10.3 Constraining a DFA model

If \(\mathbf{a}\), \(\mathbf{Z}\), and \(\mathbf{Q}\) are not constrained, the DFA model above is unidentifiable. Nevertheless, we can use the following parameter constraints to make the model identifiable:

• \(\mathbf{a}\) is constrained so that the first \(m\) values are set to zero;
  
• in the first \(m-1\) rows of \(\mathbf{Z}\), the \(z\)-value in the \(j\)-th column and \(i\)-th row is set to zero if \(j > i\); and
  
• \(\mathbf{Q}\) is set equal to the identity matrix \(\mathbf{I}_m\).

Using these constraints, the observation equation for the DFA model above becomes

\[\begin{equation} \begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \\ y_{5} \end{bmatrix}_t = \begin{bmatrix} z_{11}&0&0\\ z_{21}&z_{22}&0\\ z_{31}&z_{32}&z_{33}\\ z_{41}&z_{42}&z_{43}\\ z_{51}&z_{52}&z_{53}\end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}_t + \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end{bmatrix}_t. \tag{10.6} \end{equation}\]

and the process equation becomes

\[\begin{equation} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}_t = \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix} \begin{bmatrix} x_{1}\\ x_{2}\\ x_{3}\end{bmatrix}_{t-1} + \begin{bmatrix} w_{1} \\ w_{2} \\ w_{3} \end{bmatrix}_t \tag{10.7} \end{equation}\]

The distribution of the observation errors would stay the same, such that

\[\begin{equation} \begin{bmatrix} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end{bmatrix}_t \sim \text{MVN} \begin{pmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} r_{11}&r_{12}&r_{13}&r_{14}&r_{15}\\ r_{12}&r_{22}&r_{23}&r_{24}&r_{25}\\ r_{13}&r_{23}&r_{33}&r_{34}&r_{35}\\ r_{14}&r_{24}&r_{34}&r_{44}&r_{45}\\ r_{15}&r_{25}&r_{35}&r_{45}&r_{55} \end{bmatrix} \end{pmatrix}. \tag{10.8} \end{equation}\]

but the distribution of the process errors would become

\[\begin{equation} \begin{bmatrix} w_{1} \\ w_{2} \\ w_{3} \end{bmatrix}_t \sim \text{MVN} \begin{pmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix} \end{pmatrix}, \tag{10.9} \end{equation}\]