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I analysing time-series to see the range of an asset. I have data for GBPUSD so i could only analyze this so far.

I have used both empirical rule and percentile. Of course they give different solutions. When i take a look at monthly data, i can forecast the minimum and maximum values using percentile. For example i can say GBPUSD will be between 1.27 - 1.35 by the end of the month (95th percentile). FYI: this is not a real forcast.

Assume that, by the 29th of the month, assume that GBPUSD is very close to 1.34. However our forecast was between 1.27 - 1.35. Now the odds are close to 1.35.

The confusing part is this one. Is 1.27 still 95th percentile? It looks like not even 1%. Do we have to calculate percentile day by day for such problems? Or using empirical rule (68-95-99.7), do we have to recalculate?

I am just training and working on economical dataset. This dataset can be anything else.

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Your problem is not specific to s. If you were calculating point mean forecasts for the end of the month, but then have values on the 29th, you could still try to update your forecasts.

The most natural thing would probably be to go to daily granularity in the first place. If you want month-end forecasts, then forecast enough days out. And update your model as you get new data.

Alternatively (e.g., if you don't have a regular time series of daily data), you could consider fitting and predicting a second "correction" model, which takes the original monthly forecasts and the intra-monthly movements, possibly also a term for how far into the month we are.

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