Consistent Variable Selection for GARCH-X Models

Adriano Zanin Zambom; Beck Saunders

arxiv: 2604.25894 · v1 · submitted 2026-04-28 · 📊 stat.ME

Consistent Variable Selection for GARCH-X Models

Adriano Zanin Zambom , Beck Saunders This is my paper

Pith reviewed 2026-05-07 15:09 UTC · model grok-4.3

classification 📊 stat.ME

keywords GARCH-X modelsvariable selectionfalse discovery rateWald testsvolatility modelingconsistencyexogenous covariates

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The pith

A Wald-test FDR procedure consistently recovers the true exogenous covariates in GARCH-X volatility models.

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

The paper develops a variable selection method for GARCH-X models that identifies which external factors truly influence volatility. It combines Wald-type tests with the Benjamini-Yekutieli FDR procedure to control false positives while proving that the selected set matches the correct covariates as the sample size grows to infinity. This matters for financial modeling because including irrelevant variables or omitting relevant ones distorts volatility forecasts and risk assessments. Monte Carlo experiments under different error distributions and dependence patterns confirm the method works reliably, and the procedure is demonstrated on SP 500 returns with macroeconomic and commodity predictors.

Core claim

The proposed selection rule is consistent: with probability approaching one, it asymptotically identifies exactly the set of exogenous covariates that enter the conditional variance equation of the GARCH-X process.

What carries the argument

A multiple-hypothesis-testing rule that applies Wald statistics to each candidate covariate and retains only those that survive the Benjamini-Yekutieli FDR threshold.

If this is right

The method remains valid for high-dimensional sets of candidate exogenous series provided the FDR threshold is applied.
Volatility forecasts and risk measures can be computed from a parsimonious, correctly specified GARCH-X equation once the selection step is complete.
The same testing framework can be used to compare nested GARCH-X specifications without ad-hoc information criteria.

Where Pith is reading between the lines

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

The procedure could be adapted to other conditionally heteroskedastic models that include exogenous regressors, such as realized-volatility regressions.
In practice, analysts could re-run the selection periodically on rolling windows to detect shifts in which macro variables drive volatility.
If the regularity conditions fail in very heavy-tailed financial data, bootstrap versions of the Wald statistics might be needed to maintain the consistency guarantee.

Load-bearing premise

The GARCH-X process and the Wald statistics satisfy the regularity conditions needed for the asymptotic distribution to be valid and for the FDR control to hold under the dependence induced by the volatility dynamics.

What would settle it

In large simulated GARCH-X samples generated from a known sparse set of covariates, the procedure either includes an irrelevant covariate or drops a relevant one with positive probability that does not go to zero.

read the original abstract

In this paper we develop a consistent variable selection procedure for GARCH-X models that identifies the truly relevant exogenous covariates influencing volatility dynamics. The proposed method is based on a multiple hypothesis testing framework with Wald-type test statistics and the Benjamini-Yekutieli False Discovery Rate (FDR) procedure to control the proportion of false discoveries. We establish the consistency of the selection rule, showing that it asymptotically recovers the correct set of covariates as the sample size increases. Monte Carlo simulations across different distributions and dependence structures validate the method's accuracy and robustness. The procedure is applied to modeling the volatility of the SP 500 using macroeconomic and commodity indicators.

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.

Desk Editor's Note private letter to a colleague

The paper applies Wald tests plus fixed-q Benjamini-Yekutieli FDR to pick covariates in GARCH-X models and claims exact-set consistency, but the fixed threshold undercuts that limit.

read the letter

The paper develops a variable selection procedure for GARCH-X models. It runs Wald-type tests on each exogenous covariate and then applies the Benjamini-Yekutieli FDR rule at a fixed level to decide which ones to retain. The authors state that this rule is consistent, recovering the true set of relevant covariates with probability approaching one as sample size grows. They support the claim with Monte Carlo runs under different error distributions and dependence patterns, plus an application to S&P 500 volatility using macroeconomic and commodity indicators. The simulations and the real-data example are the parts that feel most concrete and usable right away. They give a practitioner a clear picture of how often the method picks the right variables and what the selected set looks like on actual financial series. That is the main practical contribution. The central claim of asymptotic exact recovery is the soft spot. Even granting that the Wald statistics are asymptotically normal under the GARCH-X regularity conditions, a fixed FDR level q keeps the rejection threshold bounded away from zero. The probability of at least one false positive therefore stays positive in the limit, so the probability of selecting exactly the true set cannot go to one. The abstract does not indicate any special argument that would override this standard behavior of FDR procedures. The required moment and mixing conditions for the Wald statistics under volatility clustering are also left implicit, which matters because those conditions are not automatic in GARCH-type models. This work is aimed at applied time-series people who fit volatility models and want a systematic, FDR-controlled way to screen external covariates. It is not a deep advance in either multiple testing or GARCH theory, but it packages existing tools for this setting. A referee could usefully check whether the consistency proof actually delivers exact recovery or only FDR control. I would send it to peer review.

Referee Report

3 major / 2 minor

Summary. The paper proposes a variable selection procedure for GARCH-X models that uses Wald-type test statistics combined with the Benjamini-Yekutieli FDR procedure at a fixed level q. It claims to prove that this rule is consistent in the sense that it asymptotically recovers the exact true set of relevant exogenous covariates with probability approaching 1 as sample size n grows. The claim is supported by Monte Carlo simulations under various error distributions and dependence structures, plus an empirical application to SP500 volatility modeling with macroeconomic and commodity covariates.

Significance. If the consistency result holds under the stated conditions, the method would offer a principled way to select exogenous drivers in volatility models, reducing misspecification risk in financial econometrics. The simulation validation across distributions and the real-data illustration add practical relevance, though the theoretical guarantee is the core contribution.

major comments (3)

[Abstract and theoretical results] Abstract and theoretical results: The asserted consistency (P(selected set = true set) → 1) cannot hold under the Benjamini-Yekutieli procedure at fixed q > 0. The BY threshold remains bounded away from zero with positive probability, so the probability of rejecting at least one true null stays bounded away from zero even if p-values are asymptotically uniform under the nulls. This directly contradicts the exact-set-recovery claim.
[Section on regularity conditions and Wald statistics] Section on regularity conditions and Wald statistics: The manuscript supplies no explicit verification that the Wald statistics for the GARCH-X parameters satisfy the regularity conditions needed for asymptotic uniformity of the p-values under the null (accounting for the volatility clustering and dependence induced by the GARCH-X dynamics). Without this, even FDR control is not guaranteed.
[Simulation section] Simulation section: The Monte Carlo design does not report the finite-sample frequency of exact set recovery (or the rate at which false positives vanish) as n increases, nor does it include high-dimensional regimes with many irrelevant covariates that would stress-test the consistency claim.

minor comments (2)

[Model specification] Clarify the precise form of the GARCH-X model (including lag orders and the exogenous vector) and the exact expression for the Wald statistic in the main text rather than relegating it to an appendix.
[Empirical application] In the empirical application, report the selected covariates, the realized FDR level, and a comparison against a Bonferroni or lasso-based benchmark to illustrate the practical difference.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We appreciate the referee's detailed and constructive feedback on our manuscript. The comments raise important points about the theoretical claims, supporting conditions, and simulation design, which we address point by point below. We outline revisions where appropriate.

read point-by-point responses

Referee: The asserted consistency (P(selected set = true set) → 1) cannot hold under the Benjamini-Yekutieli procedure at fixed q > 0. The BY threshold remains bounded away from zero with positive probability, so the probability of rejecting at least one true null stays bounded away from zero even if p-values are asymptotically uniform under the nulls. This directly contradicts the exact-set-recovery claim.

Authors: We thank the referee for this observation. The referee is correct: with a fixed q > 0 the BY threshold does not vanish, so the probability of at least one false rejection remains bounded away from zero and exact-set recovery with probability approaching 1 cannot be guaranteed. We will revise the abstract and the statement of the main theoretical result to clarify that the procedure controls FDR at level q and, under standard local-power conditions on the Wald tests, includes all relevant covariates with probability approaching 1, while the expected proportion of false discoveries is bounded by q. We will also note that stronger exact-recovery consistency would require q_n → 0 at a suitable rate and indicate how the proof would need to be adapted. revision: yes
Referee: The manuscript supplies no explicit verification that the Wald statistics for the GARCH-X parameters satisfy the regularity conditions needed for asymptotic uniformity of the p-values under the null (accounting for the volatility clustering and dependence induced by the GARCH-X dynamics). Without this, even FDR control is not guaranteed.

Authors: We agree that an explicit verification is required. In the revised version we will add a dedicated subsection (or appendix) that states and verifies the regularity conditions for the Wald statistics under the GARCH-X model. This will draw on existing quasi-maximum-likelihood results for GARCH processes with exogenous regressors, confirming that the score and information matrix satisfy the necessary mixing and moment conditions so that the p-values are asymptotically uniform under the null hypotheses. revision: yes
Referee: The Monte Carlo design does not report the finite-sample frequency of exact set recovery (or the rate at which false positives vanish) as n increases, nor does it include high-dimensional regimes with many irrelevant covariates that would stress-test the consistency claim.

Authors: We will expand the simulation section to report the finite-sample proportion of exact set recovery (and the decay rate of false positives) for a sequence of increasing sample sizes. We will also add a high-dimensional Monte Carlo experiment in which the number of irrelevant covariates is large relative to the relevant ones, thereby providing a more direct numerical check of the procedure's behavior under the conditions stressed by the consistency claim. revision: yes

Circularity Check

0 steps flagged

No circularity; consistency result is presented as independent theoretical derivation

full rationale

The paper claims to establish asymptotic consistency of a variable selection rule for GARCH-X models that uses Wald statistics and the Benjamini-Yekutieli FDR procedure. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or description. The consistency statement is framed as a separate theoretical result under stated regularity conditions on the GARCH-X process, rather than reducing by construction to the procedure's own fitted quantities or prior author results. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, invented entities, or non-standard axioms are described.

axioms (1)

domain assumption Standard regularity conditions for consistency and asymptotic normality of GARCH-X quasi-maximum likelihood estimators
Required for the Wald statistics to have the claimed limiting distribution under the null.

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discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Abramovich, F., Benjamini, Y., Donoho, D.L., Johnstone, I.M.: Adapting to unknown sparsity by controlling the false discovery rate. The Annals of Statistics34(2), 584–653 (2006) https://doi.org/10.1214/009053606000000074 Beyaztas, B.H., Beyaztas, U., Bandyopadhyay, S., Huang, W.-M.: New and fast block bootstrap-based prediction intervals for garch(1,1) pr...

work page doi:10.1214/009053606000000074 2006
[2]

Autoregressive Mov- ing Average Models, pp. 171–190. John Wiley & Sons, Ltd, Hoboken, New Jersey (2014).https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118856406.ch9 Francq, C., Sucarrat, G.: An equation-by-equation estimator of a multivariate log- garch-x model of financial returns. Journal of Multivariate Analysis153, 16–32 (2017) https://doi.org/10...

work page doi:10.1002/9781118856406.ch9 2014
[3]

model for volatility prediction. IMA Journal of Management Mathematics30(2), 165–185 (2018) Sucarrat, G., Grønneberg, S., Escribano, A.: Estimation and inference in univariate and multivariate log-garch-x models when the conditional density is unknown. Com- putational Statistics & Data Analysis100, 582–594 (2016) https://doi.org/10.1016/ j.csda.2015.12.00...

work page doi:10.1016/j.jspi.2024.106238 2018

[1] [1]

Abramovich, F., Benjamini, Y., Donoho, D.L., Johnstone, I.M.: Adapting to unknown sparsity by controlling the false discovery rate. The Annals of Statistics34(2), 584–653 (2006) https://doi.org/10.1214/009053606000000074 Beyaztas, B.H., Beyaztas, U., Bandyopadhyay, S., Huang, W.-M.: New and fast block bootstrap-based prediction intervals for garch(1,1) pr...

work page doi:10.1214/009053606000000074 2006

[2] [2]

Autoregressive Mov- ing Average Models, pp. 171–190. John Wiley & Sons, Ltd, Hoboken, New Jersey (2014).https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118856406.ch9 Francq, C., Sucarrat, G.: An equation-by-equation estimator of a multivariate log- garch-x model of financial returns. Journal of Multivariate Analysis153, 16–32 (2017) https://doi.org/10...

work page doi:10.1002/9781118856406.ch9 2014

[3] [3]

model for volatility prediction. IMA Journal of Management Mathematics30(2), 165–185 (2018) Sucarrat, G., Grønneberg, S., Escribano, A.: Estimation and inference in univariate and multivariate log-garch-x models when the conditional density is unknown. Com- putational Statistics & Data Analysis100, 582–594 (2016) https://doi.org/10.1016/ j.csda.2015.12.00...

work page doi:10.1016/j.jspi.2024.106238 2018