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Forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with given data structures. Most of the point and probabilistic regression-based forecast reconciliation results ground on the so called "structural representation" and on the related unconstrained generalized least squares reconciliation formula. However, the structural representation naturally applies to genuine hierarchical/grouped time series, where the top- and bottom-level variables are uniquely identified. When a general linearly constrained multiple time series is considered, the forecast reconciliation is naturally expressed according to a projection approach. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, so far it is not completely understood if and how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. Such an expression would permit to extend reconciliation definitions, theorems and results in a straightforward manner. In this paper, we show that for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a "structural-like" approach that keeps distinct free and constrained, instead of bottom and upper (aggregated), variables, establish the probabilistic forecast reconciliation framework, and apply these findings to obtain fully reconciled point and probabilistic forecasts for the aggregates of the Australian GDP from income and expenditure sides, and for the European Area GDP disaggregated by income, expenditure and output sides and by 19 countries. [ABSTRACT FROM AUTHOR]

In recent works by Yang et al. (2017a, 2017b), and Yang et al. (2019), geographical, temporal, and sequential deterministic reconciliation of hierarchical photovoltaic (PV) power generation have been considered for a simulated PV dataset in California. In the first two cases, the reconciliations were carried out in spatial and temporal domains separately. To further improve forecasting accuracy, in the third case these two reconciliation approaches were applied sequentially. During the replication of the forecasting experiment, some issues emerged about non-negativity and coherency (in space and/or in time) of the sequentially reconciled forecasts. Furthermore, while the accuracy improvement of the considered approaches over the benchmark persistence forecasts is clearly visible at any data granularity, we argue that an even better performance may be obtained by a thorough exploitation of spatio-temporal hierarchies. To this end, in this paper the spatio-temporal point forecast reconciliation approach is applied to generate non-negative, fully coherent (both in space and time) forecasts. New spatio-temporal reconciliation approaches are adopted, exploiting for the first time some relationships between two-step, iterative and simultaneous spatio-temporal reconciliation procedures. Non-negativity issues of the final reconciled forecasts are discussed and correctly dealt with in a simple and effective way. The spatio-temporal reconciliation procedures are applied to the base forecasts with forecast horizon of 1 day, of PV generated power at different time granularities (1 h to 1 day), of a geographical hierarchy consisting of 324 series along 3 levels. The normalized Root Mean Square Error (nRMSE) and the normalized Mean Bias Error are used to measure forecasting accuracy, and a statistical multiple comparison procedure is performed to rank the approaches. In addition to assuring full coherence and non-negativity of the reconciled forecasts, the results show that for the considered dataset, spatio-temporal forecast reconciliation significantly improves on the sequential procedures proposed by Yang et al. (2019), at any level of the spatial hierarchy and for any temporal granularity. For example, the forecasted hourly PV generated power by the new spatio-temporal forecast reconciliation approaches improve on the NWP 3TIER forecasts in a range from 4.7% to 18.4% in terms of nRMSE. • New approach to produce fully coherent solar forecasts both in space and time. • Complete previous discussions on deterministic solar forecast reconciliation. • Avoid negative forecasts and preserve full spatio-temporal coherence. • Significant improvement of the forecast accuracy through hierarchical forecasting. • Forecast skills over NWP hourly forecasts between 4.7% and 18.4%. [ABSTRACT FROM AUTHOR]