Drawback of GARCH

[krose]

Hi.

In both your qaunt study notes and in Jorions VaR it is mentioned that the problem with GARCH is the models nonlinearity.

I don't really understand the implications of nonlinearity. Jorion writes that it is a problem when trying to estimate the parameters. Is he talking about the weights? Can anyone give me a better explanation of this?

Thanks, Kenneth

 

[David Harper CFA FRM]

Hi Kenneth,

http://learn.bionicturtle.com/images/forum/garch11_anote.png

Yes, in the GARCH(1,1) the things that need to be estimated are the parameters/weights: alpha, beta, gamma/omega (because the others, lagged return and variance, are observed). What he means is, the parameters are a hassle to estimate. If, instead, you could use a linear regression (y=mx+b), that would be easier than using the maximum likelihood method to find the best fitting weights. Unlike running a regression, you have to optimize numerically (I have an XLS I could upload but it is outside FRM scope. So, please let me know if you'd like an XLS example of max likelihood I am glad to dress it up for display. But it won't be tested)

That said, the word "nonlinear" confused me at first b/c I thought it referred to what I perceive (in the literature) to be a bigger drawback (i.e., I am not sure everyone thinks MLE estimation is a huge difficulty): that the GARCH variance estimate is a linear function of the lagged return^2, such that it treats positive returns the same as negative returns ("innovations"). That is, because it squares the return, both "ups" and "downs" count the same. This is perceived to be a drawback, so other variations are invented to fix this flawed symmetry (linear function of lagged return^2). Now i am maybe adding to the confusion: this aspect of the models "linearity" (not non linearity!) is arguably a drawback.

But that's an aside to mention I find the "nonlinear" a tad ambiguous here. What Jorion (clearly) means is: relative to estimating params (weights) with a linear regression, estimating them via max likliehood is a hassle.

David

 

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We like an XLS example of max likelihood . Can you post them up for display

 

[krose]

Hi David

Thank you for your quick response. I would really like to se an example of max likelihood to get an idea of how big a hasle it really is.

Don't worry about adding to the confusion. It helps me understand the ups and downs of a model. How big a problem is the aspect of treating the positive and negative returns the same? Does it generally underistimate the variance of the negative returns?

Kenneth

 

[David Harper CFA FRM]

@ Kwame, Kenneth: np, I'll upload an MLE after I finish the Credit A screencast

Kenneth:

"How big a problem is the aspect of treating the positive and negative returns the same?"
I don't really know, i've only seen it more often as a drawback than Jorion's drawback that MLE is difficult. Sorry, punt to specialist...


Does it generally underistimate the variance of the negative returns?"

Yes, that is the theory because NGARCH(1,1) was invented for this idea. Under NGARCH(1,1)

variance estimate = omega + (alpha)(last variance) + (beta)(last return - [param for weight][last volatility])^2

where if "param for weight" = 0, you are back to GARCH(1,1)

but if "param for weight" > 0, then the NGARCH is clever: a recent positive move is muted (positive return - positive), but a negative move is amplified (negative - positive). With the implicit idea: a recent positive tends to lead to less volatility, but a recent negative tends to lead to greater volatilty. This N is for non-linear, so you can see why I think Jorion's non-linear is ambigous. Jorion means the GARCH function is itself non-linear (and that we can't use linear regression, but must use MLE; technically I am not sure this is even true) but NGARCH is a non-linear "solution" to this "problem" (I can't tell you how big it is) that GARCH(1,1) treats the recent return symmeterically.

David

 

[David Harper CFA FRM]

@ Kwame, krose

I attached an XLS with example of MLE. The source is (recreated from) Taylor, Asset Price Dynamics.

Note: the GARCH(1,1) is the same as GARCH(1,1) we study in FRM but the average return is included; i.e., instead of squared returns (which we allow by the simplifying assumption that average return = 0 for short horizons), it is a "proper" variance.

Then the MLE is achieved by using Excel's solver to find the log likelihood value that it greatest, by changing the params (subject to constraints: alpha > 0, persistence < 1.0 which is required for a stable GARCH process). To say again, the MLE here means: find largest cell F14 by changing F9:F12 subject to constraints [F10>0.0001 and F11 <=0.9999].

http://learn.bionicturtle.com/images/forum/mle_thumb1.png

The log likelihood is sum of cacls in column F:

=-0.5*(LN(2*PI())+LN(D19)+(E19^2))

which essentially implements Jorion's log of likelihood f() on page 225.

So, that is an example of the numerical optimization (i.e., we need solver, can't analytically find the params) which is deemed a drawback.

David

 

[krose]

Hi David

Thank you for the explanation.

I have one practical problem though. I have downloaded your example of MLE but I have a problem when I am trying to get the results you have got. The solver tells me that there is an error in either a target cell or in one of the constraints.

Here is what I do:
- I open the Excel file and then it shows the results you have found in the cells marked with yellow.
- Then I delete the numbers in the yellow cells and ask solver to to find the values again.
- Then solver tells me that there is an error in either the target cell or in one of the constraint cells. The only value it gives me is alpha and this value equals the constraint (0,0001).

Do you know what I am doing wrong?

I took a few pictures of the screen and pasted them in to a Word document for you to see. My Excel version is in danish though but is shows what I am doing.

Thanks Kenneth