Optimal hierarchical EWMA forecasting

Prediction of demand at different levels of aggregation is a crucial task in many business and industrial activities. This task may be extremely challenging when the number of time series increases together with the number of parameters governing the dynamics of the underlying model. This paper proposes theoretical and empirical contributions providing practical tools for managers needing efficient, flexible, and timely instruments. We first derive optimal results for predicting a system of time series following multivariate Exponentially Weighted Moving Average (EWMA) dynamics. Our results have relevant practical consequences. Indeed, we propose a fast EM algorithm that maximizes the Gaussian multivariate likelihood regardless of the model's dimension. Secondly, we show optimal results for the hierarchies, deriving closed-form results for the underlying parameters. Finally, using more than one hundred Walmart sales time series, we show that our approach is competitive with the optimal forecast reconciliation approach based on univariate forecasts.