Why do top-down approaches produce biased coherent forecasts?

Question

The context is forecasting hierarchical time series. Section 10.4 of "Forecasting: Principles and Practice" (2nd edition) by Hyndman & Atahnasopoulos states:

One disadvantage of all top-down approaches, including this one, is that it does not produce unbiased coherent forecasts

It references Hyndman et al. "Optimal combination forecasts for hierarchical time series" (2011) (open access). The claim can be found on p. 4 (pp. 2582), but I do not see any proof. Let me quote:

If we assume that the base (independent) forecasts are unbiased (that is, $\mathbb{E}[\mathbf{\hat Y} _n(h)] = \mathbb{E}[\mathbf{Y}_n(h)]$), and that we want the revised hierarchical forecasts to also be unbiased, then we must require $\mathbb{E}[\mathbf{\tilde Y} _n(h)] = \mathbb{E}[\mathbf{Y} _n(h)] = \mathbf{S}\mathbb{E}[\mathbf{Y}_{K,n}(h)]$. Suppose $\mathbf{\beta}_n(h) = \mathbb{E}[\mathbf{Y}_{K,n+h}|\mathbf{Y}_1,\dots,\mathbf{Y}_n]$ is the mean of the future values of the bottom level $K$. Then $\mathbb{E}[\mathbf{\tilde Y}_n(h)] = \mathbf{SP}\mathbb{E}[\mathbf{\tilde Y}_n(h)] = \mathbf{SPS}\mathbf{\beta}_n(h)$. So, the unbiasedness of the revised forecast will hold provided \begin{align} \mathbf{SPS} = \mathbf{S} \tag{5} \end{align} This condition is true for the bottom-up method with $\mathbf{P}$ given by $(3)$. However, using the top-down method with $\mathbf{P}$ given by $(4)$, we find that $\mathbf{SPS} \neq \mathbf{S}$ for any choice of $\mathbf{p}$. So the top-down method can never give unbiased forecasts even if the base forecasts are unbiased.

(Italics are mine.) Could anyone provide the intuition behind the statement in italics and maybe a sketch of a proof?

Answer 1

Think about the simplest possible non-trivial hierarchy, with one aggregate series $y_t$, and two bottom-level disaggrete series $a_t$ and $b_t$, where $y_t = a_t+b_t$. Then $$ S = \begin{bmatrix}1 & 1\\1 & 0\\0 & 1\end{bmatrix} $$ Any top-down method will have $$ P = \begin{bmatrix}p & 0 & 0\\1-p & 0 & 0 \end{bmatrix} $$ Then $$ SPS = \begin{bmatrix}1 & 1 \\p & p\\1-p & 1-p\end{bmatrix} $$ which clearly can't be equal to $S$ for any value of $p$.

This example is easily expanded to more general collections of hierarchical time series.

Another way to think about this is to note that $SP$ is a projection matrix, and forecast reconciliation methods are designed to project the forecasts onto the coherent subspace which has the same dimension as the number of bottom-level series. But for the $P$ arising using a top-down approach, the projection is onto a 1-dimensional subspace.