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arxiv: 2605.20634 · v1 · pith:GZWGDLLRnew · submitted 2026-05-20 · 📊 stat.ME · math.ST· stat.TH

New Confidence Regions for Linear Regression Parameters with Stationary-Ergodic Dependent Errors

Mous-Abou Hamadou , Martial Longla , Mathias Nthiani Muia , Mahmud Hasan This is my paper

Pith reviewed 2026-05-21 03:05 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords random smoothingjoint confidence regionsdependent errorsstationary ergodiclinear regressionasymptotic normalitylong-run variance
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The pith

A random-smoothing estimator with an auxiliary sample produces valid joint confidence regions for linear regression coefficients under stationary ergodic errors without estimating long-run variance.

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

This paper develops a method to construct joint confidence regions for the coefficients in linear regression when both the regressors and the errors exhibit stationary ergodic dependence with unspecified serial correlation. By applying random smoothing to a vector of regression and second-moment statistics using an independent auxiliary sample and a shrinking bandwidth, the resulting estimator is asymptotically normal. This permits Wald-type confidence regions and simultaneous intervals. A sympathetic reader would care because the approach sidesteps direct long-run variance estimation or the need to specify a parametric model for the dependence structure.

Core claim

Under stationarity, ergodicity, and finite second moments, the random-smoothing estimator applied to a vector of regression and second-moment statistics is asymptotically normal and yields Wald confidence regions and simultaneous confidence intervals without direct long-run variance estimation or a parametric dependence model.

What carries the argument

The random-smoothing estimator, which uses an independent auxiliary sample and shrinking bandwidth applied to the vector of regression and second-moment statistics.

If this is right

  • Wald confidence regions can be formed directly from the smoothed estimator without separate long-run variance estimation.
  • The procedure applies to diverse dependence structures including ARMA, ARFIMA, copula-based Markov chains, and fractional Gaussian noise.
  • A scaled version with data-driven bandwidth and mild truncation yields improved finite-sample stability.
  • Simulations demonstrate near-nominal coverage and competitive region volumes relative to Newey-West and MAC estimators.

Where Pith is reading between the lines

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

  • The method might reduce the practical burden of covariance estimation in other linear models with time-series dependence.
  • Extensions could test performance when the auxiliary sample size is chosen adaptively rather than fixed.
  • Similar smoothing could be applied to construct intervals for nonlinear functionals of the same stationary ergodic data.

Load-bearing premise

The regressors and errors are jointly stationary and ergodic with finite second moments, the auxiliary sample is independent of the main sample, and the bandwidth shrinks at a suitable rate.

What would settle it

If large-sample simulations or applications from jointly stationary ergodic processes show empirical coverage of the constructed regions falling substantially below nominal levels, the asymptotic normality and validity claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.20634 by Mahmud Hasan, Martial Longla, Mathias Nthiani Muia, Mous-Abou Hamadou.

Figure 1
Figure 1. Figure 1: Joint coverage of the nominal 95% ellipsoid for β under ARMA(1, 1) errors over a grid of (ϕ, θ) and under extended FGM errors over the admissible (λ1, λ2) region, for n = 1000 and Gaussian or standardized t5 margins [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Joint coverage of the nominal 95% ellipsoid for β under ARFIMA(0, d, 0) errors as a function of d, for n ∈ {1000, 5000} and Gaussian or standardized t5 margins [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Joint coverage of the nominal 95% ellipsoid for β under fractional Gaussian noise errors as a function of H, for n ∈ {1000, 5000} and Gaussian or standardized t5 margins [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagnostic plots for model (16): residuals versus fitted values and regressors, together with the residual ACF [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pairwise projections of the joint confidence region for the slope coefficients based on the full sample and the two [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
read the original abstract

We develop joint confidence regions for linear regression coefficients when the regressors and errors are jointly stationary and ergodic with unspecified serial dependence. The method applies random smoothing, using an independent auxiliary sample and shrinking bandwidth, to a vector of regression and second-moment statistics. Under stationarity, ergodicity, and finite second moments, the estimator is asymptotically normal and yields Wald confidence regions and simultaneous confidence intervals without direct long-run variance estimation or a parametric dependence model. For implementation, we introduce a scaled estimator with data-driven bandwidth selection and a mild truncation that improves finite-sample stability. Simulations under ARMA, ARFIMA, copula-based Markov errors, and fractional Gaussian noise, with Gaussian and heavy-tailed margins, show near-nominal coverage and competitive region volumes relative to Newey-West HAC and MAC. A winter Beijing PM2.5 application illustrates the procedure. Keywords: Random smoothing, Joint inference, Confidence regions, Dependent errors, Long memory, Regression inference

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 develops a random-smoothing estimator for constructing joint Wald confidence regions and simultaneous intervals for linear regression coefficients when regressors and errors are jointly stationary and ergodic with unspecified serial dependence. An independent auxiliary sample and shrinking bandwidth are applied to a vector of regression and second-moment statistics; under stationarity, ergodicity, and finite second moments the estimator is shown to be asymptotically normal, yielding the regions without direct long-run variance estimation or a parametric dependence model. A scaled version with data-driven bandwidth selection and mild truncation is proposed for implementation. Simulations under ARMA, ARFIMA, copula Markov, and fractional Gaussian noise (Gaussian and heavy-tailed) demonstrate near-nominal coverage and competitive volumes relative to Newey-West HAC and MAC; a Beijing PM2.5 application is included.

Significance. If the central asymptotic normality result holds under the stated conditions, the approach supplies a flexible, model-free alternative to HAC-based inference for time-series regression, particularly useful when dependence is complex or long-range. The avoidance of explicit long-run variance estimation and the data-driven bandwidth are practical strengths. Reproducible simulation code and coverage across diverse dependence structures (including long-memory cases) add value; the method could see use in environmental and econometric applications where standard HAC procedures are sensitive to bandwidth choice.

major comments (2)
  1. [Asymptotic theory section] Asymptotic theory section: the statement of asymptotic normality requires explicit rates on the bandwidth h_n (e.g., h_n → 0 with n h_n → ∞ or analogous) and on the auxiliary-sample size relative to the main sample; without these rates stated as assumptions or lemmas, it is impossible to confirm that the Wald-region construction follows directly from the limiting normal distribution under only stationarity, ergodicity, and second-moment finiteness.
  2. [Implementation and simulation sections] Implementation and simulation sections: the data-driven bandwidth procedure is described as scaled and truncated, yet no consistency result or rate condition is supplied showing that the selected bandwidth satisfies the requirements of the asymptotic theory; this leaves a gap between the theoretical justification and the practical estimator used in the reported coverage probabilities.
minor comments (2)
  1. [Abstract] Abstract: the claim of 'without direct long-run variance estimation' is correct but could briefly note that random smoothing still produces an implicit smoothed covariance; a one-sentence clarification would avoid potential misreading.
  2. [Simulation tables] Simulation tables: for the fractional-Gaussian-noise heavy-tailed case the reported region volumes are larger than those of MAC; adding a short discussion of this trade-off would help readers decide when the new method is preferable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Asymptotic theory section] Asymptotic theory section: the statement of asymptotic normality requires explicit rates on the bandwidth h_n (e.g., h_n → 0 with n h_n → ∞ or analogous) and on the auxiliary-sample size relative to the main sample; without these rates stated as assumptions or lemmas, it is impossible to confirm that the Wald-region construction follows directly from the limiting normal distribution under only stationarity, ergodicity, and second-moment finiteness.

    Authors: We appreciate the referee highlighting this point. The proof of Theorem 1 does rely on the bandwidth satisfying h_n → 0 and n h_n → ∞ (to ensure vanishing bias while retaining sufficient variability for the random smoothing) together with the auxiliary sample size m_n satisfying m_n = o(n) (to preserve the joint stationarity-ergodicity and independence from the main sample). These rates are used to establish the limiting normal distribution and the subsequent Wald construction. We will revise the asymptotic theory section to state these conditions explicitly as assumptions and add a short lemma that isolates the limiting result under the stated rates. This will make the passage from asymptotic normality to the confidence regions fully transparent. revision: yes

  2. Referee: [Implementation and simulation sections] Implementation and simulation sections: the data-driven bandwidth procedure is described as scaled and truncated, yet no consistency result or rate condition is supplied showing that the selected bandwidth satisfies the requirements of the asymptotic theory; this leaves a gap between the theoretical justification and the practical estimator used in the reported coverage probabilities.

    Authors: We agree that a formal consistency result for the data-driven selector would close the theoretical gap. Under the fully nonparametric stationary-ergodic setting, however, proving that the selected bandwidth automatically satisfies the required rates would demand additional structure (e.g., specific mixing coefficients or long-memory parameters) that we deliberately avoid. The scaling and mild truncation are instead constructed to keep the chosen h_n inside the admissible range for the asymptotics, and the extensive simulation study (ARMA, ARFIMA, copula Markov, fractional Gaussian noise, both Gaussian and heavy-tailed) shows near-nominal coverage. In the revision we will add a paragraph in the implementation section that explicitly acknowledges the heuristic character of the selector, states the truncation bounds, and reports supplementary simulation results that compare the data-driven choice against fixed bandwidths known to obey the rates. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives asymptotic normality of the random-smoothing estimator directly from joint stationarity, ergodicity, and finite second-moment assumptions on regressors and errors, together with an independent auxiliary sample and bandwidth shrinking at a suitable rate. Wald regions and simultaneous intervals are then obtained from the limiting normal distribution without any redefinition of the long-run variance or fitting of the target quantity to the same data. No self-citation is invoked as a load-bearing uniqueness theorem, no ansatz is smuggled via prior work, and the data-driven bandwidth is presented as a tuning parameter rather than a circular re-expression of the quantity being estimated. The derivation therefore remains self-contained against external asymptotic theory and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on stationarity and ergodicity assumptions plus the existence of an independent auxiliary sample and a shrinking bandwidth; these are domain assumptions rather than fitted parameters or new entities.

free parameters (1)
  • bandwidth
    Data-driven bandwidth selection with mild truncation for finite-sample stability; its rate must satisfy the shrinking condition for asymptotic normality.
axioms (2)
  • domain assumption Regressors and errors are jointly stationary and ergodic with finite second moments
    Invoked to obtain asymptotic normality of the smoothed statistic.
  • domain assumption Auxiliary sample is independent of the main sample
    Required for the random-smoothing construction to produce the stated limiting distribution.

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