Asymptotic Distributions of ARMA-GARCH Parameter Estimates

Summary

The discussion asks how to describe the sampling distributions of estimated autoregressive and moving-average coefficients in an ARMA-GARCH model. The answer gives the standard asymptotic result: under suitable assumptions, parameter estimates are consistent and their centered, sample-size-scaled errors converge to a normal distribution. This supports using estimated means and standard errors to assess statistical significance in sufficiently large samples.

The result does not mean that coefficients are exactly normally distributed in every finite sample. Its validity depends on the estimator, parameter space, moment restrictions, and other regularity conditions. The answer also notes that asymptotic normality can hold when the innovations themselves are not Gaussian. It points readers toward technical treatments of GARCH estimation theory, but supplies no derivation, numerical example, or model-specific standard-error calculation. Thus the exchange provides a general inferential framework rather than a complete procedure for any particular fitted model.

Key ideas

• Under regularity assumptions, estimated ARMA-GARCH parameters are consistent and asymptotically normal.
• Asymptotic normality concerns scaled estimation error as sample size grows, not necessarily the exact finite-sample distribution.
• The estimator and parameter-space conditions affect whether the asymptotic result applies.
• Innovation distributions need not be Gaussian for asymptotic normality to hold under suitable conditions.
• Standard errors and the large-sample approximation can support significance tests.

Tags

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


# 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.

## Answer by Malick (score 2, accepted)

https://quant.stackexchange.com/a/25264

> 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)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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