So I'm trying to forecast a Kalman Filter with exogenous variables in SAS but there doesn't seem to be a method that already does this so I'm coding it by myself. Since I'm no expert in the subject I decided to ask here. The problem is as follows:
I have an ARMA,
$x_t = a_1X_{t−1} + ... + a_kX_{t−m} + u_t + b_1u_{t−1} + ... + b_lu_{t−l}$
Thanks to Bent E. Sørensen in https://uh.edu/~bsorense/kalman.pdf, I know that this process can be represented in the following State Space Form:
Transition equation:
$$\alpha_t = \begin{bmatrix}a_1&1&0&...&0\\a_2&0&1&...&0\\\vdots&\vdots&\vdots&\ddots&\vdots \\a_{m-1}&0&0&...&1\\a_m&0&0&...&0\end{bmatrix} \alpha_{t-1} + \begin{bmatrix} 1\\b_1 \\ \vdots \\b_{m-2} \\ b_{m-1} \end{bmatrix} u_t$$
Measurement equation:
$$x_t = \begin{bmatrix}1&0&...&0\end{bmatrix}\alpha_t$$
Where $m=max{k,l+1}$. Now, by adding the exogenous variables,
$$x_t = \begin{bmatrix}1&0&...&0\end{bmatrix}\alpha_t + \Gamma v_t$$
Where,
$\hspace{1cm} \Gamma$ represents the unknown matrix of coefficients
$\hspace{1cm} v_t$ represents the vector of exogenous variables
So now the question is, how to estimate $\Gamma$ and, afterwards, forecast for $x_{t+1}$? Would be really grateful if someone could help or even point me in the right direction for the following steps.
Thanks in advance
PD: Sadly can't change the programming language, SAS is where I have to code it.