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arxiv: 2209.00991 · v6 · submitted 2022-08-27 · 💱 q-fin.RM · math.ST· stat.ME· stat.TH

E-backtesting

Qiuqi Wang , Ruodu Wang , Johanna Ziegel This is my paper

Pith reviewed 2026-05-24 11:33 UTC · model grok-4.3

classification 💱 q-fin.RM math.STstat.MEstat.TH
keywords e-backtestingExpected ShortfallValue-at-Riske-valuese-processesmodel-free backtestingidentification functionsrisk measures
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The pith

Unique backtest e-statistics for VaR and ES enable model-free e-processes for risk forecast validation.

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

The paper develops a model-free procedure for backtesting Expected Shortfall forecasts required under banking regulations. It introduces backtest e-statistics derived from identification functions to construct e-processes that test the accuracy of risk measure predictions. The approach requires no assumptions about the underlying data distribution. It characterizes unique forms of the statistics for both VaR and ES and studies criteria for optimal e-process construction. The method extends to other risk measures, with simulations and data analysis showing advantages over existing techniques.

Core claim

Backtest e-statistics are introduced to formulate e-processes for risk measure forecasts, and unique forms of backtest e-statistics for VaR and ES are characterized using recent results on identification functions. For a given backtest e-statistic, a few criteria for optimally constructing the e-processes are studied. The proposed method can be naturally applied to many other risk measures and statistical quantities.

What carries the argument

Backtest e-statistics, functions derived from identification functions that form the basis for constructing e-processes to test risk measure forecasts.

If this is right

  • The method yields valid model-free tests for ES forecasts without distributional assumptions.
  • It extends directly to backtesting many other risk measures and statistical quantities.
  • Criteria for optimal e-process construction improve the power or efficiency of the tests.
  • Extensive simulation studies and data analysis confirm practical advantages over literature methods.

Where Pith is reading between the lines

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

  • Regulators could use these tests to oversee bank ES forecasts with fewer modeling restrictions.
  • The technique connects e-value methods from statistics to regulatory risk management.
  • Extensions to dependent data or multi-period forecasts would be natural next steps.
  • A counterexample dataset where the e-process fails to stay bounded under correct forecasts would falsify the validity claim.

Load-bearing premise

That e-values and e-processes can be applied directly via backtest e-statistics to produce valid model-free tests for ES without requiring additional assumptions about the underlying data distribution or model.

What would settle it

A simulation or real dataset where forecasts are correct but the constructed e-process exceeds a high threshold like 20 with substantial probability under the null, violating the claimed error control.

Figures

Figures reproduced from arXiv: 2209.00991 by Johanna Ziegel, Qiuqi Wang, Ruodu Wang.

Figure 1
Figure 1. Figure 1: Realized losses and ES forecasts with a linear extending business (left panel); average [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Realized losses and ES forecasts with a non-linear business cycle (left panel); average [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Realized losses and ES forecasts with iid losses (left panel); average log-transformed e [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Top panels: percentage of detections (%) of VaR [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Negated percentage log-returns of the NASDAQ Composite index (left panel); ES [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Log-transformed e-processes testing ES0.975 with respect to the number of days for the NASDAQ index from Jan 3, 2005 to Dec 31, 2021; left panel: GREE method, middle panel: GREL method, right panel: GREM method 2005 2010 2015 −1 0 1 2 3 4 5 dates log e−process normal t skewed−t empirical st +10% ES 2005 2010 2015 −1 0 1 2 3 4 5 dates log e−process normal t skewed−t empirical st +10% ES 2005 2010 2015 −1 0 … view at source ↗
Figure 7
Figure 7. Figure 7: Log-transformed e-processes testing ES0.975 with respect to the number of days for portfo￾lio data from Jan 3, 2005 to Dec 31, 2021; left panel: GREE method, middle panel: GREL method, right panel: GREM method 2005 2010 2015 −1 0 1 2 3 4 5 dates log e−process normal t skewed−t st +10% ES 2005 2010 2015 −1 0 1 2 3 4 5 dates log e−process normal t skewed−t st +10% ES 2005 2010 2015 −1 0 1 2 3 4 5 dates log e… view at source ↗
Figure 8
Figure 8. Figure 8: Left panel: p-value trajectories testing (VaR [PITH_FULL_IMAGE:figures/full_fig_p060_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Portfolio data fitted by different distribution from Jan 3, 2005 to Dec 31, 2021; left panel: [PITH_FULL_IMAGE:figures/full_fig_p062_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Average log-transformed e-processes testing VaR [PITH_FULL_IMAGE:figures/full_fig_p064_10.png] view at source ↗
read the original abstract

In the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most important risk measure in financial regulation. One of the most challenging tasks in risk modeling practice is to backtest ES forecasts provided by financial institutions. To design a model-free backtesting procedure for ES, we make use of the recently developed techniques of e-values and e-processes. Backtest e-statistics are introduced to formulate e-processes for risk measure forecasts, and unique forms of backtest e-statistics for VaR and ES are characterized using recent results on identification functions. For a given backtest e-statistic, a few criteria for optimally constructing the e-processes are studied. The proposed method can be naturally applied to many other risk measures and statistical quantities. We conduct extensive simulation studies and data analysis to illustrate the advantages of the model-free backtesting method, and compare it with the ones in the literature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to introduce backtest e-statistics derived from identification functions to construct e-processes for model-free backtesting of VaR and ES forecasts. It characterizes unique forms of these e-statistics for VaR and ES, studies optimality criteria for e-process construction, and illustrates the approach with simulation studies and real-data analysis, asserting applicability to other risk measures.

Significance. If the central technical step holds, the work would provide a significant advance by enabling sequential, distribution-free backtesting of ES (now the regulatory standard under Basel), leveraging e-process theory for potentially anytime-valid tests. Credit is due for the systematic use of identification functions to derive the e-statistics and for the empirical comparisons.

major comments (2)
  1. [§3] §3 (characterization of backtest e-statistics for ES): the identification function for ES is joint with VaR and has mean zero under the null by construction, but the specific transformation to an e-statistic whose unconditional expectation is ≤1 for arbitrary distributions (including heavy-tailed returns with possibly infinite moments) is not derived or verified explicitly; this step is load-bearing for the model-free claim.
  2. [§4] §4 (e-process construction): the optimality criteria and validity of the resulting e-processes for ES presuppose that the backtest e-statistic satisfies the e-value property unconditionally; without an explicit proof or counterexample analysis under the paper's weakest assumption (no extra regularity), the guarantee for model-free ES backtesting remains open.
minor comments (2)
  1. [Abstract] The abstract states that 'unique forms' are characterized but does not reference the specific theorem or equation number; adding this cross-reference would improve readability.
  2. [§2] Notation for the identification functions and the resulting e-statistics could be introduced with a short table or displayed equation in §2 to aid readers unfamiliar with the recent e-process literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised highlight the need for more explicit technical details on the e-value property under minimal assumptions. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (characterization of backtest e-statistics for ES): the identification function for ES is joint with VaR and has mean zero under the null by construction, but the specific transformation to an e-statistic whose unconditional expectation is ≤1 for arbitrary distributions (including heavy-tailed returns with possibly infinite moments) is not derived or verified explicitly; this step is load-bearing for the model-free claim.

    Authors: We thank the referee for this observation. Section 3 characterizes the unique backtest e-statistics via identification functions, which are jointly defined for ES and VaR and satisfy conditional mean zero under the null by construction. The transformation to an e-statistic (non-negative with unconditional expectation ≤1) follows from standard e-value constructions applied to these mean-zero functions. We acknowledge that the explicit derivation and verification for arbitrary distributions, including heavy-tailed cases with possibly infinite moments, was not presented in full detail. In the revision we will add a dedicated paragraph (or short appendix) providing this derivation under the paper's stated weakest assumptions, confirming the model-free property. revision: yes

  2. Referee: [§4] §4 (e-process construction): the optimality criteria and validity of the resulting e-processes for ES presuppose that the backtest e-statistic satisfies the e-value property unconditionally; without an explicit proof or counterexample analysis under the paper's weakest assumption (no extra regularity), the guarantee for model-free ES backtesting remains open.

    Authors: The referee is correct that the e-process validity and optimality criteria in Section 4 rest on the backtest e-statistic being an unconditional e-value. The constructions follow directly from e-process theory once this property holds. We agree that an explicit proof (or counterexample analysis) under the paper's weakest assumptions is required to close the argument for model-free ES backtesting. In the revision we will insert a short proof of the unconditional e-value property for the ES backtest e-statistic, together with a brief discussion of the no-extra-regularity case. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external identification results to backtesting without self-referential reduction

full rationale

The paper characterizes unique backtest e-statistics for VaR and ES by invoking recent external results on identification functions, then constructs e-processes from them. No equations or steps in the provided abstract or description reduce a claimed prediction or uniqueness result to a fitted parameter or self-citation chain internal to this work. The model-free application is presented as a direct use of those external tools rather than a re-derivation that loops back on the paper's own inputs. This is the normal case of an independent application of prior mathematical results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The approach rests on the existence of identification functions from prior work and the new construction of backtest e-statistics; no free parameters are introduced in the abstract.

axioms (1)
  • domain assumption Identification functions exist for VaR and ES and can characterize unique backtest e-statistics
    Invoked to obtain the unique forms of backtest e-statistics for VaR and ES.
invented entities (1)
  • backtest e-statistics no independent evidence
    purpose: Formulate e-processes for risk measure forecasts
    Newly defined objects to enable the e-process construction for backtesting.

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