I have a model $y_t =0.5y_{t-1} + x_t +v_{1t}$ where $x_t=0.5 x_{t-1} + v_{2t}$ and $v_{1t}$ and $v_{2t}$ follow IID normal distribution ∼ $(0,1)$. I need to show if $y_t$ is weakly stationary or not.
I know that if $y_t$ is weakly stationary then
1) $E[y_t]$ is constant
2) $Var(y_t)$ is constant
3) $Cov(y_t,y_{t-s})=y_s$
I have started with one and I have got
$E[y_t]$ = $0.5E[y_{t-1}] + 0.5E[x_{t-1}]$ however I am unsure how to precede.
Any help would be appreciate
Thanks!