parameters in garch(1,1)

[young_]

Hi David,

probably someone already asked this before, but it was not straightforward to find my question. please understand..

so my question is..
How do people choose the parameters - omega, alpha, and beta- in garch(1,1) in practice? I understand the properties between them such as if you add them up, it's 1(or close), persistence and time decay...

I'm just curious when you have raw data and want to apply garch model, how would you actually pick these parameters? is there some market convention or some numbers that people start with and tweak those numbers?

Thanks in advance.

 

[David Harper CFA FRM]

Hi cypanic,

I think there are several ways (e.g., regression) to fit the equation to data, but the overwhelming favorite approach for GARCH(1,1) seems to be maximum likehood estimation (MLE), which is actually qualitatively referenced in the AIMs, because Hull reviews it in Chapter 22, where the AIM is "Explain how the parameters of the GARCH(1,1) and the EWMA models are estimated using maximum likelihood methods" (e.g. PQ at http://forum.bionicturtle.com/threads/l1-t2-107-garch-ewma-maximum-likelihood-method-mle.3944 )
... i used to have a video that reviews the MLE --> GARCH(1,1) procedure, I thought it was on youtube, but i don't seem to find it.

MLE is discussed in Hull 22.5

But I have always found Stephen Taylor's Chapter 9 more helpful specifically, http://www.amazon.com/Asset-Price-Dynamics-Volatility-Prediction/dp/0691134790
... which is the basis for my Tab 2d.1 MLE which setups a procedure, see https://www.dropbox.com/s/nwieir7c00ymysj/T2.d 2012 XLS bundle_volatility_v0906.xlsx but requires companion help, i think. I hope that helps, thanks

 

[young_]

In the spreadsheet, did you use the excel solver to get the omega, alpha, and beta? what was the function(model)?
was it something like f(alpha, beta, omega)=Log Likliehood value ?
I tried to find some formulas in the spreadsheet. maybe I'm not familiar with the excel solver...

 

[young_]

David,
I think I just needed to enter some constraints to "match" the likelihood value in the excel solver to get the estimated prams... right?

thanks.

 

[chiyui]

In the spreadsheet, did you use the excel solver to get the omega, alpha, and beta? what was the function(model)?
was it something like f(alpha, beta, omega)=Log Likliehood value ?
I tried to find some formulas in the spreadsheet. maybe I'm not familiar with the excel solver...

Yeah you can use solver to do that but it's not that easy, i think.
You maximize f(alpha, beta, omega). This is the core of MLE as David mentioned about.
But the problem is how to maximize it.

Usually, you can do this by calculus. You can treat it as a function, take the partial derivatives w.r.t all the α, β and ω and set those partials equal to zero. Then you have 3 equations with 3 unknowns (α, β and ω). And you just solve them out.
But in some cases you can't find the partial derivatives. And even you can find them, sometimes you can't solve the 3 equations. In this case, you gotta find the maximum value of f(alpha, beta, omega) by trial and error.
There're many methods you can use to perform the trial and error. Excel Solver is one of the good computer procedure to do this. You firstly input the function f(alpha, beta, omega) in one of the cells in Excel e.g. A1 (well this has more to say later, actually). Then you call out the Solver app. It will ask you to enter which cell you wanna maximize. You choose Cell A1. Then it will ask you to enter which cells you wanna change. You choose those cells you wanna put your α, β and ω answers in. And that's done.

The most difficult part is how do you input the function f(alpha, beta, omega) in Cell A1. This function is a multivariate probability density function. That means you gotta firstly determine what kind of multivariate probability density function you wanna use. To be frank I got stuck in this problem always. For instance suppose I choose the trivariate normal density. But what is the equation of trivariate normal density function? I gotta know this before I can even input it into Cell A1 in Excel!

So I might be unhelpful of this issue.

 

[young_]

thanks guys!

 

[saloni sardana]

Hi All,

I just wanted to understand if there is any interpretation of α, β parameters that we estimate in GARCH models. As in what does it mean when α >β and α <β . What does it mean for an investor who is to invest in the portfolio fitted with GARCH estimates?

Also, why can the value of α, β and ω not be negative?

It would be of great help if any of you could provide an insight into it.

Thanks a lot.