Testing Whether the ARCH Term Is Needed in a GJR-GARCH Model

Summary

The document asks whether the ARCH coefficient can be omitted from a GJR(1,1) volatility model fitted to CAC 40 data from 2005–2014. The author reports that MATLAB omitted the coefficient and warned that an active lower-bound constraint could make standard errors inaccurate. They interpret this as possible evidence that the ARCH term is unnecessary.

As a second check, the author estimates the ARCH coefficient with an MCMC method and reports a posterior mean and standard deviation of similar scale, suggesting uncertainty about whether the coefficient differs meaningfully from zero. The document does not give a formal hypothesis test, a model comparison, or the fitted parameter details needed to resolve the question. It is therefore best read as a statistical modeling question about assessing coefficient significance and model specification, rather than as evidence that the ARCH term should be removed.

Key ideas

• A constrained estimate at a parameter boundary can make standard errors unreliable.
• The author questions whether the ARCH coefficient contributes to a GJR-GARCH specification.
• An MCMC estimate is presented with uncertainty comparable in scale to its mean.
• The document does not establish a formal test or a conclusion about removing the term.

Tags

# How to decide if the ARCH coefficient is necessary in the GJR-GARCH model?


# How to decide if the ARCH coefficient is necessary in the GJR-GARCH model?












I did some analysis for CAC 40, the French market benchmark, for the period 2005-2014, and I tried to fit the data with a GJR(1,1) model in MATLAB.

Then some warning showed

> Lower bound constraints are active; standard errors may be inaccurate.

and the ARCH coefficient is disappeared in the parameters, I guess it may be the reason that the ARCH coefficient is not necessary in the GJR model.

So, I tried another method, MCMC method and get the estimation for ARCH coefficient:

$\mu$ = 0.004222282 (considered as an approximation of the true value);

$\sigma$ = 0.003775541;

They are in the same scalar and, so, it can explain that this coefficient is insignificant in the model.

But is there any method we can do a test to show that the ARCH coefficient is not necessary in the model?

Thanks in advance.

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