I have a set of equations which I have to represent in state-space form. Can someone please check if I’m doing it correctly
$ y_t= \tau_t\ + c_t, $
$ \tau_t= g_t\ + \tau_{t-1}+v_t, $
$ g_t= g_{t-1}\ + w_t, $
$ c_t= \varphi_1c_{t-1}\ + \varphi_2c_{t-2}+e_t; $
$ v_t,\ w_t,\ e_t -\ iid\ N(0,\ \ \sigma^2) $
solution:
Unobserved states are $c_t, g_t, \tau_t$
$ \beta_t = \begin{bmatrix}\tau_t&g_t&c_t\\\end{bmatrix}' $
then state equation: $ \beta_t = \begin{bmatrix}\varphi_1&\varphi_2\\1&0\\\end{bmatrix}\beta_{t-1} + \begin{bmatrix}v_t\\0\\\end{bmatrix} + \begin{bmatrix}\omega_t\\0\\\end{bmatrix} + \begin{bmatrix}e_t\\0\\\end{bmatrix} $
Observation equation: $ x_t = \begin{bmatrix}1&1&1\\\end{bmatrix}\beta_t $