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Forecast reconciliation is the post-forecasting process aimed to revise a setof incoherent base forecasts into coherent forecasts in line with given datastructures. Most of the point and probabilistic regression-based forecastreconciliation results ground on the so called "structural representation" andon the related unconstrained generalized least squares reconciliation formula.However, the structural representation naturally applies to genuinehierarchical/grouped time series, where the top- and bottom-level variables areuniquely identified. When a general linearly constrained multiple time seriesis considered, the forecast reconciliation is naturally expressed according toa projection approach. While it is well known that the classic structuralreconciliation formula is equivalent to its projection approach counterpart, sofar it is not completely understood if and how a structural-like reconciliationformula may be derived for a general linearly constrained multiple time series.Such an expression would permit to extend reconciliation definitions, theoremsand results in a straightforward manner. In this paper, we show that forgeneral linearly constrained multiple time series it is possible to express thereconciliation formula according to a "structural-like" approach that keepsdistinct free and constrained, instead of bottom and upper (aggregated),variables, establish the probabilistic forecast reconciliation framework, andapply these findings to obtain fully reconciled point and probabilisticforecasts for the aggregates of the Australian GDP from income and expendituresides, and for the European Area GDP disaggregated by income, expenditure andoutput sides and by 19 countries.

Point and probabilistic forecast reconciliation for general linearly constrained multiple time series