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We address the construction of Realized Variance (RV) forecasts by exploiting the hierarchical structure implicit in available decompositions of RV. By using data referred to the Dow Jones Industrial Average Index and to its constituents we show that exploiting the informative content of hierarchies improves the forecast accuracy. Forecasting performance is evaluated out-of-sample based on the empirical MSE and QLIKE criteria as well as using the Model Confidence Set approach.

Forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with given data structures. Most of the point and probabilistic regression-based forecast reconciliation results ground on the so called "structural representation" and on the related unconstrained generalized least squares reconciliation formula. However, the structural representation naturally applies to genuine hierarchical/grouped time series, where the top- and bottom-level variables are uniquely identified. When a general linearly constrained multiple time series is considered, the forecast reconciliation is naturally expressed according to a projection approach. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, so far it is not completely understood if and how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. Such an expression would permit to extend reconciliation definitions, theorems and results in a straightforward manner. In this paper, we show that for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a "structural-like" approach that keeps distinct free and constrained, instead of bottom and upper (aggregated), variables, establish the probabilistic forecast reconciliation framework, and apply these findings to obtain fully reconciled point and probabilistic forecasts for the aggregates of the Australian GDP from income and expenditure sides, and for the European Area GDP disaggregated by income, expenditure and output sides and by 19 countries.

Forecast reconciliation is a post-forecasting process that involves transforming a set of incoherent forecasts into coherent forecasts which satisfy a given set of linear constraints for a multivariate time series. In this paper we extend the current state-of-the-art cross-sectional probabilistic forecast reconciliation approach to encompass a cross-temporal framework, where temporal constraints are also applied. Our proposed methodology employs both parametric Gaussian and non-parametric bootstrap approaches to draw samples from an incoherent cross-temporal distribution. To improve the estimation of the forecast error covariance matrix, we propose using multi-step residuals, especially in the time dimension where the usual one-step residuals fail. To address high-dimensionality issues, we present four alternatives for the covariance matrix, where we exploit the two-fold nature (cross-sectional and temporal) of the cross-temporal structure, and introduce the idea of overlapping residuals. We assess the effectiveness of the proposed cross-temporal reconciliation approaches through a simulation study that investigates their theoretical and empirical properties and two empirical forecasting experiments, using the Australian GDP and the Australian Tourism Demand datasets. For both applications, the optimal cross-temporal reconciliation approaches significantly outperform the incoherent base forecasts in terms of the Continuous Ranked Probability Score and the Energy Score. Overall, our study expands and unifies the notation for cross-sectional, temporal and cross-temporal reconciliation, thus extending and deepening the probabilistic cross-temporal framework. The results highlight the potential of the proposed cross-temporal forecast reconciliation methods in improving the accuracy of probabilistic forecasting models.

In recent works by Yang et al. (2017a,b), and Yagli et al. (2019), geographical, temporal, and sequential deterministic reconciliation of hierarchical photovoltaic (PV) power generation have been considered for a simulated PV dataset in California. In the first two cases, the reconciliations are carried out in spatial and temporal domains separately. To further improve forecasting accuracy, in the third case these two reconciliation approaches are sequentially applied. During the replication of the forecasting experiment, some issues emerged about non-negativity and coherence (in space and/or in time) of the sequentially reconciled forecasts. Furthermore, while the accuracy improvement of the considered approaches over the benchmark persistence forecasts is clearly visible at any data granularity, we argue that an even better performance may be obtained by a thorough exploitation of cross-temporal hierarchies. In this paper the cross-temporal point forecast reconciliation approach is applied to generate non-negative, fully coherent (both in space and time) forecasts. In particular, some relationships between two-step, iterative and simultaneous cross-temporal reconciliation procedures are for the first time established, non-negativity issues of the final reconciled forecasts are correctly dealt with in a simple way, and the most recent cross-temporal reconciliation approaches are adopted. The normalised Root Mean Square Error is used to measure forecasting accuracy, and a statistical multiple comparison procedure is performed to rank the approaches. Besides assuring full coherence, and non-negativity of the reconciled forecasts, the results show that for the considered dataset, cross-temporal forecast reconciliation significantly improves on the sequential procedures proposed by Yagli et al. (2019), at any cross-sectional level of the hierarchy and for any temporal granularity., 43 pages, 16 figures

In a recent paper, while elucidating the links between forecast combination and cross-sectional forecast reconciliation, Hollyman et al. (2021) have proposed a forecast combination-based approach to the reconciliation of a simple hierarchy. A new Level Conditional Coherent (LCC) point forecast reconciliation procedure was developed, and it was shown that the simple average of a set of LCC, and bottom-up reconciled forecasts (called Combined Conditional Coherent, CCC) results in good performance as compared to those obtained through the state-of-the-art cross-sectional reconciliation procedures. In this paper, we build upon and extend this proposal along some new directions. (1) We shed light on the nature and the mathematical derivation of the LCC reconciliation formula, showing that it is the result of an exogenously linearly constrained minimization of a quadratic loss function in the differences between the target and the base forecasts with a diagonal associated matrix. (2) Endogenous constraints may be considered as well, resulting in level conditional reconciled forecasts of all the involved series, where both the upper and the bottom time series are coherently revised. (3) As the LCC procedure does not guarantee the non-negativity of the reconciled forecasts, we argue that - when non-negativity is a natural attribute of the variables to be forecast - its interpretation as an unbiased top-down reconciliation procedure leaves room for some doubts. (4) The new procedures are used in a forecasting experiment on the classical Australian Tourism Demand (Visitor Nights) dataset. Due to the crucial role played by the (possibly different) models used to compute the base forecasts, we re-interpret the CCC reconciliation of Hollyman et al. (2021) as a forecast pooling approach, showing that accuracy improvement may be gained by adopting a simple forecast averaging strategy., 33 pages, 6 figures