  In a recent paper, while elucidating the links between forecast combination
and cross-sectional forecast reconciliation, Hollyman et al. (2021) have
proposed a forecast combination-based approach to the reconciliation of a
simple hierarchy. A new Level Conditional Coherent (LCC) point forecast
reconciliation procedure was developed, and it was shown that the simple
average of a set of LCC, and bottom-up reconciled forecasts (called Combined
Conditional Coherent, CCC) results in good performance as compared to those
obtained through the state-of-the-art cross-sectional reconciliation
procedures. In this paper, we build upon and extend this proposal along some
new directions. (1) We shed light on the nature and the mathematical derivation
of the LCC reconciliation formula, showing that it is the result of an
exogenously linearly constrained minimization of a quadratic loss function in
the differences between the target and the base forecasts with a diagonal
associated matrix. (2) Endogenous constraints may be considered as well,
resulting in level conditional reconciled forecasts of all the involved series,
where both the upper and the bottom time series are coherently revised. (3) As
the LCC procedure does not guarantee the non-negativity of the reconciled
forecasts, we argue that - when non-negativity is a natural attribute of the
variables to be forecast - its interpretation as an unbiased top-down
reconciliation procedure leaves room for some doubts. (4) The new procedures
are used in a forecasting experiment on the classical Australian Tourism Demand
(Visitor Nights) dataset. Due to the crucial role played by the (possibly
different) models used to compute the base forecasts, we re-interpret the CCC
reconciliation of Hollyman et al. (2021) as a forecast pooling approach,
showing that accuracy improvement may be gained by adopting a simple forecast
averaging strategy.
