The root mean squared error (RMSE) is considered by some to be the best measure of how good a variance estimate is. You often see it quoted as:
$RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^n(\hat{\sigma_i} - \sigma_i)^2}$
Where $\hat{\sigma}$ is the estimate of the volatility while $\sigma$ is the actual volatility.
My question is: what is $\sigma$ in this case? Suppose that $\hat{\sigma}$ is the prior day's realized volatility (i.e. $RV = \sum_{t=0}^N r_t$ where $r_t$ is the 5-minute return), is the actual volatility just the next day's absolute return?
