I have the following time series $$y_t = (-1)^tx_t$$
whereby $x_t$ is an AR model, $$x_t=\frac{1}{2}x_{t-1}-\frac{1}{3}x_{t-2}+\epsilon_t$$
with $\epsilon_t \sim \text{W(0, 1)}$ , show that $y_t$ is stationary.
So, I know also by the following $x_t = y_t-y_{t-1}$, so we should have $x_{t-1}=y_{t-1}-y_{t-2}$,
Therefore, combining all of this,
$$y_t=(-1)^t\left[\frac{1}{2}\left(y_{t-1}-y_{t-2}\right) + \frac{1}{3}\left(y_{t-2}-y_{t-3}\right) \right] + \epsilon_t$$
How do I proceed from here, I have thought to put all terms with respect to $y$ on LHS, to group, and collect the difference operator to solve the roots to a cubic?