Pith. sign in

REVIEW 1 major objections 5 minor 53 references

The paper proposes a one-step temporal coherent combination that pools experts across all levels of an electricity price hierarchy, enforces aggregation constraints, and minimizes error variance; empirically it beats the best reconciled exp

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 05:07 UTC pith:AIVHHP5F

load-bearing objection A real extension of coherent forecast combination to temporal hierarchies with solid empirical support, but the unbiasedness assumption behind the optimality claim is untested and likely violated in electricity prices. the 1 major comments →

arxiv 2607.22310 v1 pith:AIVHHP5F submitted 2026-07-24 stat.AP stat.ME

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

Daniele Girolimetto This is my paper
classification stat.AP stat.ME MSC 62M20
keywords forecast combinationtemporal hierarchiesforecast reconciliationcoherent forecastselectricity price forecastingday-ahead marketcovariance shrinkagetemporal aggregation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Day-ahead electricity prices are forecast by many competing models at many granularities, from single hours to daily baseloads, and these forecasts are typically incoherent—the average of the predicted hours does not match the predicted baseload—while no single model is the most accurate everywhere. The paper tries to solve both problems in one step: it combines the forecasts of Q experts across the whole temporal hierarchy into a single forecast that satisfies the aggregation constraints and has minimum error variance among all unbiased linear combinations. The solution is a closed-form formula driven by the covariance of the experts' stacked forecast errors. Tested on four very different model classes and two day-ahead markets over four years, the combined forecast significantly outperforms both the individual base forecasts and the best single-expert reconciled forecast at almost every temporal level, with the largest gains at the coarser aggregations used for block and baseload trading. If the result holds, forecasters no longer need to pick a single model or reconcile and combine in separate steps.

Core claim

The paper's central claim is that when Q experts each forecast every level of a temporal hierarchy, one linear combination of their stacked forecasts exists that is coherent (each aggregated level equals the corresponding average of the hourly forecasts) and has minimum error variance among all unbiased linear combinations. Proposition 1 gives this combination in closed form, in structural and zero-constrained representations (equations 12 and 13), and the single-expert case Q=1 reduces to standard temporal reconciliation. The formula depends on the covariance matrix of the stacked base forecast errors, so the paper evaluates estimators that cross four correlation structures with linear and

What carries the argument

The temporal hierarchy and the coherent-combination projector built on it. The hierarchy organizes the 24 hourly prices into non-overlapping averages at coarser levels—2-, 3-, 4-, 6-, 8-, and 12-hour blocks and the daily baseload—through a fixed aggregation matrix A, summarized by the structural matrix S=[A; I]. Proposition 1's projector, e z_c = S(S_QᵀW⁻¹S_Q)⁻¹S_QᵀW⁻¹ bz (equation 12), maps the stacked base forecasts onto the coherent subspace, weighting by the inverse covariance W⁻¹ of the stacked forecast errors. This single formula carries the argument: it chooses combination weights across experts and reconciles across temporal levels at the same time, and it collapses to single-expert

Load-bearing premise

The optimality guarantee assumes every expert's base forecasts are unbiased—that the expected forecast equals the true price—so if the individual forecasts are systematically biased, the minimum-variance-unbiased property of the combined forecast is no longer guaranteed.

What would settle it

In a period with known biased forecasts, such as price-spike episodes, compute the coherent combination with and without an explicit bias correction and compare hourly MAE; if the bias-corrected version is more accurate, the unbiasedness assumption, not the combination algorithm, is carrying the guarantee. A complementary simulation: generate Q experts with exactly known covariance but nonzero means and check that the combined forecast inherits the bias.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Forecasters can pool any number of models into one coherent forecast without a separate model-selection step; in the two markets studied, the pooled forecast beats the best reconciled single expert at nearly every temporal level.
  • The largest gains appear at coarser aggregations, directly improving the block and baseload products that are traded alongside hourly prices.
  • Combining and reconciling in one simultaneous step beats the sequential alternative of averaging the experts first and then reconciling the average.
  • Estimating the covariance structure matters more than the shrinkage technique: preserving each expert's within-expert temporal error dependence is the main driver, and sophisticated nonlinear shrinkage adds little over linear shrinkage.
  • Removing any one expert, even the strongest, does not destroy the gains; the reduced pool remains comparable to or better than the best reconciled expert in most cases, though a redundant expert can add estimation noise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Inference: the method should transfer to other temporal hierarchies whose aggregates are averages or sums—electricity load, renewable generation, retail sales, or economic flows—and the testable prediction is that the one-step combination keeps beating each reconciled expert there.
  • Inference: since the optimality proof assumes unbiased base forecasts, markets with frequent price spikes or regime shifts may need a bias-correction step; a natural experiment would compare the coherent combination with and without bias correction in such periods.
  • Inference: the by-expert covariance finding suggests an operational rule—estimate each model's error dependence across aggregation levels and ignore cross-model error correlations—which, if confirmed on other datasets, greatly reduces the covariance estimation burden.
  • Inference: the closed-form projector makes a probabilistic extension natural: rather than combining point forecasts, one could combine predictive distributions and project them onto the coherent subspace, yielding coherent scenarios useful for risk management.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper develops a temporal coherent forecast combination procedure for day-ahead electricity prices. Given Q experts who each produce forecasts for all levels of a temporal hierarchy (hourly through daily aggregates), the method combines the forecasts into one vector that satisfies the temporal aggregation constraints and, under a stated unbiasedness assumption, has minimum trace error variance among linear unbiased combinations. Proposition 1 gives closed-form structural and zero-constrained expressions, with the single-expert temporal reconciliation of Athanasopoulos et al. (2017) as the Q=1 special case. The empirical study uses the four model classes from Lipiecki et al. (2026) for Germany and Spain over 2021–2024, comparing the combined forecast with base and reconciled forecasts in terms of MAE/RMSE, Diebold–Mariano tests, and MCB Nemenyi ranks. The authors report that the coherent combination outperforms the best reconciled expert at nearly every temporal level, and that the by-expert correlation structure is the key covariance-modelling choice.

Significance. If the results hold, the paper makes a useful practical contribution: it solves forecast combination and temporal reconciliation in one step, provides a closed-form solution that nests existing methodology, and gives a thorough empirical comparison on public base forecasts from four very different model classes. The empirical design is credible: the base forecasts are taken from an existing replication package, the evaluation is out-of-sample, and significance is assessed both pairwise and through multiple-comparison plots. The robustness analysis across covariance estimators, leave-one-out expert pools, and sequential alternatives is a real strength. However, the central theoretical guarantee — minimum error variance among linear unbiased combinations — relies on an unbiasedness assumption that is not tested in the empirical setting and is arguably questionable for day-ahead electricity price forecasts during 2021–2024. The empirical gains may survive the failure of this assumption, but the paper's main claim as stated is conditional on an unverified condition.

major comments (1)
  1. [Section 2.2, before Eq. (10); Proposition 1, Eqs. (12)–(13)] The derivation assumes E[bz_q] = z for every expert q. This assumption is load-bearing: it justifies the unbiasedness constraint GS_Q = I_m, and the minimum-variance claim in Proposition 1 is only among linear unbiased combinations. If the base forecasts are biased, then E[ez_c] = z + S G bias, so the combined forecast is not unbiased and Eqs. (12)–(13) do not certify minimum variance or minimum MSE. The test period 2021–2024 includes the European energy crisis and price spikes, where day-ahead price forecasts are plausibly biased. The paper does not report the bias of the base forecasts, of the reconciled forecasts, or of the combined forecast, and does not test or correct for bias. Please add an explicit bias diagnostic (e.g., mean in-sample and out-of-sample errors for each expert and for the combined forecast) and either show the assumption is approximately satisfied, or qualify the
minor comments (5)
  1. [Table 2] The superscripts 1 and 2 are explained in the note, but in the table body they appear as digits attached to the numerical values (e.g., '19.31' vs '19.312'). This makes the table difficult to read; typeset them as proper superscripts.
  2. [Section 2.4 / Table 4] The references for the nonlinear shrinkage estimators are inconsistent: the text cites Ledoit and Wolf (2022a) for lis/gis and (2022b) for qis, but Table 4 labels all three with 'Ledoit and Wolf, 2022b'. Please correct the attribution.
  3. [Section 2.4, Eq. (15)] The paper motivates shrinkage by high dimensionality, but in the application nQ = 240 and the calibration window has roughly N ≈ 1095 days, so the sample covariance is invertible. State the concentration ratio explicitly and clarify whether the shrinkage is needed for numerical conditioning and estimation error rather than singularity.
  4. [Section 3.2 / Figure 3] The claim of 'significant' improvements is based on pairwise DM tests at the 1% level without adjustment for the number of comparisons, although the MCB Nemenyi procedure partially addresses this. It would be helpful to report whether the DM conclusions survive any standard multiple-testing correction, or to rely primarily on the MCB results.
  5. [Section 4.3, Table 5] In Germany, leaving out XGB leads to a small but systematic improvement over the full pool. This is an interesting qualification to the general message that combining more experts is beneficial; the text mentions it but could connect it more explicitly to the trade-off between signal and estimation noise.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained and the empirical evaluation is genuinely out-of-sample.

full rationale

The central derivation is Proposition 1, which obtains the coherent forecast combination as the trace-minimizing linear unbiased combination. The proof is given in Appendix A and does not assume the conclusion: it solves a constrained quadratic program under the explicitly stated assumption E[bz_q] = z. The covariance matrix W is estimated from in-sample forecast errors over the calibration window (Section 2.4), while the empirical evaluation compares combined forecasts against base and reconciled forecasts over the 2021-2024 test period using base forecasts taken unchanged from an external replication package (Lipiecki et al., 2026). No parameter is fitted to the test data and then reported as a prediction; the combined forecast at each test day depends only on calibration-window errors and the fixed base forecasts. The Q=1 reduction to standard temporal reconciliation is a mathematical special case, not a circular import. Self-citations to Girolimetto and Di Fonzo (2024) are used to position the framework, but the main theorem is proved in the paper and the proof is checkable from the stated assumptions. The possible violation of the unbiasedness assumption in electricity price forecasting is a correctness or robustness concern, not a circularity: it does not make the claimed output equivalent to the input by construction. Overall, no circular step meeting the required evidence standard is present.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no new entities or hand-set numerical parameters; its claims rest on the unbiasedness and covariance-estimation assumptions of the reconciliation literature.

axioms (4)
  • domain assumption Base forecasts are unbiased: E[\hat{z}_q] = z for each expert q.
    Invoked in Section 2.2 and used in Proposition 1 to justify the unbiasedness constraint GS_Q=I_m; not verified for the empirical forecasts, which are known to be biased in electricity price markets.
  • domain assumption The in-sample error covariance W estimated from calibration-window errors (Eq. 15) approximates the out-of-sample error covariance.
    Standard in forecast reconciliation; the optimal weights in Eq. (12) depend on W, so its out-of-sample validity is load-bearing.
  • domain assumption Block and baseload electricity prices are arithmetic averages of hourly prices (weights 1/k).
    Used to define the aggregation matrix A in Eq. (4)-(5); consistent with market practice.
  • standard math The covariance matrix W is positive definite and the involved matrices are full rank.
    Needed for the closed-form expressions in Proposition 1.

pith-pipeline@v1.3.0-alltime-deepseek · 28255 in / 11661 out tokens · 97862 ms · 2026-08-01T05:07:58.713799+00:00 · methodology

0 comments
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2607.22310 by Daniele Girolimetto.

Figure 1
Figure 1. Figure 1: Temporal hierarchy for the daily cycle of m = 24 hourly prices, with K = {24, 12, 8, 6, 4, 3, 2, 1}. Each level k partitions the day into Mk = 24/k non-overlapping blocks, and each block is the average of the k hourly values it spans, as in (2). Collecting them into x [k] τ = h x [k] (τ,1) . . . x [k] (τ,Mk) i⊤ ∈ RMk , and writing the high-frequency level as yτ = x [1] τ ∈ R m, the low-frequency (temporall… view at source ↗
Figure 2
Figure 2. Figure 2: The structural matrix S, the key matrix K = 1Q ⊗ In and their product SQ = KS for the hourly hierarchy, with m = 24 and n = 60. For readability, Q = 2 experts are shown, distinguished by color. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Average ranks and MCB Nemenyi test intervals, computed on the absolute (top) and squared (bottom) errors over the 1 461 test days, at the hourly (1H) and daily (24H) levels for the German and Spanish markets. TCC denotes the temporal coherent combination (shrbe) and the subscript r the reconciled forecasts (shr). The shaded area marks the confidence region of the best-ranked approach: approaches whose inte… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

53 extracted references · 2 canonical work pages

  1. [1]

    2025 , journal =

    Improving cross-temporal forecasts reconciliation accuracy and utility in energy market , author =. 2025 , journal =

    2025

  2. [2]

    2017 , journal =

    Forecasting with temporal hierarchies , author =. 2017 , journal =

    2017

  3. [3]

    2023 , journal =

    On the evaluation of hierarchical forecasts , author =. 2023 , journal =

    2023

  4. [4]

    2024 , journal =

    Forecast reconciliation: A review , author =. 2024 , journal =

    2024

  5. [5]

    2016 , booktitle =

    Sparse and smooth adjustments for coherent forecasts in temporal aggregation of time series , author =. 2016 , booktitle =

    2016

  6. [6]

    Machine learning approaches for cross-temporal forecast reconciliation with an application to Italian energy load data , author =

    Linear vs. Machine learning approaches for cross-temporal forecast reconciliation with an application to Italian energy load data , author =. 2025 , booktitle =. doi:10.1007/978-3-031-96303-2\_30 , isbn =

    work page doi:10.1007/978-3-031-96303-2 2025

  7. [7]

    2026 , journal =

    Temporal hierarchical forecast reconciliation of photovoltaic power generation from heterogeneous base models , author =. 2026 , journal =

    2026

  8. [8]

    2023 , journal =

    Spatio-temporal reconciliation of solar forecasts , author =. 2023 , journal =

    2023

  9. [9]

    2023 , journal =

    Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives , author =. 2023 , journal =

    2023

  10. [10]

    2024 , journal =

    Forecast combination-based forecast reconciliation: Insights and extensions , author =. 2024 , journal =

    2024

  11. [11]

    2024 , journal =

    Point and probabilistic forecast reconciliation for general linearly constrained multiple time series , author =. 2024 , journal =

    2024

  12. [12]

    2024 , journal =

    Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues , author =. 2024 , journal =

    2024

  13. [13]

    2024 , journal =

    Coherent forecast combination for linearly constrained multiple time series , author =. 2024 , journal =

    2024

  14. [14]

    CRAN , publisher =

    Girolimetto, Daniele and Di Fonzo, Tommaso , year =. CRAN , publisher =

  15. [15]

    arXiv , doi =

    Girolimetto, Daniele and Rombouts, Jeroen and Wilms, Ines and Yang, Yangzhuoran Fin , year =. arXiv , doi =

  16. [16]

    thief: Temporal

    Hyndman, Rob J and Kourentzes, Nikolaos , year =. thief: Temporal

  17. [17]

    2016 , journal =

    Forecasting with multivariate temporal aggregation: The case of promotional modelling , author =. 2016 , journal =

    2016

  18. [18]

    2019 , journal =

    Another look at forecast selection and combination: Evidence from forecast pooling , author =. 2019 , journal =

    2019

  19. [19]

    Stealing accuracy: Predicting day-ahead electricity prices with temporal hierarchy forecasting (

    Lipiecki, Arkadiusz and Bilińska, Kaja and Kourentzes, Nikolaos and Weron, Rafał , year =. Stealing accuracy: Predicting day-ahead electricity prices with temporal hierarchy forecasting (. International Journal of Forecasting , publisher =

  20. [20]

    2020 , journal =

    Temporal hierarchies with autocorrelation for load forecasting , author =. 2020 , journal =

    2020

  21. [21]

    2021 , journal =

    Dimensionality reduction in forecasting with temporal hierarchies , author =. 2021 , journal =

    2021

  22. [22]

    2022 , journal =

    Forecasting: theory and practice , author =. 2022 , journal =

    2022

  23. [23]

    2023 , journal =

    Forecast reconciliation in the temporal hierarchy: Special case of intermittent demand with obsolescence , author =. 2023 , journal =

    2023

  24. [24]

    2019 , journal =

    Improving the forecasting performance of temporal hierarchies , author =. 2019 , journal =

    2019

  25. [25]

    2019 , journal =

    Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization , author =. 2019 , journal =

    2019

  26. [26]

    2020 , journal =

    Optimal non-negative forecast reconciliation , author =. 2020 , journal =

    2020

  27. [27]

    2024 , journal =

    Probabilistic Forecast Reconciliation under the Gaussian Framework , author =. 2024 , journal =

    2024

  28. [28]

    2011 , journal =

    Optimal combination forecasts for hierarchical time series , author =. 2011 , journal =

    2011

  29. [29]

    1942 , journal =

    The precision of national income estimates , author =. 1942 , journal =

    1942

  30. [30]

    1978 , journal =

    The estimation of large social account matrices , author =. 1978 , journal =

    1978

  31. [31]

    2005 , journal =

    A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics , author =. 2005 , journal =

    2005

  32. [32]

    2001 , journal =

    The geometric mean, matrices, metrics, and more , author =. 2001 , journal =

    2001

  33. [33]

    2007 , publisher =

    Positive Definite Matrices , author =. 2007 , publisher =

    2007

  34. [34]

    Bernoulli , year =

    Ledoit, Olivier and Wolf, Michael , title =. Bernoulli , year =

  35. [35]

    Journal of Multivariate Analysis , year =

    Ledoit, Olivier and Wolf, Michael , title =. Journal of Multivariate Analysis , year =

  36. [36]

    2016 , journal =

    Fast computation of reconciled forecasts for hierarchical and grouped time series , author =. 2016 , journal =

    2016

  37. [37]

    Journal of Financial Econometrics , year =

    Ledoit, Olivier and Wolf, Michael , title =. Journal of Financial Econometrics , year =

  38. [38]

    The Annals of Statistics , year =

    Ledoit, Olivier and Wolf, Michael , title =. The Annals of Statistics , year =

  39. [39]

    2019 , publisher =

    Magnus, Jan R and Neudecker, Heinz , title =. 2019 , publisher =

    2019

  40. [40]

    Journal of the Operational Research Society , volume=

    The combination of forecasts , author=. Journal of the Operational Research Society , volume=. 1969 , publisher=

    1969

  41. [41]

    Handbook of Economic Forecasting , editor=

    Forecast combinations , author=. Handbook of Economic Forecasting , editor=. 2006 , publisher=

    2006

  42. [42]

    Journal of Business & Economic Statistics , volume = 13, number = 3, pages =

    Comparing predictive accuracy , author =. Journal of Business & Economic Statistics , volume = 13, number = 3, pages =

  43. [43]

    2023 , journal =

    tsutils: Time Series Exploration, Modelling and Forecasting , author =. 2023 , journal =

    2023

  44. [44]

    Forecasting natural gas prices in real time

    Baumeister, Christiane and Huber, Florian and Lee, Thomas K and Ravazzolo, Francesco. Forecasting natural gas prices in real time. Journal of applied econometrics (Chichester, England). doi:10.1002/jae.70018

    work page doi:10.1002/jae.70018

  45. [45]

    Koning, Alex J and Franses, Philip Hans and Hibon, Mich\`ele and Stekler, H O , year =. The. International Journal of Forecasting , volume =

  46. [46]

    1979 , journal =

    Corrigenda: The estimation of large social account matrices , author =. 1979 , journal =

    1979

  47. [47]

    2014 , journal =

    Electricity price forecasting: A review of the state-of-the-art with a look into the future , author =. 2014 , journal =

    2014

  48. [48]

    2021 , journal =

    Understanding forecast reconciliation , author =. 2021 , journal =

    2021

  49. [49]

    2023 , journal =

    Forecast combinations: An over 50-year review , author =. 2023 , journal =

    2023

  50. [50]

    2021 , journal =

    Forecasting day-ahead electricity prices: A review of state-of-the-art algorithms, best practices and an open-access benchmark , author =. 2021 , journal =

    2021

  51. [51]

    European Journal of Operational Research , volume = 279, number = 2, pages =

    Probabilistic forecast reconciliation with applications to wind power and electric load , author =. European Journal of Operational Research , volume = 279, number = 2, pages =

  52. [52]

    European Journal of Operational Research , volume = 306, number = 2, pages =

    Probabilistic forecast reconciliation: Properties, evaluation and score optimisation , author =. European Journal of Operational Research , volume = 306, number = 2, pages =

  53. [53]

    Journal of Service Research , volume = 28, number = 2, pages =

    Hierarchical Time Series Forecasting in Emergency Medical Services , author =. Journal of Service Research , volume = 28, number = 2, pages =