Sampling Distributions of ARMA Polynomial Roots

Summary

The document asks how to characterize the distributions of autoregressive and moving-average polynomial roots when ARMA model coefficients are estimated from data. It uses an ARMA model of order two for each component, defines the corresponding polynomials, and notes that estimated coefficients are random. The central question is whether normally distributed coefficient estimates imply a known distribution for the polynomial roots, and whether the answer changes when the model also includes GARCH volatility dynamics.

No answer, derivation, simulation, or empirical evidence is included, so the document does not establish a root distribution. In general, roots are nonlinear functions of estimated coefficients; their sampling behavior therefore does not follow directly from coefficient normality and can be sensitive to root ordering, complex roots, and proximity to repeated roots. Comparing ARMA with ARMA-GARCH would also require specifying estimation assumptions and how uncertainty in the conditional variance model affects coefficient estimates. The text is useful as a statistical inference question, but it leaves these modeling and inferential steps unresolved.

Key ideas

• AR and MA polynomial roots are nonlinear functions of estimated model coefficients.
• Normal coefficient estimates do not by themselves determine a simple root distribution.
• Root uncertainty can be especially sensitive when roots are complex, repeated, or close together.
• The comparison between ARMA and ARMA-GARCH depends on estimation assumptions and variance-model uncertainty.
• The document poses the inference problem but provides no derivation or empirical result.

Tags

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


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.