# Scaling a normal distribution while using EWMA

I have a time series of daily data and am assuming each point in the time series is normally distributed. If I have a distribution of the daily data and want to scale this to cover a month (30 days) I believe I just scale the standard deviation by square root of 30. I think this assumes the data points are independently distributed. If I use an exponentially weighted moving average to calculate the standard deviation can I still scale the daily data to monthly by multiplying by square root 30?

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On the other hand, consider why are you using EWMA: you probably have reasons to think that your variance is not constant. If variance was constant then you should use all data with constant weighting to get best statistical efficiency. Therefore you would be better off fitting a heteroschedastic model such as GARCH and then considering its forecast distribution at 30 days, which in this case would NOT be just the $\sqrt{T}$ extrapolation from the 1-day variance, but would also include additional uncertainty in the future variance evolution (plus other components too).