  When forecasting time series with a hierarchical structure, the existing
state of the art is to forecast each time series independently, and, in a
post-treatment step, to reconcile the time series in a way that respects the
hierarchy (Hyndman et al., 2011; Wickramasuriya et al., 2018). We propose a new
loss function that can be incorporated into any maximum likelihood objective
with hierarchical data, resulting in reconciled estimates with confidence
intervals that correctly account for additional uncertainty due to imperfect
reconciliation. We evaluate our method using a non-linear model and synthetic
data on a counterfactual forecasting problem, where we have access to the
ground truth and contemporaneous covariates, and show that we largely improve
over the existing state-of-the-art method.
