GARCH

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Hi David

1)
In one of yur practise questions,
it's been mentioned that GARCH is a normal mixture distribution...

Why is GARCH called a distribution...Isn't it a model to forecast future volatility....where does distribution come into play here..

2)
What are the ways to estimate parameters for EWMA ,ARCH. Can Maximum likelihood estimate be used for these two models as well...




Regards

 

[David Harper CFA FRM]

Hi Vinoth,

1. can you reference the question? ... because I hope the question says/implies that normal mixture can be a type (variation) of GARCH?

FRM assigns the "basic" GARCH(1,1) which we might technically call the "symmetrical normal GARCH(1,1)" ... this is often what is meant by GARCH. There are many variations on this GARCH(1,1), including assymetrical GARCH and normal mixture GARCH(1,1) which basically updates the variance by blending two GARCH(1,1)s each with their own params. So, bottom line: normal mixture GARCH is a type of GARCH

In regard to "Why is GARCH called a distribution…Isn’t it a model to forecast future volatility….where does distribution come into play here.."
Yes, you are correct, GARCH is a model to both estimate current volatility (note: it cannot be currently observed!) and project future volatility. The normal distribution is a built-in distributional assumption; i.e., in GARCH, the return is assumed to be conditionally normal. The model could, for example, replace the normal with another distribution and then it becomes another type of ARCH(m) model.

2. MLE is the popular method to estimate GARCH params (although technically I think OLS regression can be used).
As for MLE for EMWA, Carol Alexander suggests EWMA does not lend itself to statistical parameterization, and cites this as an advantage of GARCH over EWMA:
"So which is the best value to use for the smoothing constant? How should we choose? This is not an easy question. Statistical methods may be considered: for example, could be chosen to minimize the root mean square error between the EWMA estimate of variance and the squared return. But more often is often chosen subjectively. This is because the same value of has to be used for all elements in a EWMA covariance matrix, otherwise the matrix is not guaranteed to be positive semi-definite. If the value of lambda is chosen subjectively the values usually range between about 0.75 (volatility is highly reactive but has little persistence) and 0.98 (volatility is very persistent but not highly reactive)."
source: MRA volume II, Carol Alexander, page 122

Hope that's helpful, David