You look at daily volatility and you want to turn it into an annual number. That is easy, just multiply by the square root of 252 and you are done. Well, we sort of know that this may not be exact, but we like the idea that there can be a quick adjustment, yet investor should understand what the risks are with making this simple transformation. See, "Converting 1-day volatility to h-day volatility: scaling by square root of h is worse than you think".
Whether you can scale volatility is based on the assumption that the daily returns are iid; that is, the series is independent and identically distributed random variables. If there is a trend component or mean-reversion, then the series is not iid and the scaling rule will not give you an appropriate estimate. In simple terms, if there is mean reversion than actual volatility will be lower than scaled volatility. The idea that actual volatility will be different from scaled volatility has been used as a test of market efficiency.
In the simple case, if some time series follows a GARCH process, then the scaling will not work. For example, if a series is GARCH (1,1), then volatility today will be related to past volatility and past errors. The memory will change the scale effect. This was explicitly modeled by Drost-Nijman who show that a short-term GARCH will scale to a GARCH process. You cannot avoid the memory.
The long and short of what this math says that temporal aggregation of a GARCH process may cause volatility fluctuations to decline while the scaling rule will magnify volatility fluctuations. Because you may get different quality volatility forecasts through scaling, it seems that if you want to look at longer volatility, then calculate the longer volatility and if possible, avoid scaling.
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