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you need to use the forecast for both the mean and sigma. It should look something like this: forecast = ugarchforecast(modelfit, n.ahead = 1, data = mydata); sigma(forecast); fitted(forecast) Then plug these values into the equation: \begin{align} \hat{VaR}_{0.99,T|T-1}&=\hat{\mu}_{T|T-1} + \hat{\sigma}_{T|T-1} * q_{0.99} \end{align} where $T$ is ...
The standard answer to your question would be to do the maximum likelihood estimation. When you say "plug in $\sigma$" you can show that the sample estimate of $\sigma$ is actually the maximum likelihood estimate of $\sigma$ for the normal distribution. If I can assume that your data are IID then what you do is use your distribution with parameters ...