What are the values of $(a,b)$ so that this weakly stationary process is well-defined?

Question

I'm solving this exercise from previous final exam of Time Series. Because tomorrow is my exam, it's too late for me to send the professor an email.

Could you explain on what does it mean for $X$ to be well-defined in this case? I think the question have mentioned that $X$ is weakly stationary process and thus already well-defined.

Answer 1

This is an ARMA(1,1) process. I think the question b) requires you to find the coefficient values consistent with stationarity. You can write this as

$$(1-aB)X_t=(1-bB)\epsilon_t$$

As explained e.g. here this is stationary if the roots of $P(z)=1-az$ lie outside the unit circle. The roots solve

$$P(z)=0$$

or

$$1-az=0$$

or

$$z=\frac{1}{a}$$

So you need $|\frac{1}{a}| > 1$ or $-1<a<1$.