Comparing GARCH Variance Fits with ARMA Conditional Means
Summary
The document explains how to visualize a GARCH variance estimate when returns also have an ARMA conditional mean. To assess the GARCH component while treating the ARMA specification as given, compare the estimated conditional variance with squared mean-adjusted returns: the observed return minus its ARMA forecast, squared. Raw squared returns include variation around zero as well as the conditional mean, so they do not isolate the residual variance that GARCH is intended to model. The answer describes ARMA as modeling the mean and GARCH as modeling conditional standard deviation.
The comparison depends on the evaluation goal. If the ARMA mean model is assumed correct and the aim is to inspect GARCH fit, use squared residuals against the variance estimate. If the aim is to assess the joint model, compare full return forecasts with observed returns. The document does not prescribe a formal goodness-of-fit test or discuss noisy variance proxies, so these plots are diagnostic rather than conclusive evidence of model adequacy.
Key ideas
- GARCH models the conditional variance around the conditional mean specified by the ARMA component.
- For a GARCH-focused diagnostic, compare variance estimates with squared ARMA residuals.
- Raw squared returns can mix conditional-mean effects with residual variance.
- For joint assessment, compare forecasts from the full ARMA-GARCH model with observed returns.
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# What's the correct graphical comparison in a GARCH fit?
# What's the correct graphical comparison in a GARCH fit?
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 by Fr1 (score 0, accepted)
https://quant.stackexchange.com/a/41522
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.