  Many real-life applications involve simultaneously forecasting multiple time
series that are hierarchically related via aggregation or disaggregation
operations. For instance, commercial organizations often want to forecast
inventories simultaneously at store, city, and state levels for resource
planning purposes. In such applications, it is important that the forecasts, in
addition to being reasonably accurate, are also consistent w.r.t one another.
Although forecasting such hierarchical time series has been pursued by
economists and data scientists, the current state-of-the-art models use strong
assumptions, e.g., all forecasts being unbiased estimates, noise distribution
being Gaussian. Besides, state-of-the-art models have not harnessed the power
of modern nonlinear models, especially ones based on deep learning. In this
paper, we propose using a flexible nonlinear model that optimizes quantile
regression loss coupled with suitable regularization terms to maintain the
consistency of forecasts across hierarchies. The theoretical framework
introduced herein can be applied to any forecasting model with an underlying
differentiable loss function. A proof of optimality of our proposed method is
also provided. Simulation studies over a range of datasets highlight the
efficacy of our approach.
