# Can GARCH(1,1) be applied to homoskedastic time series when comparing against heteroskedastic time series?

I would like to compare a daily trade w.t.d. \$index over the period of 10 years+ versus the shorter term (170 days) using GARCH(1,1).

The short term data derives from the 10 year+ data. The trouble is that the 10 year time series is heteroskedastic, whereas the shorter data is not.

Can I apply GARCH(1,1) on the shorter term data as well even though it's not heteroskedastic (on the basis that it would be heteroskedastic if they were to be measured over a long period of time, and that the shorter period data derived from the longer period data)?

-
 Welcome to the website, @Charlie. And by "compare" you mean what? – StasK May 12 '12 at 5:32 Compare - I meant comparing the Arch and Garch coefficients for example. 10y+ is the long-term overall norm and to see what different characteristic this particular short period may show in the results. It's not a like-for-like comparison, but it's more about any deviation/anomalies from the norm in that particular period. – Charlie May 12 '12 at 10:01 It might be better to move this to quant.stackexchange.com – John Aug 12 '12 at 4:57

If there is no heteroskedasticity in the "short" sample (or, in other words, the sample size of the "short" sample is insufficient to detect the heteroskedasticity), then it is the same as saying that the (heteroskedasticity) autoregression coefficients are equal to zero. This is the model nested in any (G)ARCH model, and thus your comparison becomes the comparison of the (G)ARCH coefficients "long" sample with zero (which, as you indicated, is rejected). If there are any mean/shift parameters, you can compare them straightforwardly, too.

-
 This absolutely makes sense. I must say I got confused as I ran the GARCH equations for these shorter, homoscedastic periods, it printed the results complete with coefficients - even though it should've been zero as you said. Thank you very much for clarifying this. – Charlie May 13 '12 at 2:37

I am credulous that if there is no heteroskedasticity in the short series, you have not to use GARCH model to fit your data. If you want to convey the meaning that the short series have no heteroskedasticity, you can make a ARCH-LM test, which easily make on the statistics software like EViews, and get an answer that the long series have heteroskedasticity but the short one haven't. This process can get the answer you want.

-
 Thank you for your reply. I have in the end come up with a comparison between like-for-like longer periods, one of which includes the shorter period (which displays significant volatility) and the other sample that ends just prior to that volatile period. – Charlie May 14 '12 at 21:17 Sorry I lost the rest of my reply here - I just wanted to say that I reached the same conclusion using ARCH LM, too. Thank you. – Charlie May 14 '12 at 21:34

As StasK's answer mentions, a short-term Garch-like volatility model can be embedded in a long-term forecast. A better comparison might be to compare a Garch(1,1) model to an FIGarch model. In the equity markets which you seem to be focusing on, the partial autocorrelation function tends to deteriorate slowly. This is indicative of a fractionally integrated volatility process. By contrast, a Garch(1,1) process tends to have volatility quickly revert back to the mean. So it is possible that the Garch(1,1) process is picking up a homoskedastic process when more data would reveal that volatility exhibits long-memory. It would be a mistake to remove this data from consideration.

In addition, as a result of the different properties of the variety of volatility models, the time horizon is very important for volatility forecasts. For instance, a one-day ahead volatility forecast is going to be very similar to the estimate of today's volatility. However, a quarter or a year head volatility forecast might be substantially different.

-