GARCH type model - quasi-maximum likelihood estimation

[cf_mstr]

Hi all, simple question:



For a GARCH type model fitting, when we do the Jarque-Bera test to confirm our data does not have a Gaussian distribution, we then use the Quasi-maximum likelihood estimation for our parameters optimization. The formula I have for it is this:



[img]/User%20Files/11615/quasimentbeau.PNG[/img]



Where gamma (r) is a gamma function (r(n)=(n-1)!) , squared sigma our conditional variance and epsilon the log return of the period.

So from what I understand, Nu (v) is our degree of freedom, that we also optimize.

For out of sample usage, we optimize Nu at each step before optimizing our model parameters. Is this correct?

[cf_mstr]

Also, instead of using a Student-t, we can use a generalized normal distribution. My understanding is that both will have practically the same performance. Is that right?



Thanks

[cf_mstr]

I am guessing that the phynance GOD that is able to answer those questions could also answer this one:



I am using a GARCH (1,1) model for 04 jan 2010 to 08 dec 2015 of EURUSD daily close log return data.



The data distribution is clearly not normal (kurtosis=4.53, skewness=-0.27, Jarque-Bera test statistic= 132)



QQ plot:

[img]/User%20Files/11615/qqplot.PNG[/img]



When estimating the parameters with MLE, we have a=0.0416 b=0.9568 w=1.51E-07 MLE=4306



Then, I want to use a QMLE with GED as follow:

[img]/User%20Files/11615/GEDd.PNG[/img]

[cf_mstr]

Asking this question in two posts since there seem to be a problem. When I post it as one message it cuts out parts. I am guessing it has something to do with the screenshots.



Anyway here's the rest of the question:



When v = 2, we have a standard normal distribution.

When v is less than 2, we have a distribution with thicker tails. As mentioned by Nelson (1991) (Conditional Heteroskedasticity in Asset Returns: A New Approach)



However, when I optimize the degrees of freedom, it gives v = 3.66 Then optimizing parameters we have a=0.0396 b=0.9604 w=4.72E-07 QMLE=6188



The optimization result for the degree of freedom doesn't make sense to me.

If the data distribution is leptokurtic, shouldn't the optimal degree of freedom reflect that and be less than 2 ?



Also note: When estimating parameters with QMLE, v=2, the results are the same as when estimating with MLE, which tells me that my formulas are probably good