Likelihood-based inference in temporal hierarchies

We consider the importance of correctly specifying the variance–covariance matrix to allow information to be shared between aggregation levels when reconciling forecasts in a temporal hierarchy. We propose a novel framework for parametric modelling of the variance–covariance matrix, along with an iterative algorithm for maximum likelihood estimation. The covariance between aggregation levels can be modelled by aggregating the lower-level errors and disaggregating information from the higher levels. Using the likelihood approach, statistical inference can be applied to identify a parsimonious parametric structure for the variance–covariance matrix. We test and discuss different structures for how forecast errors are connected across aggregation levels and present a framework for simplifying these structures using Wald and likelihood-ratio tests. We evaluate the proposed method in a simulation study and through an application to day-ahead electricity load forecasting and find that it performs well compared to optimal shrinkage estimation.