Is it possible to calculate the prediction intervals for top down, bottom up, and middle out reconciliation of hierarchical time series?

Question

I have read in several places (heres one) that we can not calculate prediction intervals for the classical reconciliation approaches, top down, middle out, and bottom up, and hence optimal reconciliation is more useful in that sense. But whats confusing to me is that I think I have seen the intervals in practice. SAS forecast studio for example provides forecasts using those techniques and they claim to provide the reconciled confidence intervals. So to be more specific the questions I have are:

• Is it possible to calculate the prediction interval for td, mo, and bu forecasts?
  • If so, how can I do it? (R code would be very helpful)
  • If not, what is meant when papers say there are no intervals with these methods?

Thanks in advance for any help that can be offered!

Answer 1

Yes, it is possible to produce prediction intervals for these old methods, using the approach described in our papers. A convenient textbook intro might be https://otexts.com/fpp3/reconciliation.html. The prediction intervals can be computed assuming a normal distribution with means equal to the point forecasts and variances obtained from the diagonal elements of the $V_h$ matrix.

The talk you reference was explaining that prediction intervals were not possible before this theory was developed.

I've no idea what SAS does to compute forecast intervals, but they may have implemented this approach.

You can get bottom up prediction intervals in R using the fable package as demonstrated in this example: https://otexts.com/fpp3/prison.html. Top-down and middle-out forecasts will also soon be available in the fable package in a future release.