Empirical Daily Stock Volatility

Question

Currently I am very confused about the concept of daily volatility. First a quick background to the question. For a university paper I need to analyze several stock return volatilities (daily) using the GARCH Models. I understand that GARCH allows me to calculate a daily volatility based on the parameters and historical values. Nevertheless I would also like to compare the GARCH volatility estimations to the empirical volatilities.

The problem here is that I understand volatility as a measure of average squared deviation from the mean over a certain period. This would mean that in order to calculate volatility I would need a number of data points and not just one (daily closing prices). I have seen other papers compare GARCH volatility to the empirical volatility so I know that it is possible.

I just fail to understand how the empirical daily volatility based on one observation is calculated. If someone could explain this I would be very grateful!

Thanks

Answer 1

I understand volatility as a measure of average squared deviation from the mean over a certain period.

This sounds like conditional variance averaged over the chosen time period. Meanwhile, a GARCH model considers the conditional variance at a given time point (a period of one).

You can take the squared residual from your conditional mean model as an estimator of the conditional variance. Aside from any imprecisions of the conditional mean model, it will be an unbiased although noisy (high-variance) estimator of the true underlying conditional variance. (There is no point in talking about consistency of such an estimator, as the estimator uses a fixed sample size of one.)

What the other studies might be doing is using intraday data to produce realized volatility to which the fitted or predicted volatility from a GARCH model is compared.