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Forecast Encompassing Tests for the Expected Shortfall

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Pith's one-line read The paper introduces three forecast encompassing tests for Expected Shortfall, proves the tests stay valid when the risk model is misspecified, and shows one variant needs only the ES forecasts banks already report.

desk verdict The paper gives the first forecast encompassing tests for Expected Shortfall; the joint and auxiliary variants are solid, but the strict ES test's advertised robustness rests on an unproven negligibility claim that the simulations do not fully support. read the letter →

arxiv 1908.04569 v3 pith:LOBZEIKW submitted 2019-08-13 q-fin.RM econ.EMmath.STstat.TH

Timo Dimitriadis , Julie Schnaitmann This is my paper · ORCID
classification q-fin.RMecon.EMmath.STstat.TH MSC 62P0562F0362M10
keywords forecastencompassingExpectedShortfallValueatRiskjointlossfunctionelicitabilitymisspecificationrobustinferencecombinationstatisticaltests
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Expected Shortfall (ES) is the risk measure the Basel III rules make banks report, but it has no loss function of its own, so forecast evaluation for it has lagged behind. This paper builds on the fact that ES is jointly elicitable with Value at Risk and constructs three encompassing tests—tests that ask whether one forecast already contains the information in any combination of two forecasts: a joint test for the VaR–ES pair, an auxiliary test that tests only the ES parameters, and a strict ES test that uses ES forecasts alone, which is the situation regulators face under current reporting rules. The paper proves that under mild regularity conditions the Wald statistics of all three tests converge to a chi-squared distribution, and it develops the asymptotic theory under model misspecification because the strict test feeds ES forecasts into the quantile link and can therefore be misspecified. An extensive simulation study across GARCH, GAS, and ES-CAViaR data generators finds that the tests are approximately correctly sized in large samples—the empirical sizes at $n=500$ are inflated, but shrink toward the nominal level as $n$ grows—and have good power, with the strict test nearly matching the auxiliary test that has access to more information. Applied to daily returns of IBM stock, the S&P 500, and the DAX 30, the tests show ES forecast combinations frequently beat the stand-alone models for the single stock, while the broad indices benefit less.

What carries the argument

The central object is the zero-homogeneous joint loss function for the pair (VaR, ES), $\rho(Y,q,e) = -\frac{1}{e}\left(e - q + \frac{(q-Y)\mathbf{1}\{Y\le q\}}{\alpha}\right) + \log(-e)$, introduced by Fissler and Ziegel (2016). Because the ES alone is not elicitable, this loss supplies the moment conditions and the semiparametric regressions $Y_{t+1} = g_q(\hat{q}_t,\beta) + u^q_{t+1}$ and $Y_{t+1} = g_e(\hat{e}_t,\eta) + u^e_{t+1}$, through which the optimal combination weights are estimated by M-estimation. A general link function $g$ maps two competing forecasts and parameters into a linear or nonlinear forecast combination; the null hypothesis that forecast 1 encompasses forecast 2 is $\theta^* = \theta_0$, meaning the weight on forecast 2 is zero and the weight on forecast 1 is one. The strict ES test sets the quantile link to $g_q(\hat{e}_t,\beta)$, using ES forecasts in place of VaR forecasts, and the misspecification-robust asymptotic theory in Propositions 2.8 and 2.9 plus Theorem 2.10 is what keeps the Wald statistics $\chi^2$ under the null.

What would settle it

Simulate a data-generating process with pronounced time-varying skewness so the ratio of ES to VaR moves substantially, generate two forecasts where one truly encompasses the other, run the strict ES test at $n=5000$, and check the empirical rejection rate; if it stays far above the 10% nominal level or the estimated weight on the encompassing forecast is not near one, the paper's negligibility argument fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that forecast encompassing for the ES can be tested through M-estimation of a joint VaR–ES regression, and that the resulting Wald tests are valid even when the regression model is misspecified. Theorem 2.10 states that under Assumption 2.7 the test statistics of all three variants converge to a chi-squared distribution with degrees of freedom equal to the number of restricted parameters: four for the joint test, two for the auxiliary and strict tests under linear link functions. The asymptotic theory generalizes the joint quantile–ES M-estimator developed in earlier work to potentially misspecified nonlinear link functions, and uses a misspecification-robust covariance estimator assembled from the nid estimator of the density quantile and the scl-sp estimator of the truncated variance. The simulation evidence belongs to the claim: all three tests show approximately correct size and good power across GARCH, VaR/ES GAS, GAS-t, and ES-CAViaR DGPs, and the strict ES test behaves almost identically to the auxiliary test even though it uses no VaR forecasts, which supports the paper's argument that the misspecification from substituting ES forecasts for quantile forecasts is negligible in realistic financial settings.

Load-bearing premise

The load-bearing premise is that putting ES forecasts where the quantile forecasts belong in the strict test does not materially shift the estimated combination weights away from $(1,0)$ when one forecast truly encompasses the other; the paper argues this effect is negligible rather than proving it, and its own VaR/ES GAS simulation still rejects about 13.5% of the time at a 10% nominal level when $n=5000$.

Editorial extensions

If this is right

  • Regulators who receive only ES forecasts, as under Basel III reporting rules, can run pairwise encompassing tests without needing the underlying VaR forecasts.
  • When both directional hypotheses are rejected, the estimated combination weights from the joint regression provide a direct way to build a combined ES forecast, and the empirical application shows such combinations frequently beat stand-alone models for the IBM stock.
  • The tests are implemented for linear, affine, and nonlinear link functions, so practitioners can choose the forecast combination formula that fits their setting.
  • Because the strict and auxiliary tests behave almost identically in the simulations, the strict test loses little by not using VaR forecasts and can replace the auxiliary test when only ES forecasts are reported.
  • The misspecification-robust asymptotic theory covers nonlinear models, which was not previously available for joint VaR–ES M-estimation and supports flexible parametric forecast combination.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The strict ES test's clean behavior probably relies on the near proportionality of VaR and ES that holds for scale-type daily return models; in asset classes with strongly time-varying higher moments, the quantile misspecification could be larger than the 13.5% rejection observed here, so a cautious user would validate the test on the target data before trusting it.
  • The same joint-loss machinery transfers to other jointly elicitable functionals, such as range value at risk or the mean–variance pair, which the paper notes as future work; the misspecification-robust M-estimation theory would need to be reworked for each new functional.
  • A direct empirical check of the strict test's key assumption is to compare its estimated combination weights with those from the correctly specified joint model on datasets where both VaR and ES forecasts are available; systematic divergence would quantify the information lost by dropping VaR forecasts.
  • Conditional encompassing tests that use instruments beyond the forecasts themselves would require extending the theory to overidentified GMM with nonsmooth moments, which the paper leaves open; if supplied, such tests could identify which variables drive the misspecification.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper proposes three forecast encompassing tests for Expected Shortfall (ES) based on the FZ0 joint loss function: a joint VaR-and-ES test, an auxiliary ES test, and a strict ES test that requires only ES forecasts. The authors develop misspecification-robust asymptotic theory for the M-estimator of the combination parameters, derive Wald tests for linear and nonlinear link functions, and investigate finite-sample size and power in simulations calibrated to GARCH, GAS, and CAViaR DGPs. An empirical application to IBM, S&P 500, and DAX returns illustrates the use of the tests for forecast selection and combination.

Significance. If the results hold, the paper makes a useful contribution to risk management by providing the first encompassing tests for ES, and the strict ES test is especially valuable because Basel III requires banks to report only ES forecasts. The misspecification-robust asymptotic theory for nonlinear link functions extends prior work, and the simulations are extensive and based on realistic DGPs. However, the central claim that the strict ES test is robust to quantile misspecification rests on a negligibility argument that is not formally proved, which limits the strength of the headline result.

major comments (2)
  1. [Section 2.2 and Theorem 2.10] The strict ES encompassing test in Definition 2.6 tests the restriction η*=η0, where η* is the pseudo-true parameter defined in (2.15). The economic null of interest, that forecast e1 encompasses e2, requires that whenever e1 genuinely encompasses e2, the pseudo-true parameter satisfies η*=(1,0). The paper states after Definition 2.6 that the misspecification bias is 'negligible' but does not provide a proof or a set of sufficient conditions. Theorem 2.10 only gives a chi-squared limit for the test statistic under the hypothesis η*=η0; it does not establish that the economic null implies η*=η0. The simulation results in Table 1 are not decisive: for the VaR/ES GAS DGP at n=5000, the strict test shows rejection rates of 13.5% and 10.6% at the 10% level for the two hypotheses, which could reflect a small asymptotic bias, though the H1 size point corresponds to a correctly specified DGP (the 1F GAS model has colinear VaR and ES). Please provide a formal analysis of the pseudo-true parameters under misspecification, or clearly redefine the null as testing η*=η0 and discuss the economic interpretation accordingly.
  2. [Section 2.3, covariance estimation] The covariance matrix estimator Ω̂ relies on the approximation Ft(gq_t(β*))≈α, which is exact when the quantile link is correctly specified but not in general. For the strict ES test under misspecification, the term Ft(gq_t(β*))-α in Eq. (2.22) is nonzero, and the approximation is justified only by a heuristic 'the degree of misspecification is small' argument. Theorem 2.10 assumes that Ω̂-Ω_n converges to zero, but the consistency of the nonparametric estimators (nid and scl-sp) under the misspecification conditions of Assumption 2.7 is not established. The authors should either prove consistency of Ω̂ or provide a reference that does so in this setting.
minor comments (4)
  1. [Table 1 and notes] The column labels 'Str ES', 'Aux ES', 'VaR ES', and 'VaR' are potentially confusing; the 'VaR ES' column refers to the joint VaR-and-ES test, but the abbreviation is not defined in the table note. I suggest renaming the column to 'Joint VaR/ES' for clarity.
  2. [Equation (2.24)] The displayed expression is split across lines and the bracket opened at the end is closed only in the following display; please reformat to make the expression self-contained.
  3. [Section 4 and Assumption 2.7] The paper uses a fixed forecasting scheme with one-time in-sample estimation, but the asymptotic theory in Section 2 does not explicitly account for estimation error in forecast parameters. The paper cites Giacomini and White (2006), but it should be stated more clearly that the tests are intended for fixed forecast methods rather than models with estimated parameters.
  4. [Section 3.2, Table 1] With 2000 Monte Carlo replications and a nominal size of 10%, the binomial standard error is about 0.9%, so the difference between 13.5% and 10% is just under four standard errors. Reporting standard errors or confidence bands for the size estimates would help the reader judge the significance of the deviations.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation detected: the tests are built on external elicitability results, the asymptotic theorems are proved in the paper, and the strict-test robustness gap is an unproven empirical claim rather than a circular reduction.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The joint VaR/ES loss function is taken from Fissler and Ziegel (2016), an external result, and the encompassing null hypotheses are defined as standard parameter restrictions. Theorem 2.10 is derived, not assumed: Propositions 2.8 and 2.9 are proved in Appendix A under explicit mixing and moment conditions in Assumption 2.7. The M-estimation framework extends Patton et al. (2019) and Dimitriadis and Bayer (2019) to misspecified nonlinear settings, and the proofs are carried out in the paper rather than imported by citation. Self-citations to Dimitriadis and Bayer (2019) and Bayer and Dimitriadis (2020) provide methodological ingredients such as covariance estimators and the joint regression setup, but these are not used to assert a conclusion that is then relabeled as a prediction; the central asymptotic distribution result is proven in the manuscript. Simulation DGPs use calibrated parameters from external sources, including Patton et al. (2019), Taylor (2019), and Creal et al. (2013), not free parameters fitted to the target result. The strict ES test's robustness under quantile misspecification is asserted rather than proved: the paper states, 'The potential model misspecification might bias the pseudo-true parameters and challenge the interpretability of the test decision, but we argue that this effect is negligible for this setup,' and Table 1 shows some size distortion at 13.5% for a 10% nominal level under the VaR/ES GAS DGP at n=5000. That is a missing proof and a correctness risk, not a circular identification of a fitted input with a prediction. No equation in the paper reduces to its own input by construction, and the strict test's parameter restriction is a definitional null hypothesis rather than a derived empirical claim presented as novel evidence.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities, forces, or dimensions. The free-parameter list is empty because the test procedure itself has no fitted constants; the DGP parameters in simulations are calibrated to real financial data from prior work. The axioms are the standard M-estimator regularity conditions and two explicit pragmatic assumptions about the strict test's misspecification behavior.

assumptions (3)
  • domain assumption Assumption 2.7(a)-(i): strong mixing, compact parameter space, unique pseudo-true minimizer with uncorrelated score, smooth bounded densities, bounded moment conditions, and separation of link functions.
    Standard regularity conditions for the consistency and asymptotic normality of the M-estimator, invoked in Propositions 2.8 and 2.9 and Theorem 2.10.
  • ad hoc to paper The misspecification bias of the strict ES test's pseudo-true parameters is negligible in realistic financial settings.
    Section 2.2, after Definition 2.6: 'we argue that this effect is negligible for this setup.' This is load-bearing for the interpretation of the strict ES test, and it is supported only by simulations, not by an analytic bound.
  • ad hoc to paper In covariance estimation, the conditional distribution function at the pseudo-true quantile is approximated by alpha, namely Ft(gq_t(beta*)) approximately equals alpha.
    Section 2.3, covariance estimation paragraph: 'As the degree of misspecification in the investigated financial time series is small, we approximate Ft(gq_t(beta*)) approximately equals alpha.' This approximation enters the misspecification-robust covariance estimator.

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Pith. "Pith review of Forecast Encompassing Tests for the Expected Shortfall." pith.science (2026). https://pith.science/paper/LOBZEIKW

@misc{pith2026190804569,
  author       = {Pith},
  title        = {Pith review of: Forecast Encompassing Tests for the Expected Shortfall},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LOBZEIKW}},
  note         = {Machine review of arXiv:1908.04569}
}
read the original abstract

We introduce new forecast encompassing tests for the risk measure Expected Shortfall (ES). The ES currently receives much attention through its introduction into the Basel III Accords, which stipulate its use as the primary market risk measure for the international banking regulation. We utilize joint loss functions for the pair ES and Value at Risk to set up three ES encompassing test variants. The tests are built on misspecification robust asymptotic theory and we investigate the finite sample properties of the tests in an extensive simulation study. We use the encompassing tests to illustrate the potential of forecast combination methods for different financial assets.

Figures

Figures reproduced from arXiv: 1908.04569 by the authors.

Figure 1
Figure 1. This figure shows power curves (empirical rejection frequencies) for the encom [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. This figure shows power curves (empirical rejection frequencies) for the encom [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. This figure shows power curves (empirical rejection frequencies) for the three [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: This figure shows power curves (empirical rejection frequencies) for the encom [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]

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