arXiv subjects

Daniele Girolimetto

Publications and source records attributed to Daniele Girolimetto.

16 recordsLinked to original sources

arXiv2026

Thick as THieFs: Temporal coherent forecast combination for day-ahead electricity prices

Day-ahead electricity prices are forecast by many competing models and at several temporal granularities, from hourly prices to block and baseload products. The resulting forecasts suffer from two distinct problems: forecasts produced at different granularities are incoherent, as the aggregates do not match the averages of their components, and no single model is the most accurate in every market, period and level. We develop a temporal coherent combination approach that addresses both problems in one step, pooling the forecasts that the competing experts produce at all the levels of a temporal hierarchy into a single forecast that satisfies the aggregation constraints and has minimum error variance among the linear unbiased combinations. Since the optimal solution depends on a high-dimensional error covariance matrix, we compare different estimators obtained by crossing correlation structures with linear and nonlinear shrinkage approaches. Using the base forecasts of four model classes for the German and Spanish day-ahead markets, the combined forecasts significantly outperform both the base forecasts and the best reconciled expert at nearly every temporal level.

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arXiv2026

FoReco and FoRecoML: A Unified Toolbox for Forecast Reconciliation in R

Forecast reconciliation has become key to improving the accuracy and coherence of forecasts for linearly constrained multiple time series, such as hierarchical and grouped series. Yet, comprehensive software that jointly covers cross-sectional, temporal, and cross-temporal reconciliation has so far been lacking. The R packages FoReco and FoRecoML address this gap by offering a comprehensive and unified framework. The packages respectively implement classical and regression-based linear reconciliation approaches, and non-linear approaches based on machine learning for cross-sectional, temporal and cross-temporal frameworks. Designed for accessibility and flexibility, these packages provide sensible default options that allow new users to apply reconciliation methods with minimal effort, while still giving expert users full control to explore state-of-the-art extensions through customized settings. With this dual focus, FoReco and FoRecoML are versatile tools for practitioners and researchers working on forecast reconciliation.

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arXiv2026

Multivariate GARCH and portfolio variance prediction: A forecast reconciliation perspective

We assess the advantage of combining univariate and multivariate portfolio risk forecasts with the aid of forecast reconciliation techniques. In our analyzes, we assume knowledge of portfolio weights, a standard for portfolio risk management applications. With an extensive simulation experiment, we show that, if the true covariance is known, forecast reconciliation improves over a standard multivariate approach, in particular when the adopted multivariate model is misspecified. However, if noisy proxies are used, correctly specified models and the misspecified ones (for instance, neglecting spillovers) turn out to be, in several cases, indistinguishable, with forecast reconciliation still providing improvements. The noise in the covariance proxy plays a crucial role in determining the improvement of both the forecast reconciliation and the correct model specification. An empirical analysis shows how forecast reconciliation can be adopted with real data to improve traditional GARCH-based portfolio variance forecasts.

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arXiv2025

Forecast reconciliation with non-linear constraints

Methods for forecasting time series adhering to linear constraints have seen notable development in recent years, especially with the advent of forecast reconciliation. This paper extends forecast reconciliation to the open question of non-linearly constrained time series. Non-linear constraints can emerge with variables that are formed as ratios such as mortality rates and unemployment rates. On the methodological side, Non-linearly Constrained Reconciliation (NLCR) is proposed. This algorithm adjusts forecasts that fail to meet non-linear constraints, in a way that ensures the new forecasts meet the constraints. The NLCR method is a projection onto a non-linear surface, formulated as a constrained optimisation problem. On the theoretical side, optimisation methods are again used, this time to derive sufficient conditions for when the NLCR methodology is guaranteed to improve forecast accuracy. Finally on the empirical side, NLCR is applied to two datasets from demography and economics and shown to significantly improve forecast accuracy relative to relevant benchmarks.

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arXiv2025

Energy load forecasting using Terna public data: a free lunch multi-task combination approach

We propose a quick-and-simple procedure to augment the accuracy of 15-minutes Italian load forecasts disaggregated by bidding zones published by Terna, the operator of the Italian electricity system. We show that a stacked-regression multi-task combination approach using Terna and daily random walk naive forecasts, is able to produce significantly more accurate forecasts immediately after Terna publishes on its data portal the energy load measurements for the previous day, and the forecasts for the current day.

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arXiv2025

Forecasting Italian daily electricity generation disaggregated by geographical zones and energy sources using coherent forecast combination

A novel approach is applied for improving forecast accuracy and achieving coherence in forecasting the Italian daily energy generation time series. In hierarchical frameworks such as national energy generation disaggregated by geographical zones and energy sources, independently generated base forecasts often result in inconsistencies across the constraints. We deal with this issue through a coherent balanced multi-task forecast combination approach, which combines unbiased forecasts from multiple experts while ensuring coherence. Applied to the daily Italian electricity generation data, our method shows superior accuracy compared to single-task base and combined forecasts, and a state-of-the-art single-expert reconciliation technique, demonstrating to be an effective approach to forecasting linearly constrained multiple time series.

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arXiv2024

Balancing Accuracy and Costs in Cross-Temporal Hierarchies: Investigating Decision-Based and Validation-Based Reconciliation

Wind power forecasting is essential for managing daily operations at wind farms and enabling market operators to manage power uncertainty effectively in demand planning. This paper explores advanced cross-temporal forecasting models and their potential to enhance forecasting accuracy. First, we propose a novel approach that leverages validation errors, rather than traditional in-sample errors, for covariance matrix estimation and forecast reconciliation. Second, we introduce decision-based aggregation levels for forecasting and reconciliation where certain horizons are based on the required decisions in practice. Third, we evaluate the forecasting performance of the models not only on their ability to minimize errors but also on their effectiveness in reducing decision costs, such as penalties in ancillary services. Our results show that statistical-based hierarchies tend to adopt less conservative forecasts and reduce revenue losses. On the other hand, decision-based reconciliation offers a more balanced compromise between accuracy and decision cost, making them attractive for practical use.

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arXiv2024

Coherent forecast combination for linearly constrained multiple time series

Linearly constrained multiple time series may be encountered in many practical contexts, such as the National Accounts (e.g., GDP disaggregated by Income, Expenditure and Output), and multilevel frameworks where the variables are organized according to hierarchies or groupings, like the total energy consumption of a country disaggregated by region and energy sources. In these cases, when multiple incoherent base forecasts for each individual variable are available, a forecast combination-and-reconciliation approach, that we call coherent forecast combination, may be used to improve the accuracy of the base forecasts and achieve coherence in the final result. In this paper, we develop an optimization-based technique that combines multiple unbiased base forecasts while assuring the constraints valid for the series. We present closed form expressions for the coherent combined forecast vector and its error covariance matrix in the general case where a different number of forecasts is available for each variable. We also discuss practical issues related to the covariance matrix that is part of the optimal solution. Through simulations and a forecasting experiment on the daily Australian electricity generation hierarchical time series, we show that the proposed methodology, in addition to adhering to sound statistical principles, may yield in significant improvement on base forecasts, single-task combination and single-expert reconciliation approaches as well.

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arXiv2024

Insights into regression-based cross-temporal forecast reconciliation

Cross-temporal forecast reconciliation aims to ensure consistency across forecasts made at different temporal and cross-sectional levels. We explore the relationships between sequential, iterative, and optimal combination approaches, and discuss the conditions under which a sequential reconciliation approach (either first-cross-sectional-then-temporal, or first-temporal-then-cross-sectional) is equivalent to a fully (i.e., cross-temporally) coherent iterative heuristic. Furthermore, we show that for specific patterns of the error covariance matrix in the regression model on which the optimal combination approach grounds, iterative reconciliation naturally converges to the optimal combination solution, regardless the order of application of the uni-dimensional cross-sectional and temporal reconciliation approaches. Theoretical and empirical properties of the proposed approaches are investigated through a forecasting experiment using a dataset of hourly photovoltaic power generation. The study presents a comprehensive framework for understanding and enhancing cross-temporal forecast reconciliation, considering both forecast accuracy and the often overlooked computational aspects, showing that significant improvement can be achieved in terms of memory space and computation time, two particularly important aspects in the high-dimensional contexts that usually arise in cross-temporal forecast reconciliation.

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arXiv2023

Point and probabilistic forecast reconciliation for general linearly constrained multiple time series

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.

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arXiv2023

Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues

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 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, the results highlight the potential of the proposed methods to improve the accuracy of probabilistic forecasts and to address the challenge of integrating disparate scenarios while coherently taking into account short-term operational, medium-term tactical, and long-term strategic planning.

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arXiv2023

Exploiting Intraday Decompositions in Realized Volatility Forecasting: A Forecast Reconciliation Approach

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.

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arXiv2022

Enhancements in cross-temporal forecast reconciliation, with an application to solar irradiance forecasts

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.

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arXiv2021

Forecast combination based forecast reconciliation: insights and extensions

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.

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arXiv2020

Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives

Forecast reconciliation is a post-forecasting process aimed to improve the quality of the base forecasts for a system of hierarchical/grouped time series (Hyndman et al., 2011). Contemporaneous (cross-sectional) and temporal hierarchies have been considered in the literature, but - except for Kourentzes and Athanasopoulos (2019) - generally these two features have not been fully considered together. Adopting a notation able to simultaneously deal with both forecast reconciliation dimensions, the paper shows two new results: (i) an iterative cross-temporal forecast reconciliation procedure which extends, and overcomes some weaknesses of, the two-step procedure by Kourentzes and Athanasopoulos (2019), and (ii) the closed-form expression of the optimal (in least squares sense) point forecasts which fulfill both contemporaneous and temporal constraints. The feasibility of the proposed procedures, along with first evaluations of their performance as compared to the most performing `single dimension' (either cross-sectional or temporal) forecast reconciliation procedures, is studied through a forecasting experiment on the 95 quarterly time series of the Australian GDP from Income and Expenditure sides considered by Athanasopoulos et al. (2019).

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arXiv2020

Fully reconciled GDP forecasts from Income and Expenditure sides

We propose a complete reconciliation procedure, resulting in a 'one number forecast' of the GDP figure, coherent with both Income and Expenditure sides' forecasted series, and evaluate its performance on the Australian quarterly GDP series, as compared to the original proposal by Athanasopoulos et al. (2019).

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