I would like to ask help concerning the utilization of GARCH(p,q) models to identify volatility. Suppose that I have daily closing prices of 6 financial sectors spanning several years, and I am interested in identifying which sector is the most and the least volatile (in terms of return). I have modeled their volatility using GARCH(p,q) and is now wondering how to use the estimated model to identify their volatility. Can I do the comparison by simply comparing the orders (perhaps, p) of their models? Or the coefficients?

P.S. I have a computation of historical volatility and am planning to supplement my findings with fitted GARCH(p,q) models. Can I do this?


In the very begining I advice you to model always linear effects in the time series (ARMA models). Then you add a model which investigate ARCH effects (GARCH family).
When you have done the models estimation part It is advised to check if residuals of the models do not show any dependiencies ( close to normal distribution, independent).
In another step you calculculate the volatility of your time series using the calculated models and you can calculate their statistics ( min, max, average volatility). You can also calculate forecasts of the volatilities and compare them. It is worth checking the plots and analize how often do we have volatility clustering in our time series.
Analizing the parameters of the model can be problematic becase you can get different models for different time series and how will you measure and compare them objectively.

  • $\begingroup$ Could your clarify if "In another step you calculculate the volatility of your time series" means you run the GARCH model on the residuals of the ARIMA? That is, after you removed the mean return from the series? $\endgroup$ – mugen Sep 6 '15 at 16:47
  • $\begingroup$ No. You run an Arma Garch model on returns and then you calculate volatility from the garch model in your software or manualy in Excel or in some Programming language using calculated parameters. $\endgroup$ – Robert Szóstakowski Sep 6 '15 at 17:49
  • $\begingroup$ Why should we need an ARMA model if we want to fit a GARCH model? $\endgroup$ – Ric Sep 7 '15 at 9:09
  • $\begingroup$ Yes you are right. If somebody wants to model only volatility he doesn't need to use Arma models, but when somebody wants to detect all dependiencies and have i.i.d rests It is better to use at Arma-Garch models $\endgroup$ – Robert Szóstakowski Sep 7 '15 at 12:28

Volatility is a difficult object and it is not always clear what we mean when we use the word volatility.

I would make the following distinction as a first step:

  • historical volatility: measuring the ex-post volatility of an asset/market/sector. You pick an observation period of interest (e.g. 3 months up to 3 years). You pick a frequency (often daily or weekly returns), calculate the standard deviation and annualize these numbers to make volatilities comparable (multiply by $\sqrt{250}$ if you use daily data, take $\sqrt{50}$ for weekly. Doing this you can find the time series that was most volatile in the past.

  • ex-ante volatility: you want to predict/forecast volatility. Then do the above and chose an observation period together with a frequency and then fit a (G)ARCH model or something similar. Such models give you a forecast of each asset's volatility for the coming period in the future. Doing this you can estimate the time series that could be most volatile in the future.

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    $\begingroup$ Can you clarify on how to forecast the volatility through a (G)ARCH model? I mean, I understand how the historical volatility is determined. How about ex-ante? $\endgroup$ – math_stat_enthusiast Sep 7 '15 at 10:04
  • $\begingroup$ If you look e.g. here en.wikipedia.org/wiki/… then you see the equation of a GARCH model. on the lhs there is $\sigma^2_t$ and on the rhs there are terms up to $t-1$. So: fit the model (to get the coefficients), plug in the terms of the past and you get an estimate of $\sigma^2_t$. $\endgroup$ – Ric Sep 8 '15 at 6:57

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