Choosing Historical or GARCH Volatility for Cross-Sectional Models
Summary
The document compares rolling historical standard deviation with a GARCH(1,1) volatility estimate for use as a regressor in a cross-sectional study. GARCH models conditional variance as a function of a baseline level, recent squared residuals, and prior conditional variance. Under stationarity, its long-run expected variance is determined by the baseline and the persistence parameters, provided their sum is below one.
The accepted response favors simple standard deviation when the aim is to describe historical data rather than forecast, citing lower computational cost. Another response cautions that neither measure is universally superior: modeling persistent heteroskedasticity may help when it is present, and an exponentially weighted moving average is mentioned as another option. The discussion is brief and offers no empirical comparison, so the appropriate estimator depends on the study's goal, volatility dynamics, and whether prediction is involved.
Key ideas
- Historical standard deviation summarizes dispersion over a chosen sample window.
- GARCH(1,1) models conditional variance using recent squared residuals and prior variance.
- The long-run expected GARCH variance has a finite expression when the persistence parameters sum to less than one.
- Simple standard deviation may be practical for historical description because it is less computationally intensive.
- GARCH or EWMA may be considered when volatility clustering matters, especially for forecasting.
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Full text
# Should I use GARCH volatility or standard deviation in cross-sectional regression?
# Should I use GARCH volatility or standard deviation in cross-sectional regression?
I want to do a cross-sectional study where the historical, medium-long run volatility of some return series (call it $R_t$) is included as a regressor. Which of the following two estimates of volatility is superior in this context?
$$\text{Option 1}$$
Of course, the simple standard deviation of historical returns over some window.
$\boxed{\text{std.dev.}(R_t) = \sqrt{E[(R_t-E[R_t])^2]}}$
$$\text{Option 2}$$
Let's set up the GARCH(1,1) as an example of an alternative;
- Mean equation:
$R_t = \mu + \epsilon_t$
$\epsilon_t = z_t \sigma_t$
$z_t \sim N(0,1)$, $\epsilon_t \sim N(0,\sigma_t)$
- Variance equation:
$\sigma_t^2 = \omega + k_1 \epsilon_{t-1}^2 + k_2 \sigma_{t-1}^2$
Then we have that $E[\sigma_t^2] = \omega + k_1 E[\epsilon_{t-1}^2] + k_2 E[\sigma_{t-1}^2]$
$\implies E[\sigma_t^2] = \omega + k_1 E[\sigma_t^2] + k_2 E[\sigma_t^2]$
$\implies \boxed{E[\sigma_t^2] = \frac{\omega}{1-k_1-k_2}}$
## Answer by Matt Wolf (score 2, accepted)
https://quant.stackexchange.com/a/4722
I would recommend to use simple standard deviation (among the 2 options you offered). You are performing time series analysis of historical data points, you are not forecasting. Thus, why exposing yourself to a much more computationally intensive method?
May I also point you to a related (not duplicate) thread: Why are GARCH models used to forecast volatility if residuals are often correlated?
## Answer by Bob Jansen (score 1)
https://quant.stackexchange.com/a/4726
Neither of the options is strictly superior over the other. I agree with Freddy about the disadvantages of GARCH. On the other hand, correcting for heteroskedasticity can help your model and forecasts* if it is present and persistent. Whether GARCH is your best choice is debatable. You could look at other sources to determine the volatility or, as an option 3, use EWMA on the data you already have to estimate volatility.
- I assume you want to do forecasts at some point.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.