# Difference between using ARMA and GARCH to model volatility?

I just went through the following work http://epublications.bond.edu.au/cgi/viewcontent.cgi?article=1199&context=ijbf (mirror) and was wondering what is the difference between model (9) on page 12 which models volatility as a function of past volatility and past error and model (10) which is the GARCH(1,1) model. Both seem to be of the same nature to me, please correct me if i'm wrong.

The model (9) is

$$\hat\sigma_{t+1}^2=\beta_0+\beta_1\hat\sigma^2_{t}-\alpha\varepsilon_t,$$

with $\hat\sigma_{t}=r_t^2$, where $r_t$ are the returns. Hence realy the model is

$$r_{t+1}^2=\beta_0+\beta_1r^2_{t}-\alpha\varepsilon_t,$$

Here we have the problem that the article claims that this is ARMA(1,1) model which is not entirely correct, since term $\varepsilon_{t+1}$ is missing. Whereas the model (10) is

$$r_t=\mu+\varepsilon_t,$$

where $\varepsilon_t\sim N(0,h_t)$ and

$$h_t=\beta_0+\beta_1h_{t-1}+\alpha\varepsilon_{t-1}^2.$$

So although algebraically there are some similarities it is evident that models are different.

For one, the there is no guarantee that forecasted volatility would be positive for the first model (note that $\varepsilon_t$ is not squared in the model).

• The author probably missed the error term t+1 in the equation.Just for confirming ht is the same as sigma^2 right?ht is volatility? – ankc Oct 7 '13 at 18:37
• In this article no, these two quantities are different. – mpiktas Oct 8 '13 at 1:50