How to calculate forecast error (confidence intervals) for ongoing periods?

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I often need to forecast for future periods in monthly series of data.

Formulas are available to calculate the confidence interval at alpha for the next period in the time series, but this never includes how to treat the second period, and third, etc.

I’d visually imagine that if any forecast was graphed with upper and lower confidence intervals, generally those intervals should exponentially increase or decrease against the mean forecast, as uncertainty is a cumulative force.

Let’s say I had unit sale of
Apr = 10 May = 8 June = 11 July = 13
and no other context such as seasonality or population data

We need to forecast (albeit blindly) August, September, October.

What method would you use?
and more importantly here, how will you measure confidence for September and October?

Sorry that this might be a simple question for some experts – I have been digging far for a clear answer, and I’m sure this is something all amateurs like me would love to understand.

Answer

There are so many narrow aspects calculating prediction intervals: data generating process and the model used to described this process (time series model, regression model), is your data stationary (for this type your conclusion is wrong as stationary data is not tending to run far from its mean value) or explosive (for an integrated process you will see something that you described). I think that an excellent review by Chris Chatfield regarding Prediction Intervals will answer most of your questions.

Regarding unit sales:

  • since you have a short forecasting interval you may try to forecast by exponential smoothing (in R it is the ets() function from forecast)
  • another option would be to model it like ARIMA process (the same library has auto.arima())
  • in micro-econometrics, however, regression models are preferable to a-theoretic ones, but in the short run they not necessarily beat the first two

Both cases has formulas to calculate the prediction intervals and are discussed in above-mentioned review (commonly the normality of the residuals is assumed, but this is not a crucial assumption).

Attribution
Source : Link , Question Author : Nick , Answer Author : Dmitrij Celov

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