}\) is the matrix of covariate-effect parameters.

\[\begin{equation} y_t = \mathbf{Z}_t \mathbf{x}_t + \boxed{\mathbf{D} \mathbf{d}_t} +v_t. \end{equation}\]