I have made a combined forecast of a univariate time series. I used ETS, ARIMA and STL and take a simple average of these means and get a combined point forecast 12 months ahead. My question is. How to calculate a 95% prediction interval for the combined point forecast?
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One way is working out what the probability distribution of the mean exactly, which can sometimes be quiet difficult.
Another way is assuming that the mean is approximately gaussian due to the central limit theorem. Because N=3 and because you don't know the true variance of your sample, you can use a t-distribution.
In practice, calculate the mean and sample standard deviation, use N-1 degrees of freedom on a t-distribution percent point function (python/scipy has st.t.interval) and compute the 95% C.I. for your mean.
Source: https://en.wikipedia.org/wiki/Student%27s_t-distribution
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$\begingroup$ Thank you for answer. I am wondering whether a simple average of the 3 prediction intervals are appropriate as a combined prediction interval for the combined point forecast $\endgroup$– HdmasMay 5, 2020 at 11:26