Say one has finished estimating a correctly specified GARCH(1,1) on a daily time series and now wants to evaluate the accuracy of the one step ahead forecasts what steps or tests could one do?
I understand that the MSE
$$MSE = \frac{1}{N} RSS = \frac{1}{N} \sum (\hat{\sigma}_i -\sigma_i)^2$$
can be computed where $N$ is the number of samples and $\hat{\sigma}_i$ is the estimated one step ahead volatility. Because we do not know the realized volatility $\sigma_i$ we can use the squared return of that day as proven here.
But is the one step ahead predictor not already defined as the value $\hat{\sigma}$ of the volatility that minimizes the MSE? So why compute this measure if it going to be the minimum across models anyway?
Moreover do I understand correctly that the practical way to compute the $MSE$ for the one step ahead forecast is:
- Estimate the correctly specified model on your data (returns) except the last data point.
- Compute the one step ahead forecast of your model.
- Repeat step 1 but take away an other data point.
Doing this process $N$ times one obtains $N$ $\hat{\sigma}_i$ that can then by utilized in the given formula. Am I getting this right?
Also I understand from this paper (Bollerslev 1998) that utilizing the squared daily return to approximate the realized volatility leads to noise. For example a better estimate of realized daily volatility would be the sum of 30 minutes squared returns of that day. Is this correct?
So to get a "better" $MSE$ I could substitute every $\sigma_i$ with the sum of 30 minutes squared returns of that day instead of simply the daily squared return? Is this how you use realized volatility to evaluate the goodness of your forecasts?
I will end this rambling by asking for a good reference in evaluating the accuracy of the forecasts using realized volatility because it is obvious that I am very confused. I can't seem to find a good text on my own.
Because I have asked a lot of questions in a confused fashion I will put a bounty of 50 reputation so you might want to wait to answer (but you could comment to give me a reference straight away if you happen to know one).