# 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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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?

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@Jase, care to explain the unselecting of the answer and chosing a different answer which only adds that Garch models help in forecasting when you clearly stated that you do not look to forecast? I do not have an issue with the fact that you are at liberty to chose the best answer for you, I just do not follow your rational to chose an answer that agrees with mine and adds the forecasting point which you made clear you do not look to perform plus me pointing out that my answer does not pertain to forecasting volatility. Confused!!! – Matt Wolf Dec 16 '12 at 9:08
add'l info: OP chose this answer then switched to a different one, was just curious of his/her thought process and was not trying to influence his final decision. – Matt Wolf Dec 16 '12 at 10:25
Freddy I have been thinking about this for a while. I am still not sure. The reason I switched answers originally is because Bob made me consider EWMA when I had previously not considered it, and also because I don't view computational burden as an issue because I'm using low frequency data (meaning that, in my perception, your resulting recommendation lacked any substance when it came to my specific application). – Jase Dec 16 '12 at 11:28
ok, but then maybe you want to consider including a description of your specific application next time so people can give you a more targeted answer. Also, you do not need to rush marking an answer if you are unsure. Just my 2 cents. – Matt Wolf Dec 16 '12 at 13:27
Yeah I should have. I think we've talked about it enough and we should get on with our lives :-). Thank you for your contribution (just because I changed correct answer doesn't mean I didn't find it helpful!) – Jase Dec 16 '12 at 13:47

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.
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