# How to check whether a given ARIMA (p, d, q) process is stationary or not?

I know that a finite MA process $$X_t = \Theta(B)Z_t$$ is always stationary.

Also, whether an AR(p) process is stationary or not can be verified by checking the roots of $$\Phi(B)=0$$ where the process is $$\Phi(B)X_t=Z_t$$.

Consequently, we can also check if an ARMA(p, q) process is stationary or not by checking the roots of $$\Phi(B)=0$$ where the process is $$\Phi(B)X_t=\Theta Z_t$$.

But how to check if an ARIMA (p,d,q) process is stationary or not?

Let the ARIMA process be $$\Phi(B).(1-B)^d.X_t=\Theta(B).Z_t$$

Do I need to compute the roots of $$\Phi(B).(1-B)^d=0$$ or just the roots of $$\Phi(B)=0$$ in order to determine stationarity?

• $\Phi(B).(1-B)^d=0$ already has $d$ roots equal to $1$. That's why we pull them out. To further check whether still any unit root is left you just need to check the roots of $\Phi(B)=0$. – Dayne Oct 16 '19 at 5:07