# How to measure the true underlying daily volatility from daily data?

I am looking at how well GARCH, GJR-GARCH and EGARCH capture the volatility dynamics before, during and after the financial crisis. I also compute out-of-sample forecasts and examine if there is one model that does well in all samples or not.

However, I'm kind of stuck in my search for a benchmark. I do the same analysis for stock returns and I use $y_t = y_{t-1} + \varepsilon_t$ as my benchmark. However, for volatility and I do not know what to use because I have daily data and I would like to stick with this frequency. I know that realized volatility is $$RV=\sum_{i=1}^tr_t^2$$ but I do not know how to create a plot/benchmark (like a random walk for stock returns) to compare with the models that I picked.

As far as I know, there is no solution to your problem. That is, technically it is impossible to measure the underlying daily variance when you only have one observation per day. You have a daily series of $T$ days where the variance may be different on each day and there may be no connection between the variance on day $t$ and the variance on day $s$ for any $(t,s)\in 1, \dots, T$ where $t\neq s$. Thus effectively you can treat your series as $T$ separate random variables, unless you are willing to assume some structure that connects them. And it is clearly impossible to measure the variance of a random variable from just one realization of that variable, without making further assumptions.
If you were willing to make some assumptions, you could start from assuming that the expectation is zero on each day. Then you would be technically able to calculate the sample variance which would be, on any day $t$, $$\widehat{Var}_t(x_t) = (x_t-0)^2 = x_t^2.$$ This would be a very noisy measure of variance, so it would not be a "good" proxy for the true variance. Unfortunately, you can only get this far without making further assumptions or collecting data at a higher frequency.