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I'm new to GARCH, but I've got daily data of TV Ratings. I've been trying to forecast this for future, and a quick background - the data is non-stationary, has high seasonality (weekly, monthly & yearly).

I've tried UCM, but forecasts for weekly data using UCM are easier to handle, and daily level of forecasts aren't making the cut. Which is when I turned to GARCH to see if I can quickly get some high level estimates into the future.

I'm stuck with trying to get the forecasts for both the "conditional mean" and the "conditional variance" for t periods in the future. I've got the estimates for the GARCH(1,1) model, but I'm stuck trying to forecast the series into the future.

                y(t) = constant + AR(1)coeff*y(t-1) + u(t)
                h(t) = ARCH0 + ARCH1*u(t-1) + GARCH1*h(t-1)

But I haven't been able to implement this to get the predicted conditional mean & variance values, and haven't really found a perfect step-by-step guide anywhere yet! I'm new to GARCH, so, I'm guessing this might be pretty basic, but any help would be much appreciated.


Tags: conditional-mean, forecasting, garch

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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.




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


Hope this helps.


Clive Jones

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?



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.

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.


Kevin Gray


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