Kalman Filter likelihood function with constant state

I am trying to compute the likelihood function for the following model, where the state $$\begin{bmatrix} \pi_t \\ x_t \\ i_t \end{bmatrix}$$ is measured directly:

$$\begin{bmatrix} \pi_t \\ x_t \\ i_t \end{bmatrix} = \begin{bmatrix} a_{\pi} \\ a_x \\ a_i \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix} \begin{bmatrix} \pi_{t-1} \\ x_{t-1} \\ i_{t-1} \end{bmatrix} + \begin{bmatrix} \varepsilon_{\pi} \\ \varepsilon_x \\ \varepsilon_i \end{bmatrix}$$

All instructions for using the Kalman filter to estimate the parameters of this model have required the following state and measurement equations: $$\xi_{t+1} = A_0\xi_t + Cw_{t+1} \\ y = G x_t + v_t$$

To this end, I have then re-written the above model in the following state-space form:

$$\begin{bmatrix} 1 \\ \pi_t \\ x_t \\ i_t \end{bmatrix} = \ \begin{bmatrix} 1 & 0 & 0 & 0 \\ a_{\pi} & b_{11} & b_{12} & b_{13} \\ a_x & b_{21} & b_{22} & b_{23} \\ a_i & b_{31} & b_{32} & b_{33} \end{bmatrix} \begin{bmatrix} 1 \\ \pi_{t-1} \\ x_{t-1} \\ i_{t-1} \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \varepsilon_{\pi} \\ \varepsilon_x \\ \varepsilon_i \end{bmatrix}$$

However any time I try to compute the likelihood function, I am told that there is no way to compute the variance of the forecast of the state because the variance-covariance matrix is singular. The vcov matrix $$\Gamma_0$$ is defined by:

$$vec(\Gamma_0) = (I - A_0 \otimes A_0)^{-1}(C\otimes C)vec(\Sigma)$$

How does one elicit the likelihood function of the first system then?

• Hi: are the a's and the b's known ? If so, then I don't see why you get singularity of the variance covariance matrix ? Also, I don't see why you have kronecker products for the vcov matrix ? I think there's a simpler formula but it's been a while since I looked at the kalman fitler updates. Is there special about this KF that makes you need kronecker products ? Commented Jun 20, 2020 at 12:15
• The a's and b's are what I am trying to estimate by computing the likelihood function. stats.stackexchange.com/questions/251067/… This defines the variance of a stationary VAR process and why I use kronecker products. The Kalman filter requires an an initial distribution of the states, defined by an expectation $\begin{pmatrix} \pi_0 \\ x_0 \\ i_0 \end{bmatrix}$ and a variance-covariance of that state. Commented Jun 20, 2020 at 23:22
• Hi Jacob: Maybe the singularity has something to do with your updating equation where you're trying to mimic the variance of a VAR. It seems like a regular KF ( that's why I asked about a's and b's ) so can't you just use the update for the system variance in the KF updating equations. For estimates of $v_t$ and $w_t$, you can use the likelihood decomposition. See Harvey's blue book ( structural models and the kalman filter, 1990 ) for details of how that works. Commented Jun 22, 2020 at 1:46