  The practical importance of coherent forecasts in hierarchical forecasting
has inspired many studies on forecast reconciliation. Under this approach,
so-called base forecasts are produced for every series in the hierarchy and are
subsequently adjusted to be coherent in a second reconciliation step.
Reconciliation methods have been shown to improve forecast accuracy, but will,
in general, adjust the base forecast of every series. However, in an
operational context, it is sometimes necessary or beneficial to keep forecasts
of some variables unchanged after forecast reconciliation. In this paper, we
formulate reconciliation methodology that keeps forecasts of a pre-specified
subset of variables unchanged or "immutable". In contrast to existing
approaches, these immutable forecasts need not all come from the same level of
a hierarchy, and our method can also be applied to grouped hierarchies. We
prove that our approach preserves unbiasedness in base forecasts. Our method
can also account for correlations between base forecasting errors and ensure
non-negativity of forecasts. We also perform empirical experiments, including
an application to sales of a large scale online retailer, to assess the impacts
of our proposed methodology.

Methodology;Econometrics;Machine Learning