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

TL;DR

This paper develops a new structural-like forecast reconciliation method for linearly constrained multiple time series, enabling coherent point and probabilistic forecasts, demonstrated on GDP data for Australia and Europe.

Contribution

It introduces a generalized reconciliation formula for linearly constrained series, extending existing structural approaches to broader data structures.

Findings

Successfully reconciled Australian GDP forecasts from income and expenditure data.

Achieved coherent European GDP forecasts disaggregated by sectors and countries.

Extended forecast reconciliation theory to general linear constraints.

Abstract

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.