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.