# distribution of AR, MA coefficients estimation in ARMA-GARCH models

could anyone give me an information about distributions of AR and MA coefficients via estimation? So, for example, I have ARMA(1,1)-GARCH(1,1) model with the same AR(1) and MA(1) parameters estimations. So, I know "mean" for it and std, but what's the distribution of each?

Hope, my question isn't dummy.

Thank you.

Normally distributed and that's why the two first moments are sufficient to infer their statistical significance.

Proof are rather technical (and sometimes are not specific to time-series models) and mainly depends of:

• The estimation method employed ( QMLE, Least Squares, Moment, Whittle...)

• The parameter space

• Moment restrictions

• ...

These proofs demonstrate, under assumptions, that estimated parameters are consistent ($\hat{\theta} \rightarrow \theta$) and asymptotic Normal ($\sqrt{n}(\hat{\theta}-\theta)\rightarrow N(0,\sigma)$). This is true even if innovations are not Gaussian distributed.

You can have a look to:

• A Tour in the Asymptotic Theory of GARCH Estimation by Christian Francq, Jean-Michel Zakoïan (Handbook of Financial Time Series)