# 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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Yes you can: you're still getting a one day stddev estimate, which refers to the very same quantity, just obtained by another route with different precision and calculation method. How to project to 30 days is not affected from the origination of such quantity.

What instead would affect scaling are e.g. different distributional assumptions, or autocorrelation. See also Meucci 2011 - Visualizing the Propagation of Risk - Square-Root Rule, Covariances and Ellipsoids and Meucci 2010 - Annualization and General Projection of Skewness, Kurtosis and All Summary Statistics.

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).

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thanks for the clear answer – Python_Noob Jul 10 '13 at 12:24
Updated with some info on the context. – Quartz Jul 10 '13 at 12:39
how would daily sddev scale to the monthly equivalent with GARCH? We would no longer be talking about a normal distribution would we? – Python_Noob Jul 10 '13 at 12:54
I don't know if there could be an equivalent scaling in closed form. But you could always simulate the fitted GARCH and estimate (semi)empirically. – Quartz Jul 10 '13 at 17:11