Producing 'structurally reconciled' forecasts (e.g. revenue = price * volume) I recently learned about Hierarchical Forecasting and that this principle can be applied across hierarchies, cross-sectional groups or temporal aggregations. Independent Forecasts are created and the reconciled.
In my current problem, some hierarchies are present, but there are also structural relations between the time series, that should result in coherent forecasts as well. As a specific example there are revenue, price and volume time series. The revenue forecast should correspond to the product of the price and volume forecasts.
Either the revenue forecast fixed (top down) and price and volume should be reconciled with it or all three series are forecasted, but in a coherent manner. Both approaches would seem meaningful.
 A: This is an interesting question.
First off, if prices can be set, and all we are forecasting is unit sales and total revenue, then the problem is straightforward: simply use the prices in the summation matrix, instead of ones. Then, for instance, we can reconcile unit sales forecasts $\hat{y}_1$ and $\hat{y}_2$ of two different products with prices $p_1$ and $p_2$ with a total revenue forecast $\hat{r}$ by using the summation matrix
$$ S=\begin{pmatrix} 1 & 1 \\ p_1 & 0 \\ 0 & p_2 \end{pmatrix}. $$
Using similar logic, we can address "mixed" hierarchies, where some nodes stand for total unit sales, and others for total revenue. Or we could do multiple runs to do a scenario analysis of forecasts, for different price scenarios.
The problem is that this approach considers the entries in the summation matrix as fixed. They will not be changed during the reconciliation step.
One alternative would be to work on a log scale. Since revenue is price times unit sales, $r=py$, we have
$$ \log r = \log (py) = \log p+ \log y. $$
So we could, for instance, stack unit sales and price forecasts for a product, and reconcile those, on a log scale.
One problem with this is that we can't add the revenue from multiple products any more. Unless, that is, we are willing to subject the separate revenue forecasts (from a log reconciliation as above) and a total revenue forecast to its very own reconciliation step. And perhaps then plug the results back into the bottom level log reconciliation. It might be worthwhile trying this iterative approach and seeing whether it at least converges.
Incidentally, one advantage of the log approach is that we don't need to worry about negative prices or unit sales forecasts.
The other problem is that reconciliation may not make a lot of sense on a log scale. Point forecasting is equivalent to extracting a single one-number summary from the (frequently only implicit) full density forecast (Kolassa, 2020). The only common functional that is additive, and where reconciliation thus makes sense, is the expectation. As such, our reconciliation on a log scale will go a way towards getting us a reconciled unbiased forecast, but only on the log scale. But the back-transformation will break the sum consistency, whether we just use $\exp$ and get a biased back-transform, or include the bias correction by including an estimate of the variance.
It may well be that your underlying issue is best addressed through the kind of price scenario analysis described above.
