I'm interested in model a GARCH for a series. The original series is $y_t$ (price index of a stock market), which has a unit root. So I created the returns: $x_t = \ln(y_t) - \ln(y_{t-1})$.
Now, I'm confused about the fact of using $\lvert x_t\rvert$ for my GARCH. Why can I use absolute value, I'm thinking, that because I want to model volatility I'm just interested in how the series deviates from its mean in a period of time?
If $x_t$ is a series of logarithmic returns, the standard GARCH(1,1) model for it takes the form
$$ \begin{aligned} x_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 x_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \end{aligned} $$
where $\varepsilon_t$ is $i.i.d.(0,1)$ random variable. (It is straightforward to extend the model order to obtain GARCH($p$,$q$).) If you care about the volatility (measured by standard deviation or variance) of the returns, no absolute values get involved.
A GARCH model that involves absolute values is absolute value GARCH (AVGARCH or TS-GARCH) due to Taylor and Schwert (Bollerslev, 2009, p. 30) and is formulated as follows:
$$ \begin{aligned} x_t &= \sigma_t \varepsilon_t, \\ \sigma_t &= \omega + \alpha_1 |x_{t-1}| + \beta_1 \sigma_{t-1}. \end{aligned} $$
The model is less sensitive to large errors as compared to the regular GARCH model.
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