Point and probabilistic forecast reconciliation for general linearly constrained multiple time series
Hierarchical forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with cross-sectional/temporal/cross-temporal data structure. In both theoretical and empirical frameworks, most of the point and probabilistic hierarchical forecast reconciliation results move from the classic reconciliation formula valid for the structural representation of a hierarchical time series (Hyndman and Athanasopoulos, 2021, ch. 11). However, this formula holds for genuine hierarchical/grouped time series, sharing both the top and the bottom level variables. When a general linearly constrained multiple time series is considered, the projection approach reconciliation formula (van Erven and Cugliari, 2015) gives a general solution. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, it is not obvious if and how a structural-like reconciliation formula may be derived for a general, not genuinely hierarchical time series. Such an expression would permit to extend definitions, theorems and results found by Panagiotelis et al. (2020) for probabilistic forecast reconciliation in a rather straightforward manner.In this paper, we show that even for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a structural approach that keeps distinct free and basic, instead of bottom and upper (aggregated), variables. Then, we extend the definition of probabilistic forecast reconciliation to a general linearly constrained multiple time series. Finally, we apply the results to obtain a ‘one number forecast’ for the Australian GDP from Income and Expenditure Sides, in both point and probabilistic settings.References
Hyndman, R.J., Athanasopoulos, G. (2021), Forecasting: principles and practice (3rd ed.), Melbourne, OTexts.
Panagiotelis, A., Gamakumara, P., Athanasopoulos, G., Hyndman, R.J. (2020), Probabilistic forecast reconciliation: properties, evaluation and score optimisation, Monash University, Department of Econometrics and Business Statistics, Working Paper 26/20.
Van Erven, T., Cugliari, J. (2015), Game-theoretically Optimal Reconciliation of Contemporaneous Hierarchical Time Series Forecasts, in Antoniadis, A., Poggi, J.M., Brossat, X. (eds.), Modeling and Stochastic Learning for Forecasting in High Dimensions, Berlin, Springer, 297-317.