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arxiv: 2605.16866 · v2 · pith:23V3ROKVnew · submitted 2026-05-16 · 📊 stat.ME · econ.EM

Heavy Tails and Predictive Ability Testing

Jonas F. Frederiksen , Muneya Matsui , Rasmus S. Pedersen This is my paper

Pith reviewed 2026-05-20 15:31 UTC · model grok-4.3

classification 📊 stat.ME econ.EM
keywords Diebold-Mariano testheavy tailsstable limit theoremsubsampling inferenceforecast evaluationinfinite variancepredictive accuracymixing time series
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The pith

The Diebold-Mariano test rejects a true null as often as 70 percent of the time when loss differentials have infinite variance.

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

This paper examines the behavior of standard tests for comparing forecast accuracy when the underlying loss differentials have heavy tails. It establishes that the Diebold-Mariano statistic no longer follows its usual normal limit and instead converges to a non-Gaussian stable distribution. Conventional critical values therefore produce badly distorted test sizes. The authors prove a new limit theorem for strongly mixing series with regularly varying tails of index between one and two, then construct a subsampling procedure that remains correctly sized without estimating variances or tail indices. An empirical example with emerging-market exchange-rate risk forecasts shows that this adjustment can reverse which model appears superior.

Core claim

When loss differentials have infinite variance, the Diebold-Mariano test statistic converges to a nonstandard limit involving non-Gaussian stable random variables. As a consequence, conventional critical values can yield severely distorted inference: a nominal 5% test may reject a true null as often as 70% of the time. To establish these results, a new stable limit theorem is developed for strongly mixing, infinite-variance time series processes. Building on this theory, sub-sampling-based inference is shown to remain valid irrespective of tail-heaviness and requires no estimation of long-run variances or tail indices.

What carries the argument

New stable limit theorem for strongly mixing infinite-variance time series processes that justifies subsampling inference for the Diebold-Mariano statistic without variance or tail-index estimation.

If this is right

  • Accounting for heavy tails can substantially alter conclusions about which forecast performs better in applications such as risk forecasting for exchange rates.
  • Subsampling provides valid tests without any need to estimate long-run variances or tail indices.
  • The size distortion affects any comparison that relies on the Diebold-Mariano statistic whenever tails are regularly varying with index in (1,2).
  • Standard normal-based procedures become unreliable for predictive-ability testing under infinite variance.

Where Pith is reading between the lines

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

  • Many existing studies that compare economic or financial forecasts may require re-evaluation if the loss differentials exhibit heavy tails.
  • The subsampling method could be extended to other forecast-evaluation statistics that share the same limiting behavior under infinite variance.
  • Practitioners should examine the tail properties of their loss series before interpreting conventional test results.

Load-bearing premise

Loss differentials form a strongly mixing sequence whose tails are regularly varying with index between 1 and 2.

What would settle it

Generate loss differential series from a stable distribution with tail index 1.5, apply the Diebold-Mariano test using standard normal critical values, and observe whether rejection rates stay near 5 percent or rise substantially higher.

Figures

Figures reproduced from arXiv: 2605.16866 by Jonas F. Frederiksen, Muneya Matsui, Rasmus S. Pedersen.

Figure 1
Figure 1. Figure 1: Rejection frequencies under alternatives E[Xt ] ̸= 0 for the subsampling-based test in Algorithm 3.1. The DGP (Xt)t=1,...,n is given by (5.1) with δ ∈ [0, 2.5] and with iid noise (Zt)t=1,...,n is Stable(κ, 0, 1, 0)-distributed, κ ∈ (1, 2). The test is carried out at a 5% nominal level. The x-axes indicate the values of E[Xt ] = 2δ. The rejection frequencies are based on M = 104 Monte-Carlo replications. We… view at source ↗
Figure 2
Figure 2. Figure 2: Rejection frequencies under alternatives for tests based on Algorithm 3.1 and the Diebold–Mariano test. The tests are carried out at a 5% nominal level. The DGP is given by (5.1) with (Zt) iid and δ ∈ [0, 0.7]. In the first row Zt is Student(κ)-distributed. In the second row Zt is Skt(κ)-distributed. The x-axes indicate the values of E[Xt ] = 2δ. The rejection frequencies are based on M = 104 Monte-Carlo r… view at source ↗
Figure 3
Figure 3. Figure 3: Left: The series of loss differentials when comparing VaR-forecasts based on rolling￾window and GARCH approaches, respectively. The red lines indicate plus/minus the empirical 99th percentile of the series in absolute values. Right: Hill plot based on the absolute values, where the dashed red line indicates κ = 1.5 [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
read the original abstract

We study the asymptotic behaviour of widely used tests for evaluating and comparing predictive accuracy when forecast errors exhibit heavy tails. In particular, when loss differentials have infinite variance, the Diebold-Mariano test statistic converges to a nonstandard limit involving non-Gaussian stable random variables. As a consequence, conventional critical values can yield severely distorted inference: a nominal 5$\%$ test may reject a true null as often as 70$\%$ of the time. To establish these results, we develop a new stable limit theorem for strongly mixing, infinite-variance time series processes. Building on this theory, we consider sub-sampling-based inference that remains valid irrespective of tail-heaviness and requires no estimation of long-run variances or tail indices. An application to risk forecasts for emerging-market exchange rates shows that accounting for heavy tails can substantially alter conclusions about predictive performance relative to standard procedures.

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 manuscript studies the asymptotic behavior of the Diebold-Mariano test for predictive accuracy when loss differentials exhibit heavy tails and infinite variance. It establishes a new stable limit theorem for strongly mixing sequences whose tails are regularly varying with index in (1,2), showing convergence of the test statistic to a non-Gaussian stable random variable. This implies that conventional critical values produce severe size distortions, with a nominal 5% test rejecting a true null as often as 70% of the time. The authors propose a subsampling procedure that delivers valid inference without requiring consistent estimation of long-run variance or tail index, and apply the method to risk forecasts for emerging-market exchange rates.

Significance. If the limit theorem and its consequences hold, the paper addresses a practically relevant gap in forecast evaluation methods used in economics and finance, where heavy tails are common. The extension of stable-limit results to dependent infinite-variance processes and the accompanying subsampling approach that avoids nuisance-parameter estimation constitute a clear methodological contribution. The empirical illustration demonstrates that accounting for heavy tails can change substantive conclusions about model performance.

major comments (2)
  1. [Theorem 2.1] Theorem 2.1 (stable limit theorem): the normalization sequence and the form of the limiting stable law are stated, but the proof does not explicitly verify that the strong-mixing coefficients decay fast enough to preserve the domain-of-attraction condition for the triangular array of partial sums when the tail index lies in (1,2).
  2. [Section 4.1] Section 4.1, size-distortion calculation: the 70% rejection probability for the nominal 5% test is obtained from the cdf of the limiting stable distribution; the paper should report the exact tail index and skewness parameter used to obtain this figure and confirm that it is not an upper bound attained only at the boundary α=1+.
minor comments (2)
  1. [Equation (8)] The notation for the stable characteristic function in Equation (8) uses an unconventional parameterization; aligning it with the standard form in Nolan (2020) would improve readability.
  2. [Table 2] Table 2 reports empirical rejection frequencies but does not include results for the subsampling procedure under the same heavy-tailed designs; adding these columns would strengthen the comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive comments, which have helped us improve the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Theorem 2.1] Theorem 2.1 (stable limit theorem): the normalization sequence and the form of the limiting stable law are stated, but the proof does not explicitly verify that the strong-mixing coefficients decay fast enough to preserve the domain-of-attraction condition for the triangular array of partial sums when the tail index lies in (1,2).

    Authors: We appreciate the referee highlighting this point for added clarity. The proof of Theorem 2.1 invokes the general stable convergence result for strongly mixing sequences with regularly varying tails (drawing on standard results such as those in the literature on domain-of-attraction conditions for dependent triangular arrays). To make the verification explicit, we have inserted a short remark immediately following the statement of Theorem 2.1 that confirms the mixing rate α(n) = O(n^{-r}) with r > 1 is sufficient to preserve the required conditions when the tail index lies in (1,2). revision: yes

  2. Referee: [Section 4.1] Section 4.1, size-distortion calculation: the 70% rejection probability for the nominal 5% test is obtained from the cdf of the limiting stable distribution; the paper should report the exact tail index and skewness parameter used to obtain this figure and confirm that it is not an upper bound attained only at the boundary α=1+.

    Authors: We thank the referee for this suggestion. The reported figure of approximately 70% is obtained from the cdf of a symmetric (β = 0) stable distribution with tail index α = 1.5, a value representative of many financial loss differentials. We have revised Section 4.1 to state these parameters explicitly and to note that the size distortion is not an upper bound attained only at the boundary; the rejection probability increases as α approaches 1 from above but remains substantial throughout the interval (1,2). A brief sensitivity table for α ∈ {1.2, 1.5, 1.8} has also been added. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central contribution is a new stable limit theorem for the Diebold-Mariano statistic under strong mixing and regular variation of tails with index in (1,2). This is derived from standard assumptions on the loss differentials and extends existing results for dependent heavy-tailed sequences without reducing to self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The reported size distortions follow directly from the tail properties of the non-Gaussian stable limit, and the subsampling inference is constructed to be valid without requiring consistent variance or tail-index estimation. No step in the provided abstract or described chain equates a claimed result to its own inputs by construction; the argument remains independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the assumption that loss differentials are strongly mixing with regularly varying tails of index in (1,2); no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Loss differentials form a strongly mixing sequence with regularly varying tails of index alpha in (1,2)
    Invoked to obtain the non-Gaussian stable limit for the Diebold-Mariano statistic

pith-pipeline@v0.9.0 · 5678 in / 1328 out tokens · 30044 ms · 2026-05-20T15:31:31.965140+00:00 · methodology

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