In this paper, we propose a forecast reconciliation approach that can keep the base forecasts of specific levels or multiple nodes from different levels immutable after reconciliation. The proposed method is flexible and general enough to allow for expert judgement in choosing the immutable series. We prove that the proposed method can produce unbiased reconciled forecasts as long as the base forecasts are unbiased, and the equality constraints do not go beyond the boundary conditions.
Monte Carlo simulations and two empirical applications show the superiority of the proposed method over unconstrained forecast reconciliation. In particular, constrained forecast reconciliation shows promising results when series are noisy, and decision-makers have limited knowledge about the underlying data generating processes of lower levels in the hierarchy. The application to sales data from a major Chinese online retailer shows the potential of the proposed method in reconciling the forecasts of a high-dimensional hierarchy where careful judgement is used in selecting immutable time series.
There are several valuable directions worthy of further investigation. First, while we show results for cross-sectional scenarios we assert that the framework is suitable for both cross-sectional and temporal hierarchies. As such, the framework can also be extended into the cross-temporal data as in Kourentzes & Athanasopoulos (2019). Second, we select the set of immutable series based on features about the time series (their intermittence, and the length of training data). It is worth exploring whether the theoretical properties of the time series can be used to automate this process. Third, while the proposed approach focuses on point forecast reconciliation, reconciling probabilistic forecasts with some immutable forecasts opens another avenue for future research. Due to the diversity between different forecasting methods, and the heterogeneity across different datasets and application contexts, such extensions have the potential to make our proposed constrained forecast reconciliation a useful tool to improve forecast accuracy in a number of key applications.