Should I use GARCH volatility or standard deviation in cross-sectional regression?

Question

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?

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$$\text{Option 1}$$

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 1

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 2

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.