(G)ARCH: Squared Residuals vs Absolute Residuals I estimated a GARCH model to forecast the variance of a variable conditional on past information. I evaluated the forecast by comparing the squared forecast error, i.e. the squared value by which the conditional mean model forecast missed the actual value, with the forecasted conditional variance.
It turned out to be a terrible forecast on average (I did this for a multitude of time-series, i.e. I am working in a panel setting). Specifically, a small number of extremely bad forecasts in terms of the aforementioned evaluation appear to "make it bad" on average.
I then used the absolute values of the residuals as my dependent GARCH model variable instead of squared values, as the classic GARCH approach suggests. I compared the forecast values resulting from this model with the absolute value by which the conditional mean model forecast missed the actual value. It turned out to be much better on average.
Now my question is: Is it generally fine to use absolute values of the residuals instead of squared values in a GARCH model? From my understanding, instead of modelling the conditional variance one would then simply model the conditional standard deviation. Is this correct? Thank you in advance.
 A: A standard GARCH model would be something like
\begin{aligned}
x_t &= \mu_t+\varepsilon_t, \\
\varepsilon_t &= \sigma_t z_t, \\
\sigma_t^2 &= \omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2, \\
z_t &\sim i.i.d.(0,1)
\end{aligned}
for some conditional mean $\mu_t$, e.g. $\mu_t=\mu$ or $\mu_t=\varphi_1 x_{t-1}$.
You substitute $\sigma_t^2=\omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2$ with $\sigma_t=\omega+\alpha_1|\varepsilon_{t-1}|+\beta_1\sigma_{t-1}$, and this is fine.
Regardless of which model you use, when you are evaluating it, you will likely get a better fit from regressing the fitted (or predicted) $\hat\sigma_t$ on the absolute valus of residuals (or prediction errors) $|\hat\varepsilon_t|$ than from regressing $\hat\sigma_t^2$ on $\hat\varepsilon_t^2$.* If you are going to compare different models, make sure to compare fit in an apples-to-apples way. What you should not do is compare the fit of $\hat\sigma_t$ on $|\hat\varepsilon_t|$ from one model with the fit of $\hat\sigma_t^2$ on $\hat\varepsilon_t^2$ from another model; that would be an apples-to-oranges type of comparison.
*There are also other ways of evaluating the fit of the models. Proper scoring rules is a relevant keyword.
