My student (co-supervised with Nikos Kourentzes), Kandrika F. Pritularga has written a paper on “Stochastic coherency in forecast reconciliation”, which has been recently published in International Journal of Production Economics (here it is). This paper contributes to the field of hierarchical forecasting, the main issue of which is that the forecasts produced on different levels of hierarchy typically do not agree with each other (lower levels do not add up to the higher ones). In order to resolve this issue, Hyndman et al. (2011) proposed a technique to reconcile the forecasts, making them coherent (meaning that they no longer have aforementioned issue and agree with each other on different levels). I do not want to go into technical details of this, I would refer readers to the post of Nikos Kourentzes on a related topic. The main idea behind this is to estimate a matrix that would reconcile forecasts produced on different levels of hierarchy.
And here is the main issue with this approach: it assumes that the estimated matrix does not have any variability (it is sort of known). This is obviously not true, and as the sample size changes, the values in the matrix would change as well. This implies that there is an uncertainty around the reconciliation mechanism and thus the forecasts produced using any reconciliation technique are not just coherent, they are “stochastically coherent” (the term proposed by Kandrika). This means that the coherency condition would change with the change of the sample size. This has serious implications, because the more complex the reconciliation technique is, the more variable forecasts will be due to increased estimation uncertainty, leading to more variable accuracy. We show in our paper that techniques, such as MinT shrinkage, will on average produce more accurate point forecasts at the cost of their increased variability (they will fail in some cases), while simpler reconciliation techniques, such as weighted least squares, would perform slightly worse in terms of average accuracy, but would also have lower variability (thus they will not fail as much as more complex approaches). This research shows that there is no “golden standard” in hierarchical forecasting and that the reconciliation technique should be selected based on the specific practical needs.