  Forecast reconciliation is a post-forecasting process aimed to improve the
quality of the base forecasts for a system of hierarchical/grouped time series
(Hyndman et al., 2011). Contemporaneous (cross-sectional) and temporal
hierarchies have been considered in the literature, but - except for Kourentzes
and Athanasopoulos (2019) - generally these two features have not been fully
considered together. Adopting a notation able to simultaneously deal with both
forecast reconciliation dimensions, the paper shows two new results: (i) an
iterative cross-temporal forecast reconciliation procedure which extends, and
overcomes some weaknesses of, the two-step procedure by Kourentzes and
Athanasopoulos (2019), and (ii) the closed-form expression of the optimal (in
least squares sense) point forecasts which fulfill both contemporaneous and
temporal constraints. The feasibility of the proposed procedures, along with
first evaluations of their performance as compared to the most performing
`single dimension' (either cross-sectional or temporal) forecast reconciliation
procedures, is studied through a forecasting experiment on the 95 quarterly
time series of the Australian GDP from Income and Expenditure sides considered
by Athanasopoulos et al. (2019).
