I need to determine whether or not the following time series models are Causal and Stationary or not. My current understanding of Causal is that (for an AR(2) for example) the roots of $1 - p_1x - p_2x^2$ lie outside of the unit circle.
a)$Y_t= Z_t + 27Z_{t-6}$
b)$Y_t= 2Y_{t−1} + Z_t$
c)$Y_t= \frac 14Y_{t−1} + \frac 34Y_{t−2} + {3\over 16}Y_{t−3} + Z_t$
d)$Y_t= \frac 14Y_{t−1} + \frac 34Y_{t−2} + {3\over 16}Y_{t−3} + Z_t + 12Z_{t−12}$
e)$Y_t= Y_{t−3} + Z_t$
Any help would be greatly appreciated!
You are correct, the characteristic polynomial must have all its roots outside the unit circle in order for the Time Series process to be stationary. Take the case for a), the Time Series process can be rewritten as:
$$ Y_t = (1+27B^6)Z_t. $$
Thus we have to solve the following polynomial:
$$ 1+27z^6=0 $$
which in turn gives
$$ z^6=-\frac{1}{27}. $$
Now all that is left is to find the roots. Notice that one of the roots is $z=\frac{i}{\sqrt{3}}$, which lies inside the unit circle, and thus the Time Series process is not stationary.