Stationarity for an AR(2) process

Asked Oct 09 '19 at 23:59

Active Oct 10 '19 at 07:08

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How can I show that the following AR(2) process is stationary

$X_t = X_{t-1} + cX_{t-2}+Z_t$, provided -1 < c < 0 ?

I represented the series as $\Phi(B)X_t = Z_t$ and then tried to find out the roots of $\Phi(B)=0$ but couldn't figure out the solution.

Edit: The roots of $\Phi(B)=0$ are $\cfrac{-1\pm\sqrt{1+4c}}{2c}$

edited Oct 10 '19 at 03:04

asked Oct 09 '19 at 23:59

exAres

Do you know that this process is stationary and want to prove it or are you asking if it is stationary and why? To me, it's not obvious why this is stationary. The term $X_{t−1}$ adds complexity... – Vitali Avagyan Oct 10 '19 at 00:08
Well, under the condition -1 – exAres Oct 10 '19 at 00:13
Thanks for that, I would have never guessed this was stationary by just looking at the terms :) – Vitali Avagyan Oct 10 '19 at 00:20
What are the roots of $\Phi(B)$? – Dilip Sarwate Oct 10 '19 at 01:45
I have updated the question to mention roots of $\Phi(B)=0$. – exAres Oct 10 '19 at 03:02
1

A general solution can be found [here](https://stats.stackexchange.com/questions/118019/a-proof-for-the-stationarity-of-an-ar2/145652#145652). For your case, just put $\phi_1=1$ and $\phi_2=c$. – Dayne Oct 10 '19 at 07:10

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