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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.

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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)
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