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For an $AR(p)$ process $ Y_t = a_1Y_{t-1}+a_2Y_{t-2}+...+a_qY_{t-q}$ :

Is having the coefficients $|a_1|,....,|a_p| < 1$ just a necessary condition for stationarity, or is it sufficient as well?

Christoph Hanck
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Skander H.
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1 Answers1

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It is certainly not sufficient, try the AR(2) process $$ Y_t = 0.5Y_{t-1}+0.5Y_{t-2}+u_t $$ for which not all roots of the equation $1-0.5z-0.5z^2$ are outside the unit circle:

> polyroot(c(1, -0.5, -0.5))
[1]  1-0i -2+0i

Your condition is, moreover, also not necessary: try $$ Y_t = 1.1Y_{t-1}-0.9Y_{t-2}+u_t, $$ whose characteristic polynomial has all roots outside the unit circle:

> Re(polyroot(c(1, -1.1, 0.9)))^2+Im(polyroot(c(1, -1.1, 0.9)))^2
[1] 1.111111 1.111111

Here are the real and imaginary parts together with the unit circle:

enter image description here

library(plotrix)
plot(Re(polyroot(c(1, -1.1, 0.9))),Im(polyroot(c(1, -1.1, 0.9))),asp=1,ylim=c(-1,1),xlim=c(-1,1),col="red",pch=19)
draw.circle(0, 0, 1)

A necessary condition that I would argue is useful is given by $$\sum_{j=1}^pa_j<1,$$ see e.g. here, p. 54.

Christoph Hanck
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  • Are you saying $Y_t = 1.1Y_{t-1}-0.9Y_{t-2}+u_t$ is stationary? – Skander H. Jan 19 '18 at 08:04
  • Yes, because all roots are outside the unit circle. Try plot(arima.sim(list(ar=c(1.1,-0.9)),n=1000)). – Christoph Hanck Jan 19 '18 at 08:29
  • @Alex, effectively, the large negative coefficient on the second lag "undoes" the large positive on the first lag, so that, overall, the process does not become explosive. – Christoph Hanck Jan 22 '18 at 12:03
  • @ChristophHanck The necessary condition you are citing at the bottom of your reply is exactly a condition I have been trying to establish here https://math.stackexchange.com/questions/4441327/explanation-reference-request-for-necessary-and-sufficent-conditions-for-polynom . Any chance you could point me to a reference that proves this result, alternatively hint at how I could proceed? I tried to check out the reference in the link you provided, but I could not locate the result (and I do not speak german). Thanks a lot if you have anything to add! – Thomas Fjærvik May 02 '22 at 21:29