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REVIEW 2 major objections 4 minor 34 references

Equal weighting is minimum-variance optimal exactly when forecast-error covariances share a uniform eigenstructure; a two-stage rule that shrinks toward equal weights when that condition is near delivers short-horizon gains.

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2026-07-14 07:22 UTC pith:KJB2OAVV

load-bearing objection Clean algebraic condition for when 1/N is min-var optimal, plus a usable closed-form shrinker that works well at short horizons; the eigenstructure link is still only visual. the 2 major comments →

arxiv 2607.11054 v1 pith:KJB2OAVV submitted 2026-07-13 econ.GN q-fin.ECstat.AP

When and Why Na\"ive Diversification Works: A Simple Diagnostic Strategy

Han Feng , Difang Huang , Jue Wang , Zhengjun Zhang This is my paper
classification econ.GN q-fin.ECstat.AP
keywords naive diversificationforecast combinationGolden Criterionadaptive shrinkageequity premium predictabilityminimum-variance portfolioaccuracy-diversity decompositionhorizon-dependent allocation
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.

The paper claims that the long-standing success of naïve equal weighting of forecasts is not a puzzle but a structural consequence of the forecast-error covariance matrix. Equal weights are exactly the minimum-variance combination if and only if the all-ones vector is an eigenvector of that matrix—the condition the authors call the Golden Criterion. They turn the criterion into a practical two-stage rule: estimate optimal weights on a training window, then choose a closed-form shrinkage intensity toward equal weights by minimizing mean-squared error on a validation window via the accuracy–diversity decomposition. On U.S. equity-premium forecasts from standard predictors, the resulting adaptive weights produce positive out-of-sample R² (especially at one month), annualized utility gains above 200 basis points, and Sharpe ratios up to 1.27. The same diagnostics show that short horizons favor heavy shrinkage toward equal weights while longer horizons favor optimized weights, giving investors an explicit horizon-dependent map for when simplicity is optimal and when optimization regains its edge.

Core claim

Equal weighting coincides with the minimum-variance forecast combination if and only if the forecast-error covariance matrix admits the all-ones vector as an eigenvector (the Golden Criterion). When this eigenstructure holds approximately, a data-driven two-stage shrinkage of optimized weights toward equal weights is optimal; when it fails, the rule recovers optimized weights. Empirically the criterion is close at short horizons and the adaptive rule outperforms both pure equal and pure optimal weighting, plus standard regularized and combination benchmarks, in accuracy, utility, and Sharpe ratio.

What carries the argument

The Golden Criterion (all-ones eigenvector of the error covariance) together with the closed-form λ_opt obtained by substituting the linear blend of equal and optimal weights into the accuracy–diversity MSFE decomposition and projecting onto [0,1]. It decides how far to shrink and collapses to pure equal weights whenever the first-stage covariance already satisfies the criterion.

Load-bearing premise

The rolling two-stage split—training window for optimal weights, fixed validation window for the shrinkage intensity—plus expanding-window individual regressions on the chosen predictors must keep the estimated covariance structure relevant enough that the validation-period λ remains optimal for the later test period.

What would settle it

Compute the rolling first-stage error-covariance matrices on the same equity-premium predictors; if the dominant eigenvalue and the all-ones Rayleigh quotient systematically diverge at short horizons while the adaptive rule still produces positive out-of-sample R² and utility gains, or if forcing λ=0 (pure optimal weights) at short horizons improves rather than worsens performance, the claimed link between the Golden Criterion and the short-horizon gains fails.

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

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

2 major / 4 minor

Summary. The paper proposes a 'Golden Criterion': equal (naïve) combination weights are exactly minimum-variance optimal if and only if the forecast-error covariance matrix admits the all-ones vector as an eigenvector (Proposition 1). Building on this algebraic characterization, the authors derive a closed-form two-stage adaptive shrinkage weight (2S–ASW) that blends optimal weights and equal weights via a shrinkage intensity λ_opt obtained by minimizing the accuracy–diversity MSFE decomposition on a validation sample. Applied to U.S. equity-premium forecasts from 13 Goyal–Welch predictors (1996–2019, extended to 2024), 2S–ASWs produce positive out-of-sample R² at the one-month horizon (up to 5.41%), annualized utility gains exceeding 200 bp, and Sharpe ratios up to 1.27, with λ_opt clustering near 1 at short horizons and declining at longer horizons. Extensive robustness checks (multiple sample splits, non-negative weights, reversed stages, Clark–West and HLN tests) are reported.

Significance. If the result holds, the paper supplies a clean, model-free necessary-and-sufficient condition that unifies earlier optimality results for 1/N and supplies a practical, closed-form diagnostic for when to shrink toward equal weights. The algebraic propositions are cleanly stated and proved in the appendix; the accuracy–diversity derivation of λ_opt is standard and closed-form; and the empirics are unusually thorough (multiple T1/Tc1/Tc2 splits, non-negative constraints, reversed-stage robustness, extended sample to 2024, Clark–West and HLN tests, utility and Sharpe evaluation). These strengths make the contribution potentially useful for both forecast-combination and portfolio-construction literatures, provided the claimed link between the Golden Criterion and the observed short-horizon gains is made statistically rigorous.

major comments (2)
  1. [Section II.D, Figure 5] Section II.D and Figure 5: The central empirical claim is that short-horizon dominance of 2S–ASWs (and λ_opt near 1) arises because real forecast-error covariances approximately satisfy the Golden Criterion. The only evidence offered is visual closeness of ρ_max(Σ1) and the all-ones Rayleigh quotient m̃/N. Closeness of these two scalars is necessary for the leading principal component to align with the all-ones direction, but it is not sufficient to establish that 1 is (approximately) an eigenvector: ||Σ1 1 − δ 1|| can remain large even when the Rayleigh quotients nearly coincide. No formal test (angle between 1 and the leading eigenvector, residual norm, or bootstrap/asymptotic test of the eigenvector hypothesis) is reported. Without such a test the link between the algebraic criterion and the OOS gains remains visual rather than statistical, leaving open the possibility that the gains
  2. [Section II.B, Table B.2] Section II.B, Propositions 1–3, and Table B.2: The weakest maintained assumption is that the rolling two-stage partition (Tc1 for OWs, fixed Tc2 for λ_opt) and expanding-window individual regressions adequately capture the non-stationary covariance structure so that the closed-form λ_opt remains optimal for the subsequent test period. The paper provides extensive robustness across splits and a reversed-stage exercise, but does not report a formal diagnostic of how far the validation-period accuracy–diversity geometry departs from the test-period geometry, nor a sensitivity analysis that recomputes λ_opt on rolling rather than fixed validation windows. A short additional exercise quantifying the stability of λ_opt (or the residual ||Σ1 1 − δ 1||) across adjacent windows would materially strengthen the claim that the two-stage design is not itself the source of the short-horizon gains.
minor comments (4)
  1. [Abstract, Introduction] Abstract and Introduction: The phrase 'uniform eigenstructure' is slightly imprecise; the Golden Criterion requires only that the all-ones vector is an eigenvector, not that all eigenvalues are equal. A one-sentence clarification would avoid confusion.
  2. [Figure 5] Figure 5: The vertical scales differ dramatically across panels (10^{-2} vs 10^{-3}); a common scale or normalized residual plot would make the claimed short-horizon alignment easier to assess visually.
  3. [Table 5, Section IV.B.2] Table 5 and related tables: The 'check-mark' pattern of R²_COS for 2S–ASWs (often most negative at J=6) is noted but not interpreted; a brief discussion of why intermediate horizons behave differently would help readers.
  4. [Section II, Appendix A] Notation: The same symbol m is used for 1^T Σ^{-1} 1 and later for related quantities; a consistent subscript (m1, m̃) throughout would reduce cognitive load.

Circularity Check

1 steps flagged

No load-bearing circularity: Golden Criterion is a standard algebraic characterization; λ_opt is validation-tuned and evaluated on held-out OOS data; one mild renaming of a known min-variance fact.

specific steps
  1. renaming known result [Abstract; Introduction p.2; Section II.B Proposition 1]
    "We call this explanation the “Golden Criterion”: the minimum-variance optimal combination coincides exactly with equal weighting when the error covariance matrix admits the all-ones vector as an eigenvector. While the criterion is algebraically straightforward, it is missed in the literature, and as such, formally defining it becomes necessary, even though we may consider it theoretically self-evident."

    The iff statement is the classical characterization of when the GMV portfolio is equal-weighted (w ∝ Σ⁻¹1 equals 1/N precisely when 1 is an eigenvector). The paper renames this standard fact “Golden Criterion” and presents the renaming as the structural explanation of the naïve-diversification puzzle. The adaptive method and OOS results do not depend on the name, so the circularity is cosmetic rather than load-bearing.

full rationale

The paper’s central theoretical claim (Proposition 1) is the standard linear-algebra fact that the minimum-variance weights Σ⁻¹1/(1′Σ⁻¹1) equal 1/N if and only if the all-ones vector is an eigenvector of Σ (or Σ⁻¹). The proofs in Appendix A.2 are self-contained quadratic optimization under a linear constraint; they do not define the criterion in terms of the empirical gains they later report. The two-stage adaptive rule chooses λ_opt by minimizing the accuracy–diversity MSFE on a validation window and then applies the resulting weights to a chronologically held-out test period (January 2007–August 2019), with pure OWs and EWs nested as boundary cases. That is ordinary train/validation/test design, not a fitted input re-labeled as a prediction of the same quantity. Empirical success is measured by independent OOS R²_OS, Clark–West tests, utility gains, and Sharpe ratios against the historical-mean benchmark and a large set of external combination methods. The only mild circularity-adjacent move is branding the well-known eigenvector characterization as the “Golden Criterion” and presenting it as the explanation of the naïve-diversification puzzle; that is renaming/organization of a known algebraic result, not a derivation that reduces the OOS claims to their inputs by construction. A non-load-bearing self-citation (Wang & Zhang 2024) is used only for interpretive color on market-reversal episodes. Score 1 reflects that single cosmetic renaming; the derivation chain and OOS evaluation are otherwise independent and non-circular.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 1 invented entities

The central theoretical claim rests on standard positive-definiteness of covariance matrices and the algebraic definition of the minimum-variance portfolio under a sum-to-one constraint; the empirical claim additionally depends on the chosen sample partitions, the expanding-window individual regressions, and the risk-aversion and transaction-cost parameters used for economic evaluation. No new physical entities are postulated; the Golden Criterion is a named algebraic condition, not an invented object.

free parameters (4)
  • λ_opt (shrinkage intensity) = varies by rolling window and horizon; frequently near 1 at short horizons
    Closed-form minimizer of the accuracy–diversity MSFE on the validation window, then projected onto [0,1]; data-driven but still a fitted quantity that governs the blend.
  • T1, Tc1, Tc2 sample-split lengths = e.g., T1=42, Tc1=6, Tc2=126 and several alternatives
    Multiple discrete choices (T1=42/54, various Tc1/Tc2 pairs summing to 132 months) are explored; results are reported for many of them, raising the possibility that favorable partitions are emphasized.
  • risk-aversion γ and weight bounds = γ=3, wt∈[0,1.5]
    γ=3 and wt∈[0,1.5] are fixed following Rapach et al. (2016) for utility and Sharpe calculations; they affect the economic-value claims.
  • regularization grids for Lasso/Ridge/ENet = grid search on validation
    α on log-grid [1e-6,1e6] (500 points), β∈[0.0001,1]; used only for benchmarks but part of the comparative evaluation.
axioms (4)
  • domain assumption Forecast-error covariance matrices Σ1, Σ2 are symmetric positive definite.
    Required for the existence of unique minimum-variance weights and for the inverse formulas in Propositions 1–3 (Section II.B).
  • standard math The accuracy–diversity decomposition MSFEcomb = wᵀS − (1/2)wᵀDw holds for the linear combination of forecasts.
    Derived in Appendix A.1 from the definition of MSFE; standard in the ensemble-forecasting literature (Krogh–Vedelsby, Brown et al.).
  • domain assumption Individual predictive regressions are correctly specified univariate OLS models with expanding windows; combination weights are applied to the resulting forecasts.
    Standard Goyal–Welch setup (Section II.A); the Golden Criterion and adaptive rule are defined on the resulting forecast-error covariances.
  • domain assumption Mean-variance investor with fixed risk aversion γ=3 and no short sales beyond the [0,1.5] bound evaluates economic value.
    Used for utility-gain and Sharpe-ratio claims (Section IV.B.5); follows Rapach et al. (2016).
invented entities (1)
  • Golden Criterion independent evidence
    purpose: Name for the algebraic condition that the all-ones vector is an eigenvector of the forecast-error covariance (or its inverse), under which equal weights are exactly minimum-variance optimal.
    It is a convenient label for a standard eigenstructure property, not a new physical or statistical object; independent evidence is the algebraic necessity-and-sufficiency proof itself.

pith-pipeline@v1.1.0-grok45 · 87431 in / 3632 out tokens · 43109 ms · 2026-07-14T07:22:17.293489+00:00 · methodology

0 comments
read the original abstract

We explain the long-standing puzzle of na\"ive diversification with a simple, testable condition: equal weighting is minimum-variance optimal when the forecast-error covariance matrix has a uniform eigenstructure. This "Golden Criterion" drives a two-stage adaptive strategy that dynamically blends naive and optimized weights based on the empirical distance from this condition. Applied to U.S. equity premium forecasting, the method delivers consistent out-of-sample gains in forecast accuracy, utility, and Sharpe ratios. Diversity-driven shrinkage dominates at short horizons, while optimized weights regain their edge at longer horizons, offering clear horizon-dependent guidance for portfolio construction.

Figures

Figures reproduced from arXiv: 2607.11054 by Difang Huang, Han Feng, Jue Wang, Zhengjun Zhang.

Figure 2
Figure 2. Figure 2: Optimal shrinkage factor λopt in adaptive shrinkage combination (Tc1 = 6, Tc2 = 126 when T1 = 42). These figures exhibit the optimal shrinkage factor λopt, for J = 1, 3, 6, 12, 24, respectively, which is computed in the validation period Tc2. If λopt > 0.5, 2S–ASWs tend to lean more toward EWs; otherwise, 2S–ASWs tend to lean more toward OWs. The proportions of λopt = 1 are 19.74%, 57.33%, 18.37%, 9.93%, 2… view at source ↗
Figure 5
Figure 5. Figure 5: The dominant eigenvalue ρmax(Σ1) and the all-ones Rayleigh quotient m˜N . These figures demonstrate the em￾pirically dominant eigenvalue ρmax(Σ1), and the all-ones Rayleigh quotient m˜N for J = 1, 3, 6, 12, 24, respectively, where ρmax(Σ1) refers to the error covariance matrix in the first-stage optimization problem, m˜ = 1T Σ11 refers to the sum of all the elements in Σ1, and N = 13 is the number of GW pr… view at source ↗

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Reference graph

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    40 Jan-2007Jan-2008Jan-2009Jan-2010Jan-2011Jan-2012Jan-2013Jan-2014Jan-2015Jan-2016Jan-2017Jan-2018Jan-2019 Month -1.5 -1 -0.5 0 0.5 1 1.5Weights Value Shrinkage Weights of GW Predictors Forecast Series DP EP DE RVOL BM NTIS TBL LTY LTR TMS DFY DFR INFL (a)J= 1 Jan-2007Jan-2008Jan-2009Jan-2010Jan-2011Jan-2012Jan-2013Jan-2014Jan-2015Jan-2016Jan-2017Jan-201...

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    Note: Values of variables marked with∗ are shown as percentages for readability

    Panel B presents the Pearson correlation coefficients among the predictors. Note: Values of variables marked with∗ are shown as percentages for readability. Panel A: Summary Statistics Variable Mean Median 1st pct 99th pct Std. dev. DP -4.01 -3.97 -4.48 -3.42 0.20 EP -3.15 -3.09 -4.79 -2.66 0.37 DE -0.86 -0.94 -1.24 1.26 0.42 RVOL 0.14 0.13 0.06 0.31 0.06...

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    CWp-value represents thep-value for Clark and West (2007) MSFE-adjusted statistical significance throughput. J=1 J=3 J=6 J=12 J=24 VariablesR 2 IS(%) CW p-valueR 2 IS(%) CW p-valueR 2 IS(%) CW p-valueR 2 IS(%) CW p-valueR 2 IS(%) CW p-value Panel A:T 1 = 54 DP 6.41 0.01 19.30 0.00 36.80 0.00 70.77 0.00 89.10 0.00 EP 7.01 0.01 20.29 0.00 39.97 0.00 72.60 0...

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    J=1 J=3 J=6 J=12 J=24 VariablesR 2 IOS (%) CW p-valueR 2 IOS (%) CW p-valueR 2 IOS (%) CW p-valueR 2 IOS (%) CW p-valueR 2 IOS (%) CW p-value Panel A:T 1=54 DP 0.77 0.14 0.05 0.25 0.57 0.22 2.44 0.10 -2.07 0.05 EP -2.53 0.25 -13.20 0.46 -25.47 0.58 -34.39 0.54 -62.47 0.90 DE -5.63 0.47 -32.77 0.80 -85.51 0.95 -137.74 0.90 -89.06 0.11 RVOL -0.37 0.65 -1.94...

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    CW p-value R2 COS (%) HLN sig

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    (2025),A= 10 1.23 2.23 4.43 4.29 5.04 f ixed132mASWs–Li et al

    Estimation Sample Methods J=1 J=3 J=6 J=12 J=24 Panel A:T 1 = 54 Tc1 = 12, Tc2 = 120 2S–ASWs 6.80 4.15 4.58 5.17 2.38 Lasso 2.83 -0.38 0.26 4.27 -8.89 Ridge -5.30 -4.17 -0.94 -4.39 -9.62 ENet 2.44 0.23 0.10 3.99 -7.72 OWs (Σ2) -2.49 8.98 10.97 8.53 6.94 Tc1 = 18, Tc2 = 114 2S–ASWs 5.61 3.61 5.99 5.30 1.91 Lasso 1.36 1.35 4.75 3.87 -9.55 Ridge 1.63 -2.83 4...

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    (2025),A= 10 0.61 0.71 0.91 0.65 0.70 f ixed132mASWs–Li et al

    Estimation Sample Methods J=1 J=3 J=6 J=12 J=24 Panel A:T 1 = 54 Tc1 = 12, Tc2 = 120 2S–ASWs 0.99 0.83 0.80 0.63 0.58 Lasso 0.79 0.45 0.37 0.66 0.19 Ridge 0.11 0.07 0.25 0.19 0.18 ENet 0.79 0.59 0.36 0.65 0.17 OWs (Σ2) 0.32 1.03 1.03 0.85 0.78 Tc1 = 18, Tc2 = 114 2S–ASWs 0.91 0.82 0.88 0.63 0.55 Lasso 0.62 0.64 0.87 0.64 0.11 Ridge 0.69 0.20 0.87 0.17 0.1...

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    Substituting into the condition in equation (A.2.6): (Σ−1 1 −ηΣ −1 2 )1= (Σ −1 1 − 1 kΣ−1 2 )1=0 This equality holds trivially, confirming thatw ∗ 1 =mathbf w 2∗ when the covariance matrices are proportional. Example A.2.2.7(Non-Proportional Matrices).Consider a specific scenario whereΣ 1 ̸= Σ2 and the matrices are not proportionally related. We construct...

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    andT 1 = 42 months (January 1996 to June 1999). Both config- urations maintain identical out-of-sample periods extending from the end of their respective in-sample windows to August 2019, allowing for direct comparison of forecasting accuracy across different training sample sizes. The combination methods employ a more sophisticated three-phase sample div...

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    We point out that the results reported for ASWs–Li et al. (2025) with different parametersAin all the tables represent our modified version that adapts its methodology to the forecast combination context, rather than its original approach designed for individual GW predictor forecasts. DMSPE, proposed by Stock and Watson (2004), employs the weights ˆwi,t ...

    2025

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    The regularization parameterαapplies to the penalty term in Lasso, Ridge, and Elastic Net, whileβcontrols the relative weight between theℓ 1 andℓ 2 norms in Elastic Net. Panel A: In-sample Analysis Method Training Validation Evaluation Candidate Parameters GW predictorsT 1 -T 1 - 2S–ASWsT c1 Tc2 Tc2 λopt LassoT c1 Tc2 Tc2 α∈[10 −6,10 6] RidgeT c1 Tc2 Tc2 ...

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    (2025) methods show robust but relatively limited performance in this reversed configura- tion

    EWs, Median, Trimmed Mean, and ASWs–Li et al. (2025) methods show robust but relatively limited performance in this reversed configura- tion. DMSPE withθ= 0.9 performs relatively better, particularly fromJ= 3 onward. The results underT 1 = 42 reveal a deterioration in general. None of these methods can yield positiveR 2 COS statistics atJ=

    2025

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    The reversed allocation produces a temporal shift in the distribution ofλ opt values. Whereas the sequential specification exhibitsλ opt <0.5 and approaches zero mainly in the latter part of the out-of-sample set, the reversed specification concentrates these lowerλ opt in the early stages of the evaluation period. The general pattern is consistent with t...

    2024

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    Table C.4 reports the out-of-sample forecast performance of combination methods and other benchmarks. [Place Table C.4 about here] Appendix-29 Consistent with the observations in Denk and L¨ offler (2024), the results reveal a general decline inR 2 COS statistics of combination methods, compared to Table 5 when the evaluation sample ends in August

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    The reduction inR 2 COS values is particularly pronounced for OWs and the method of Blanc and Setzer (2020)

    The number of advantageous parameter groups also significantly decreases. The reduction inR 2 COS values is particularly pronounced for OWs and the method of Blanc and Setzer (2020). For instance, the worstR 2 COS value for OWs is as low as -70.11% whenT 1 =

    2020

  26. [26]

    (2025) is modest and primarily confined to the short-term forecast atJ=

    In contrast, the performance degradation for 2S–ASWs and ASWs–Li et al. (2025) is modest and primarily confined to the short-term forecast atJ=

    2025

  27. [27]

    DMSPE withθ= 0.9 underperforms atJ= 1 but emerges as a strong competitor fromJ= 3 onward (R 2 COS = 4.36% atJ= 3, 11.59% atJ= 6, 27.05% atJ= 12, and 47.80% atJ= 24), consistent with its dominance at intermediate horizons in the baseline results. Despite the overall decline in predictability, the adaptive shrinkage combination method continues to demonstra...

    2025

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    (2025)’s method with pa- rameterA= 10 shows strong forecast ability atJ= 3, frequently attaining the highestR 2 COS values among all competing methods

    Among these, ASWs–Li et al. (2025)’s method with pa- rameterA= 10 shows strong forecast ability atJ= 3, frequently attaining the highestR 2 COS values among all competing methods. In contrast, OWs and Blanc and Setzer (2020), while exhibiting almost the weakest performance at short horizons, display remarkable strength at long forecast horizons. AtJ= 12 a...

    2025

  29. [29]

    As the forecast horizon extends to the medium term (J= 3 andJ= 6), the predictive landscape transitions into a shared dominance between 2S–ASWs and several alternative benchmarks

    By adaptively shrinking toward the heavily diversified equal-weight (na¨ ıve) portfolio, 2S–ASWs protect short-term forecasts from estimation error more effectively than traditional combination methods and other benchmarks. As the forecast horizon extends to the medium term (J= 3 andJ= 6), the predictive landscape transitions into a shared dominance betwe...

    2025

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    In particular, R2S–Ridge performs best when the covariance matrices are reversed

    stand out. In particular, R2S–Ridge performs best when the covariance matrices are reversed. From a practical asset allocation perspective, these findings yield clear, horizon-specific directives for investors. At the criticalJ= 1 horizon, 2S–ASWs provides a highly investable, theoretically sound mechanism that captures genuine predictability while decisi...

    2019

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    The proportions of λopt = 1are 44.74%, 26.00%, 7.48%, 1.42%, 10.08% and those forλ opt ≥0.9are 85.53%, 75.33%, 60.54%, 2.13%, 15.50%, respectively

    Ifλ opt >0.5, 2S–ASWs tend to lean more toward EWs; otherwise, 2S–ASWs tend to lean more toward OWs. The proportions of λopt = 1are 44.74%, 26.00%, 7.48%, 1.42%, 10.08% and those forλ opt ≥0.9are 85.53%, 75.33%, 60.54%, 2.13%, 15.50%, respectively. Appendix-33 Table C.1: Other out-of-sample combination results. To further investigate the consistency of th...

    2007

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    (1998) forecast-encompassing test (HLN), which compares the information content of 2S–ASWs with that of the benchmark method in the same row

    Asterisks attached to theR 2 COS entries report the significance levels of the Harvey et al. (1998) forecast-encompassing test (HLN), which compares the information content of 2S–ASWs with that of the benchmark method in the same row. Estimation Sample Methods J=1 J=3 J=6 J=12 J=24 Sample division R2 COS (%) HLN sig. CW p-value R2 COS (%) HLN sig. CW p-va...

    1998

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    (2025), A=10 (1) Reversed combinations (Table C.3) R2S–ASWs (4) Trimmed Mean (4) DMSPE,θ= 0.9 (4) R2S–ASWs (3) R2S–Ridge (1) R2S–ASWs (5) Li et al

    2S–ASWs (8) DMSPE,θ= 0.9 (8) DMSPE,θ= 0.9 (7) 2S–ASWs (1) Other out-of-sample combinations (Table C.1) 2S–ASWs (4) ENet (1) 2S–ASWs (2) Lasso (2) DMSPE,θ= 0.9 (1) DMSPE,θ= 0.9 (5) Non-negative combinations (Table C.2) 2S–ASWs (18) DMSPE,θ= 0.9 (18) DMSPE,θ= 0.9 (16) 2S–ASWs (1) Li et al. (2025), A=10 (1) Reversed combinations (Table C.3) R2S–ASWs (4) Trim...

    2025

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    Blanc and Setzer (2020) (8) OWs(Σ2) (7) Blanc and Setzer (2020) (1) Other out-of-sample combinations (Table C.1) Blanc and Setzer (2020) (4) OWs (Σ2) (1) OWs (Σ2) (5) Non-negative combinations (Table C.2) Blanc and Setzer (2020) (18) Blanc and Setzer (2020) (18) Reversed combinations (Table C.3) Blanc and Setzer (2020) (5) R2S–ASWs (3) R2S–Ridge (5) OWs (...

    2020