# 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 $E[\sigma_t^2] = \omega + k_1 E[\epsilon_{t-1}^2] + k_2 E[\sigma_{t-1}^2]$

$\implies E[\sigma_t^2] = \omega + k_1 E[\sigma_t^2] + k_2 E[\sigma_t^2]$

$\implies \boxed{E[\sigma_t^2] = \frac{\omega}{1-k_1-k_2}}$

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