I'm working on these problems and think I figured out most of the steps, but am stuck near the end as I don't understand how to relate my roots back to the unit circle in order to determine stationarity/causality.

For each of the following models:

Express them using the backshift operator, state the autoregressive or moving average polynomial and use it to determine whether they are causal stationary or invertible.

$$(1)X_t = 1.8X_{t-1}-0.81X_{t-2}+\epsilon_t$$ $$(2)X_t=\epsilon_t-1.2\epsilon_{t-1}+0.35\epsilon_{t-2}$$

For (1) expressed using backshift is $$(1-1.8B+0.81B^2)X_t = \epsilon_t$$ The root of this polynomial is $$z_{1,2}=\frac{1.8±\sqrt{0}}{1.62}=1.111$$

So now I think I need to relate this number to the unit circle somehow? I don't know where to go from here.


if $|z| ≠1$, a stationary solution exists. If $|z|>1$, AR(p) is causal. In this case, $$|z| = 1.111 > 1$$ Which means that it is causal stationary. Is this correct?

I also managed to express (2) using backshift and got the roots $(2, \frac{1}{0.7})$ - but also do not know where to go from here, would the calculation be very different from (1) because this is a MA model? Or would it still be pretty similar to solving (1)?

EDIT: Still having some trouble with this one, trying to figure out invertibility

Thanks heaps!!


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