Point and probabilistic forecast reconciliation for general linearly constrained multiple time series
Time series can often be naturally disaggregated by various nested and/or crossed attributes of interest (Hyndman and Athanasopoulos, 2021, ch. 11). As an example, sales data can be disaggregated by product categories, and then by product subcategories, down to Stock Keeping Unit (SKU). Alternatively, sales data can be disaggregated by geographic divisions. Following Panagiotelis et al. (2020a), a hierarchical/grouped time series is a linearly constrained multiple time series consisting of a collection of time series that follows one or more hierarchical aggregation structure. 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 (firstly shown by Athanasopoulos et al., 2009). 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, Di Fonzo and Girolimetto, 2021) gives a general solution. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, we show how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. To do this, we describe how constrained multiple time series may be represented by a small number of basic variables that take the role of the bottom variables in the specific hierarchical/grouped framework. These tools would permit us to extend definitions, theorems and results found by Panagiotelis et al. (2020b) for probabilistic forecast reconciliation in a rather straightforward manner. To apply the theoretical framework, we propose a complete reconciliation procedure of probabilistic GDP forecasts, resulting in GDP forecasts coherent with both Income and Expenditure sides’ forecasted series (‘one number forecast’). The forecasting performance is evaluated on the Australian quarterly GDP series, as compared to the original proposal by Athanasopoulos et al. (2020). Our approach is applied within the same forecasting experiment, that considers ARIMA base forecasts for each time series from 1 quarter ahead up to 4 quarters ahead, using an expanding window. The base forecasts are reconciled using the R-package FoReco (Girolimetto and Di Fonzo, 2021). In general, the simultaneously reconciled probabilistic forecasts give results as good as those of Athanasopoulos et al. (2020). In addition, the newly proposed approach produces forecasts that are fully coherent with all economic constraints coming from National Accounts relationships, avoiding annoying discrepancies between the GDP forecasts from either Income or Expenditure sides.
References
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Girolimetto, D., Di Fonzo, T. (2022). FoReco: Point Forecast Reconciliation. https://cran.r-project.org/package=FoReco
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