<s>[INST] Is the time series $Y_t = \frac{1}{2} Y_{t-1} + \frac{1}{2} Y_{t-2} - \frac{1}{3} \epsilon_{t-1} + \epsilon_t$ stationary? How can I tell if the series $Y_t = \frac{1}{2} Y_{t-1} + \frac{1}{2} Y_{t-2} - \frac{1}{3} \epsilon_{t-1} + \epsilon_t$ is stationary? 
 [/INST] Letting $B$ represent the backshift operator, you can write this model in compact form as:
$$(1 - \tfrac{1}{2} B - \tfrac{1}{2} B^2) Y_t = (1 - \tfrac{1}{3} B) \epsilon_{t}.$$
The auto-regressive characteristic polynomial for the model can be factorised as:
$$\phi(B) = 1 - \tfrac{1}{2} B - \tfrac{1}{2} B^2 = (1 - \tfrac{1}{2}B)(1 - B).$$
From here it should be relatively simple to see if you have stationarity.  Do you know the required conditions for this?
</s>