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If we model asset return volatility for periods of more than one (say more than one day) there is the square-root rule which holds true under some assumptions.

On the other hand practitioners sometimes use rolling, overlapping data. Treating them as if they were non-overlapping seems wrong to me (it is wrong) - but how wrong and how can the approach be fixed?

I heard about he following modelling approach: They take a sample of $1000$ daily observations (daily returns/percentage changes) and then they build rolling $180$ day returns. Finally they look at the empirical distribution function (edf) and empirical quantiles of these rolling/overlapping returns.

Mathematically they have $(r_i)_{i=1}^{1000}$ and then they look at $$y_1 = \sum_{i=1}^{180} r_i, \quad y_2 = \sum_{i=2}^{181} r_i, \quad y_3 = \sum_{i=3}^{182} r_i, \cdots $$

The sample of the $(y_i)_{i=1}^{820}$ is a set of strongly dependent random variables. What are the properties of its edf? How does it relate to the edf of the sample of $(r_i)_{i=1}^{1000}$?

As we speak about asset returns we can assume the $(r_i)_{i=1}^{1000}$ to be serially uncorrelated but not independent. This makes a rigorous treatment difficult.

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