Suppose I have a forecasted daily volatility for K days.
How can I get the forecasted monthly volatility from the daily ?
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If you assume the underlying time series $x_t$ is not autocorrelated (which is a reasonable assumption for daily financial returns), then $$ \text{Var}( x_{t+1} + \dotsc + x_{t+K} ) = \text{Var}(x_{t+1}) + \dotsc + \text{Var}(x_{t+K}) $$ and you can substitute forecasts for the theoretical quantities: $$ \widehat{\text{Var}}( x_{t+1} + \dotsc + x_{t+K} ) = \widehat{\text{Var}}(x_{t+1}) + \dotsc + \widehat{\text{Var}}(x_{t+K}). $$ You have your daily foreasts $\widehat{\text{Var}}(x_{t+1}), \dotsc, \widehat{\text{Var}}(x_{t+K})$ with $K$ around 22 for working days or 30 for calender days; this allows you to obtain the monthly forecast $\widehat{\text{Var}}( x_{t+1} + \dotsc + x_{t+K} )$.
Meanwhile, in presence of autocorrelation you would have $$ \text{Var}( x_{t+1} + \dotsc + x_{t+K} ) = \text{Var}(x_{t+1}) + \dotsc + \text{Var}(x_{t+K}) + \sum_{i=1}^K \sum_{j=1}^K \text{Cov}(x_{t+i},x_{t+j}) $$ and a corresponding expression for forecasts in places of theoretical quantities.
answered Aug 25, 2016 at 16:10
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$\begingroup$ Is this true also if the prediction were performed with a weighted system according to which more recent returns influence more?.(Like garch) $\endgroup$– DonbeoCommented Aug 25, 2016 at 16:12
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$\begingroup$ @Donbeo, The expression holds for variance forecasts in general. If you know that the forecasts are somehow biased or correlated, then you could probably try obtaining a more accurate combination than the above. But in a simple setting this holds for generic forecasts thus also GARCH $\endgroup$ Commented Aug 25, 2016 at 16:14