What's the correct graphical comparison in a GARCH fit?

Question

Suppose that the stationary series $r_t$ is well fitted by an $ARMA(p,q)+c$ and $GARCH(r,s)$ model, where $GARCH(r,s) = \sigma_t ^2$

If in the testing sample I have to graphically compare the estimated $GARCH (r,s)$ with the actual conditional variance series in order to better visualize the goodness of fit, is it more useful (and maybe correct) directly comparing the $GARCH(r,s)$ series with the $r^2$ series (as actual conditional variance approximation), or the $\sigma_t = \sqrt{\sigma_t^2} = \sqrt{GARCH(r,s)}$ series with the absolute values of $r_t$ ? Or is it equal?

Answer 1

If the series has a significant ARMA structure on the conditional mean, if you want to evaluate the only GARCH specification assuming ARMA is fine, then you have to display GARCH estimates against the $(r - ARMA_{forecast})^2$ because via GARCH you are trying to estimate the conditional variance of return around a conditional mean represented by ARMA. So you can’t plot the square of $r$ series against the only GARCH structure because you are missing the conditional mean represented by ARMA. Remember that daily returns $r$ are generally representable as $r= \mu_r+ \sigma_r \cdot innov$ where $innov$ is assumed to be a standard normally distributed $N\left(0;1\right)$. ARMA is used for the conditional mean structure and GARCH for the conditional standard deviation. So your forecast is $r_{forecast}=ARMA+\sqrt{Garch}\cdot innov$, not $r_{forecast}=\sqrt{Garch} \cdot innov$ as you assume if you plot $r^2$ against GARCH.

However, in my answer here I am assuming that you are taking it for sure that your ARMA specification is correct: If you wish to test all your GARCH and ARMA specs at the same time (i.e. your full model), then you plot your $r_{forecasted}$ against $r$. Intuitively, we could say that ARMA will try to predict the sign and GARCH will try to predict the magnitude of the return.