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Permalink Reply by Clive Vaughan Jones on August 26, 2013 at 10:02am

This is a really interesting question, Arun.

I did some search and found notes from a course in applied econometrics which explains the process of forecasting the conditional variance several steps into the future, as one of repeated substitution.

 

See http://faculty.apec.umn.edu/pglewwe/documents/ap821226.pdf

 

You can also forecast the conditional variance. That is:

Et[ht+1] = α0 + α1εt + β1ht

Using repeated substitution (see p.151 of Enders for

details) the j-step-ahead forecast for the GARCH(1,1) is:

Et[ht+j] = α0[1 + (α1+β1) + (α1+β1)2 + … + (α1+β1)j-1] + (α1+β1)jht

This can be done for other processes as well. Note that to obtain ht you need to use repeated substitution to make it a function of past εt’s.

 

The Enders reference probably is to

http://books.google.com/books/about/APPLIED_ECONOMETRIC_TIME_SERIES...

 

Hope this helps.

 

Clive Jones

http://businessforecastblog.com

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Permalink Reply by Arun on August 26, 2013 at 11:15am

The repeated substitution to find out the h(t) is the tricky part!

I'm actually fumbling at understanding the concept as well. Let me outline my question a little differently -

I've got a series y(t), and I'm trying to fit a AR(1) GARCH(1,1) to it. So the 2 equations I'm trying to estimate are -

1. The AR equation with y(t) = c1 + a*y(t-1) + e(t)

2. The Volatility equation h(t) = c2 + a1*ep(t-1) + b1*h(t-1) (which is basically synonymous to a ARMA(q,p) model on h(t), though not exactly)

*****************************************************PART II *******************************************************

SAS throws out the beta coeffs for each of the above equations. ARCH1, ARCH0, GARCH1 etc..

Which leads me to two questions -

1. Do we have to calculate h(t-1) in terms of ep(t-1) ,ep(t-2) .. ep(0)? Is this what is called repeated substitution? And this needs to be then done for every new forecast observation?

2. If I do calculate the E(h(t+j)), how can I arrive at the E(y(t)) with a relevant C.I.?

3. Finally, the forecasts for volatility E(h(t+1)) through E(h(t+j)), will they look very similar in their variance to the Var(e(t))?

Bonus question: Also, when y(t) is forecasted, and if I try to remodel the h(t) over the new series, till (t+j), will that help me understand/validate the volatility model any better? Will the errors look similar from before & after?

Thanks,

Arun

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Permalink Reply by Arun on August 26, 2013 at 11:16am

And yes, I really liked that document you've added here, it actually explains things in greater detail! But I'm yet to have the 'Eureka' moment!

More research and reading to do I guess..!

I know I'm just missing a fine point somewhere.. and guess it'll take its time! Anyway, thanks so much for your reply and the links.

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Permalink Reply by Kevin Gray on August 26, 2013 at 3:44pm

Hi Arun,

Just for future reference, GARCH Models: Structure, Statistical Inference and Financial Applications (Francq and Zakoian) provides an in-depth mathematical overview of ARCH/GARCH.  Not a "how to" cookbook but nonetheless seems to be a best seller on this topic.

Regards,

Kevin Gray

www.cannongray.com

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