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We consider the time series ARMA(2,1):

$X(t)-0.75X(t-1)+0.5625X(t-2)=Z(t)+1.25Z(t-1)$

Does $\{X(t)\}$ have stationary solution? Give the form of the solution.

At first we are looking for the roots of autoregressive polynomial. None of them is on the unit circle so the solution is stationary.

Then I don't know how to find this solution. I know it has a following form:

$X(t)=\sum_{j=-\infty}^{\infty}\Phi_jZ(t-j)$

where

$\sum_{j=-\infty}^{\infty}\Phi_jz^j=\frac{1+1.25z}{1-0.75z+0.56z}$.

SigmaMat
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  • The roots have to be outside the unit circle (don't forget to check for common roots). But you showed that. Now, to get the result, you just need to find the two polynomials. The AR polynomial is $\phi(z) = 1-0.75z+0.5625z^2$ and the MA polynomial is $\theta(z) = 1+1.25z$ and the solution to $X(t)$ is $\theta(z)/\phi(z)$. – Therkel Dec 24 '17 at 07:20

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