A hybrid approach with step-size aggregation to forecasting hierarchical time series

Hierarchical time series arise in various fields such as manufacturing and services when the products or services can be hierarchically structured. “Top-down” and “bottom-up” forecasting approaches are often used for forecasting such hierarchical time series. In this paper, we develop a new hybrid approach (HA) with step-size aggregation for hierarchical time series forecasting. The new approach is a weighted average of the two classical approaches with the weights being optimally chosen for all the series at each level of the hierarchy to minimize the variance of the forecast errors. The independent selection of weights for all the series at each level of the hierarchy makes the HA inconsistent while aggregating suitably across the hierarchy. To address this issue, we introduce a step-size aggregate factor that represents the relationship between forecasts of the two consecutive levels of the hierarchy. The key advantage of the proposed HA is that it captures the structure of the hierarchy inherently due to the combination of the hierarchical approaches instead of independent forecasts of all the series at each level of the hierarchy. We demonstrate the performance of the new approach by applying it to the monthly data of ‘Industrial’ category of M3-Competition as well as on Pakistan energy consumption data.