Calculating forecast error for a long-term volatility forecast

#1
Hi, I am trying to calculate the forecast standard error for a long-term volatility forecast of a time series.

I am simply using the unbiased estimator of the historical volatility of the time series as my forecast. The historical time series is monthly frequency with approximately 360 data points and I would like to forecast future long-term volatility over the next 5-10 years (60-120 data points).

I believe the standard error of the monthly volatility estimator is \(\frac{1}{\sqrt{2T}}\sigma\) (where \(T\) is the total number of historical data points, and \(\sigma\) is the true historical monthly volatility.

My assumption is that I simply need to multiple this by \(\sqrt{S}\) (where \(S\) is the number of future data points for the forecast). Is this correct?