# AR(p) with white noise error term -- always covariance stationary?

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

• No it is not always true. Sep 15, 2018 at 23:42
• But it is true whenever we have P roots inside the unit circle?
– 123
Sep 15, 2018 at 23:45
• 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. Sep 16, 2018 at 1:19
• 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 $(t-s)$. Sep 17, 2018 at 0:36
• 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. Sep 20, 2018 at 9:05