I am solving exercises from time series book and I am not sure if I am doing it right.
Consider ARMA(1,1) process : $y_t = y_{t-1} + 0.5u_{t-1} + u_t$. Show that errors obtained from regressing $y_t$ on $y_{t-1}$ i.e $0.5u_{t-1} + u_t$ don't satisfy orthogonality conditions, thus show that $\mathbb{E}[y_{t-1}|0.5u_{t-1} + u_t] \neq 0$
I am trying to solve it like that:
$\mathbb{E}[y_{t-1}|0.5u_{t-1} + u_t] = \mathbb{E}[y_{t-2} + 0.5u_{t-2} + u_{t-1}|0.5u_{t-1} + u_t]$
We can use independence of $y_{t-2}$ and $u_{t-2}$ from future errors and since they have zero mean:
$\mathbb{E}[y_{t-1}|0.5u_{t-1} + u_t] = \mathbb{E}[y_{t-2}] + \mathbb{E}[ 0.5u_{t-2}] + \mathbb{E}[ u_{t-1}|0.5u_{t-1} + u_t]$
$\mathbb{E}[y_{t-1}|0.5u_{t-1} + u_t] = \mathbb{E}[u_{t-1}|0.5u_{t-1} + u_t]$
How do I prove that this expression doesnt equal 0?