Forecast reconciliation: Methodological issues and applications

In recent years, there has been growing interest in the development of reconciliation techniques for forecasting multiple time series, both in academia and industries. Time series data are widely employed to predict the future behavior of phenomena such as retail sales, tourist flows, macroeconomic and climatic trends. However, forecasts of different series, linked by some constraints and derived from different sources, often exhibit discrepancies that compromise the reliability of the results. Forecast reconciliation is a post-forecasting process aimed at improving the quality of forecasts for a system of linearly constrained multiple time series. Despite notable progress, these approaches encounter various challenges, both methodological and practical. A significant issue lies in ensuring the non-negativity of reconciled forecasts, a critical consideration when forecasting variables that cannot assume negative values, such as sales or tourist flows. Another challenge involves extending reconciliation methods beyond genuine hierarchical/grouped time series, where the roles of variables are uniquely identified. This also includes more complex scenarios, combining the cross-sectional (involving multiple time series at the same frequency) and temporal (a single series at multiple frequencies) frameworks to obtain cross-temporally coherent forecasts. Additionally, dealing with large volumes of data and selecting efficient algorithms are crucial factors that must be addressed to ensure the effectiveness of reconciliation processes. This thesis aims to provide solutions, both from a methodological and computational point of view. To achieve this goal, each chapter of the thesis will present one or two specific applications, illustrating the effectiveness of the proposed methodologies in real-world contexts. Chapter 1 introduces the concept of simultaneous reconciliation of cross-sectional and temporal forecasts. This chapter presents two novel contributions: an expression for cross-temporal point forecasts and an iterative procedure that alternates reconciliation along a single dimension until convergence is achieved. Chapter 2 focuses on the application of forecast reconciliation of hierarchical photovoltaic (PV) power generation for a simulated PV dataset in California. Various aspects are investigated, including the including the non-negativity problem and forecast coherency through cross-temporal reconciliation approaches. Chapter 3 extends the approach based on forecast combination for reconciling a hierarchy, known as LCC. This method minimizes a quadratic loss function subject to an exogenous constraint imposed by the base forecast of the higher-level series within the hierarchy, whose value remains unchanged. An alternative approach is proposed, which modifies this constraint by incorporating endogenous constraints within the same context. Chapter 4 explores the reconciliation of generally constrained multiple time series, thereby extending the results obtained thus far, which have been limited to genuine hierarchical structures. Chapter 5 extends the cross-sectional probabilistic reconciliation approach to the cross-temporal framework. This chapter introduces both parametric and non-parametric approaches to draw base forecasts samples. Finally, in Chapter 6, the reconciliation of realized volatility forecasts using intra-day decompositions is addressed for the first time in a financial framework.