<p>Is it always true that an AR(p) process with a white noise error will be covariance stationary? </p>

CC BY-SA 4.0
2018-09-15T23:42:48.240Z
No it is not always true.
Andrew M
CC BY-SA 4.0
2018-09-15T23:45:02.147Z
But it is true whenever we have P roots inside the unit circle?
123
CC BY-SA 4.0
2018-09-16T01:19:33.300Z
Hi: If any of the roots of the AR(p) are outside the unit circle, then the AR(p) is not stationary in mean which means that it;s definitely not covariance stationary.
mark leeds
CC BY-SA 4.0
2018-09-17T00:36:07.427Z
I should add that an AR(p) with all roots inside the unit circle is, by definition, covariance stationary  because  the covariance of $y_{t}$ and $y_{s}$ is only a function of $(t-s)$.
mark leeds
CC BY-SA 4.0
2018-09-20T09:05:18.850Z
Yes, this is correct if we are only talking about stochastic trends but not if the process also has a deterministic trend. It could have all roots within the unit circle but if there is still a deterministic trend then the process would still not be covariance stationary.
Andrew M
<macroeconomics><econometrics><time-series>
AR(p) with white noise error term -- always covariance stationary?
123
# AR(p) with white noise error term -- always covariance stationary?

(Asked by: username_0 on 2018-09-15)

Is it always true that an AR(p) process with a white noise error will be covariance stationary?

**Tags:** macroeconomics, econometrics, time-series

> **Comments:**
> **username_1**: No it is not always true.  
> **username_0**: But it is true whenever we have P roots inside the unit circle?  
> **username_2**: Hi: If any of the roots of the AR(p) are outside the unit circle, then the AR(p) is not stationary in mean which means that it;s definitely not covariance stationary.  
> **username_2**: I should add that an AR(p) with all roots inside the unit circle is, by definition, covariance stationary  because  the covariance of $y_{t}$ and $y_{s}$ is only a function of $(t-s)$.  
> **username_1**: Yes, this is correct if we are only talking about stochastic trends but not if the process also has a deterministic trend. It could have all roots within the unit circle but if there is still a deterministic trend then the process would still not be covariance stationary.  

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