# Distribution of AR and MA polynoms roots in ARMA/ARMA-GARCH models

I have another noob question. So, for example, I have ARMA(2,2) model: $$x_{t} = \phi_{1}x_{t-1} + \phi_{2}x_{t-2} + e_{t} + \theta_{1} e_{t-1} + \theta_{2} e_{t-2}$$.

So, I have 2 polynoms: $$1 - \phi_{1}z - \phi_{2}z^{2}$$ and $$1+\theta_{1}z+\theta_{2}z^{2}$$

Their roots are: $z^{\phi}_{1}, z^{\phi}_{2}, z^{\theta}_{1}, z^{\theta}_{2}$

So, $\widehat{\phi_{i}}, \widehat{\theta_{j}}$ are random, where $\widehat{\phi_{i}}, \widehat{\theta_{j}}$ are estimations of $\phi_{i}, \theta_{j}$. I know, that all $\widehat{\phi_{i}}, \widehat{\theta_{j}}$ have normal distributions (distribution of AR, MA coefficients estimation in ARMA-GARCH models). My question: what's the distributions of AR part and MA part polynoms roots $z^{\widehat{\phi}}_{i}, z^{\widehat{\theta}}_{j}$? Does it any difference in root distributions between ARMA and ARMA-GARCH models?

Thank you.

• $\phi_{i}, \theta_{j}$ are not random, only their estimate $\hat{\phi_{i}}, \hat{\theta_{j}}$ are random variables. – Malick Apr 14 '16 at 9:04
• Yes, sorry, I'll correct it now. – Dmitriy Apr 14 '16 at 9:05
• Express the roots in terms of phis and thetas, and given the distributions of phis and thetas, you will have the distribution of roots (unfortunately, they will be nasty). – Richard Hardy Apr 14 '16 at 15:03
• It's not possible in when $N>4$ in general... I thought about this variant - make solutions for $N=2,3,4$, but u want to ring solution "in general case"... – Dmitriy Apr 14 '16 at 16:32
• "but I want to find solution "in general case"... Android is so Android... – Dmitriy Apr 14 '16 at 16:54