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It is well established that a necessary condition for stationarity of both $AR$ and $ARMA$ processes is that the coefficients of the autoregressive components sum to less than unity in magnitude.

Do we know any conditions that if taken in addition to the above suffice for guaranteeing stationarity? Of course, only non-trivial conditions are of interest such that sufficiency only holds under the combination of both.

If there are any such conditions, which of them is the least restrictive?

Steven
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1 Answers1

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The AR coefficients of ARMA processes can define a polynomial. These processes will be stationary if the (imaginary) roots of these polynomials have modulus strictly greater than 1.

See this link or this link, for instance.

FP0
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  • That would be an example of a trivial condition because it provides sufficiency regardless of whether the necessary condition holds that I gave. – Steven Jul 26 '22 at 01:50
  • Steven: the condition that you gave in your question ( sum of coefficients less than 1 ) is a heuristic version of the condition that FPO is referring to ( depending on how one defines the polynomial, the modulus can be either greater or less than 1 ), so I think the answer is that it's not possible to provide a sufficient condition that is independent of your necessary condition. But I could be wrong. Hopefully someone else can confirm or refute my claim. – mlofton Jul 26 '22 at 04:29
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    The condition I gave is not heuristic. See whuber's answer in this question for a proof – Steven Jul 26 '22 at 06:56