Why GARCH Is Fit to Returns Rather Than Absolute Returns

Summary

The document asks whether a GARCH model should be fitted to absolute log returns when the goal is to model volatility. It starts from a stock-market price index with a unit root and defines returns as differences in successive log prices. The answer clarifies that absolute returns or squared returns can be plotted as exploratory measures to look for volatility clustering, meaning intervals of relatively high or low volatility.

For fitting a standard GARCH model, the input remains the signed return series. The model’s likelihood procedure estimates the conditional variance from those returns; transforming the input to absolute values or squares is not the fitting step described here. The exchange gives a concise distinction between visual diagnosis and model estimation, but does not specify a GARCH variant, distributional assumptions, estimation details, or empirical evidence for a particular dataset.

Key ideas

• Log differences of prices are used as the return series when modeling with GARCH.
• Absolute or squared returns can help reveal volatility clustering in exploratory plots.
• A standard GARCH fit uses signed returns rather than absolute returns or squared returns as its input.
• The conditional variance is estimated as part of the model likelihood procedure.

Tags

# logarithm and absolut value in returns of stocks


# logarithm and absolut value in returns of stocks












Well, i'm interested in model a GARCH for a serie. The original serie is $y_t$ (price index of a Stock Market), which has a unit root. So i create 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 his mean in a period of time? Thank's a lot for the answers!

## Answer by simmy (score 0)

https://quant.stackexchange.com/a/26118

You can plot $\vert x_t \vert$ or $x^2$ just to see whether your data presents volatility clusters (periods in which volatility is high and other periods in which it is low). When you fit a GARCH model, you fit it simply on the $x_t$ series, not on the $\vert x_t \vert$ or $x^2$ series (as previously said in the comments): the likelihood algorithm will do the work.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.