Wild Bootstrap Inference for Linear Regressions with Many Covariates

Wenze Li

REVIEW 2 major objections 3 minor 35 references

Wild Bootstrap Inference for Linear Regressions with Many Covariates

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that a modification to the wild bootstrap—multiplying null-restricted residuals by an adjustment factor built from a cross-fit variance estimator—gives asymptotically valid t-tests in linear regressions with many…

desk verdict Promising enough for a referee, but Theorem 1 is not proven as stated: the multiplier CLT step lacks a Lindeberg condition, and the simulations use a Bernoulli-dummy design that can make the procedure undefined. read the letter →

arxiv 2506.20972 v1 pith:NQGCPMOJ submitted 2025-06-26 econ.EM

Wenze Li This is my paper · ORCID

classification econ.EM MSC 62F4062J0562E20

keywords wildbootstrapmanycovariatesheteroskedasticitylinearregressionvalidityhigh-dimensionalinferenceresidual-makermatrixcross-fitvarianceestimator

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

This paper asks whether the wild bootstrap can be made reliable for inference on one coefficient in a linear regression when the number of control variables is a large fraction of the sample size and the errors are heteroskedastic. Standard wild bootstrap is known to fail in this regime, and normal-theory variance estimators over-reject badly. The proposed fix multiplies the null-restricted residuals by an adjustment factor before drawing bootstrap weights, and the main theorem shows the bootstrap t-statistic and the true t-statistic have the same limiting distribution uniformly over the real line. If right, practitioners get a straightforward bootstrap that holds its size even when controls are 90 percent of the observations, which the simulations confirm for n=100.

What carries the argument

The load-bearing object is the adjustment factor $a_n(\beta_0)$, together with the residual-maker matrix $M_n = I_n - W(W'W)^{-1}W'$ whose diagonal entries the bootstrap divides by. The factor is chosen so that the bootstrap meat $n^{-1}\sum_i \hat v_i^2 y^*_i \acute u^*_i$ approximates the null-imposed cross-fit quantity $\tilde\Sigma_n(\beta_0)$; the $\max\{\cdot,1/n\}$ guard keeps the square root well defined when the estimated variance is not positive. The proof decomposes the difference between bootstrap and sample variance estimators into three remainder terms and shows each is conditionally $o_{p^*}(1)$ using the conditional multiplier central limit theorem, the boundedness of $(\min_i (M_n)_{ii})^{-1}$, and a small-maximum fitted-value condition on the null-restricted predictions.

What would settle it

Set $n=100$, $q_n=90$, and include among the controls a dummy variable that equals 1 for exactly one observation; at that observation $(M_n)_{ii}=0$, so the bootstrap estimator of the variance divides by zero and the method cannot be computed. Running this design across many Monte Carlo draws and recording the minimum of $(M_n)_{ii}$ would show Assumption 4 is violated, and the claimed uniform approximation cannot hold there.

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Extended reading notes

Core claim

The central claim is Theorem 1: under the paper's Assumptions 1–5, when the null hypothesis is true, $\sup_c |F_n(c)-F^*_n(c)| = o_p(1)$, so the modified wild bootstrap t-statistic is asymptotically valid even when $q_n/n$ does not shrink to zero. The modification is to generate bootstrap errors as $u^*_i = a_n(\beta_0)\omega^*_i \tilde u_i(\beta_0)$, where $\tilde u_i$ are null-restricted residuals and $a_n(\beta_0)=\sqrt{\max\{\tilde\Sigma_n(\beta_0),1/n\}/\hat\Sigma_n(\beta_0)}$. This adjustment makes the bootstrap version of the cross-fit-style variance estimator line up with its sample counterpart, and the proof shows that conditional on the data the bootstrap t-statistic converges to a standard normal in probability while the actual t-statistic does the same. A direct consequence is that percentile-t bootstrap tests control size under many covariates and heteroskedasticity, a combination for which the paper argues no proven-valid bootstrap existed before.

Load-bearing premise

The procedure divides by the diagonal entries of the residual-maker matrix and requires every one of them to stay bounded away from zero; if any observation is completely or almost completely pinned down by the controls, the bootstrap variance estimate is undefined or unstable and the theorem's conditions fail.

Editorial extensions

If this is right

For empirical work, the procedure gives t-tests and confidence intervals for a treatment effect that remain correctly sized when controls are numerous and errors are heteroskedastic, with no need to know whether $q_n/n$ is small or large.
In the simulations with $n=100$, the modified bootstrap holds null rejection frequencies between roughly 0.031 and 0.051 across $q_n/n = 0.1$ to $0.9$, while the best normal-based estimator reaches 0.172 and HC0/HCK reach 0.581.
When $q_n/n \to 0$ the adjustment factor converges to 1, so the procedure reduces to the standard null-imposed wild bootstrap.
The percentile-t bootstrap is recommended over the percentile version, and the paper sketches a score-bootstrap extension for vector coefficients and linear hypotheses.
The panel fixed-effects simulations show the same size control when the many covariates are group dummies, with bootstrap rejection frequencies near nominal even at 50 groups.

Reading between the lines

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

A likely practical extension is to clusters: applying the same adjustment factor within cluster blocks would be natural, but the theorem's independence assumption does not cover cluster dependence, so that extension would need fresh proof.
The guard $\max\{\tilde\Sigma_n,1/n\}$ suggests a direct diagnostic: practitioners can compare the adjusted and unadjusted variance estimates, and a large gap indicates the regime where normal critical values are unreliable and the bootstrap adjustment matters.
The proof's reliance on $(\min_i (M_n)_{ii})^{-1}=O_p(1)$ implies the method should be used with caution when controls include singleton indicators or nearly saturated dummies; checking the diagonal of $M_n$ before running the bootstrap would flag unstable cases.
The paper's Monte Carlo designs use Bernoulli dummies with small success probabilities, so a natural stress test is to record the minimum diagonal of $M_n$ in those same designs to see whether Assumption 4 is actually met or whether finite-sample performance is carried by the $1/n$ guard.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a look, but the main theorem has a real hole. The adjustment factor is a clean idea, and as far as the cited literature goes, this is the first proposed valid wild bootstrap for linear regressions with q_n/n not vanishing under heteroskedasticity. That alone makes it interesting. The Monte Carlo results are also striking: the bootstrap controls size where HC0, HCK, and HCA over-reject badly, and the panel extension is sensible.

The problem is the proof of Theorem 1. The conditional multiplier CLT from van der Vaart and Wellner (Lemma 2.9.5) requires a Lindeberg condition: the largest squared coefficient in the linear form, relative to the sum of squares, must vanish. The proof instead asserts max_i sigma_i*^2 = O_p(1) and uses that to bound remainder terms. That bound is not implied by Assumptions 1–5; with only fourth moments on the errors, max_i |u_tilde_i| can be of order n^{1/4}, so max_i sigma_i*^2 can grow. More importantly, after the adjustment factor cancels, the effective coefficients are proportional to vhat_i * u_tilde_i, and the Lindeberg ratio max_i (vhat_i^2 u_tilde_i^2) / sum_k (vhat_k^2 u_tilde_k^2) need not vanish. The proof also uses max_i u_tilde_i^2 = O_p(1) in an earlier step, which is a stronger claim than the assumptions justify. So the central theorem is not established as stated.

There is also a simulation design issue. The Monte Carlo uses Bernoulli dummies with pi = 0.01 or 0.02 at n = 100. That means a nontrivial fraction of replications will have a dummy that equals 1 for exactly one observation, making (M_n)_ii = 0 and the whole procedure undefined, since division by (M_n)_ii appears in Steps 2 and 4. The paper never says how those cases are handled. If they are simply included, the reported results are not evaluating the proposed method. This is a fixable reporting problem, but it is a serious gap in the evidence as presented.

On balance: the idea is genuinely new and the direction is right, but the proof gap is load-bearing. The paper deserves a referee, not because it is correct, but because the question is important and the adjustment factor is a credibly promising construction. A serious referee would need to see a verified Lindeberg condition or a counterexample, and the simulations should either use a DGP that satisfies Assumption 4 or explicitly handle the degenerate cases.

Referee Report

2 major / 3 minor

Summary. The paper proposes a modification of the wild bootstrap for inference on a scalar regression coefficient when the number of controls q_n is a non-negligible fraction of the sample size. The modification is an adjustment factor a_n(β0) that multiplies null-imposed residuals, following the construction of Jochmans (2022). The main theoretical result, Theorem 1, claims that under Assumptions 1-5 the conditional bootstrap distribution of the percentile-t statistic approximates the null distribution uniformly. Monte Carlo simulations compare HC0, HCK, HCA and the proposed wild bootstrap with Gaussian and Rademacher weights for n=100 and q_n/n up to 0.9, and report that the bootstrap controls size better than normal-based methods.

Significance. If the theorem is correct, the paper fills a genuine gap: no previously proven-valid bootstrap is available for linear regressions with both q_n/n not tending to zero and heteroskedasticity. The proposed adjustment factor is a deterministic function of the data under the null and involves no fitted parameters, which is a useful feature. The simulation evidence is suggestive, though the reported Monte Carlo design is problematic. The main concern is the proof of the bootstrap central limit theorem, which is asserted rather than verified at the key step.

major comments (2)

[Appendix B, paragraph after Eq. (7)] The conditional multiplier CLT is not established. The proof invokes Lemma 2.9.5 of van der Vaart and Wellner (1996) to conclude that \bar t_n^* converges conditionally to N(0,1), but it does not verify the Lindeberg condition for the weighted sum. Conditional on the data, \bar t_n^* is a weighted sum of the independent bootstrap weights with coefficients proportional to \hat v_i a_n(β0) \tilde u_i(β0); after normalization the relevant quantity is max_i \hat v_i^2 \tilde u_i^2 / \sum_k \hat v_k^2 \tilde u_k^2. The proof instead uses the asserted bound max_i σ_i^{*2}=O_p(1), where σ_i^{*2}=a_n^2(β0)\tilde u_i^2(β0). That bound is not implied by Assumptions 1-5: with only \max_i E[ε_i^4|X_n,W_n]=O_p(1) in Assumption 2, max_i |\tilde u_i| can be of order n^{1/4} in probability, so max_i σ_i^{*2} need not be O_p(1). The earlier variance bound for the first term of Eq. (4), n^{-2}\sum \hat v_i^4(a_n^4 E[ω_i^{*4}]\tilde u_i^4 - a_n^4\tilde u_i^4)=o_p(1), also relies on \tilde u_i^4 being controlled at the right rate, which is not stated among the assumptions. Thus Theorem 1 is not established as written; the proof needs either a direct verification of the Lyapunov/Lindeberg condition or an additional assumption that delivers max_i \tilde u_i^2=o_p(n) and max_i \hat v_i^2 \tilde u_i^2/\hatΣ_n(β0)=o_p(1).
[Section 5, Tables 1-3] The Monte Carlo DGP violates Assumptions 2 and 4 with high probability. With q_n-1 independent Bernoulli(0.02) dummy variables and n=100, at q_n/n=0.9 there are 89 dummies; the probability that at least one dummy column is identically zero is essentially one, so W has rank deficiency and Assumption 2 fails. Moreover, singleton dummy cells are common, producing (M_n)_ii=0 and making \acute u_i = \tilde u_i/(M_n)_ii undefined, which violates Assumption 4. The paper does not state how such replications were treated. The simulation evidence should be recomputed with a design that enforces full rank and min_i(M_n)_ii>0 (for example, by redrawing W until those conditions hold), and the effective number of replications should be reported.

minor comments (3)

[Section 3, Eq. (3)] The notation \acuteΣ_n(β0) and \hatΣ_n(β0) is used in the definition of a_n(β0) before the reader is told that \acuteΣ_n(β0) is the null-imposed analogue of the meat in \acuteΩ_n; consider defining both quantities directly after the display.
[Appendix B, displays after Eq. (4)] The proof mixes o_{p*}(1) and o_p(1) for bootstrap quantities; for readability, state explicitly in each display whether the high-probability statement is under P or under P^* in probability.
[Tables 1-3, q_n/n=0.9 row] In all three tables the HC0 and HCK rejection frequencies are identical at q_n/n=0.9 (0.581, 0.574 and 0.583 respectively); this is likely a symptom of rank-deficient dummy columns, and the tables or the surrounding text should explain this coincidence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bootstrap validity proof rests on external multiplier CLT and prior variance-estimator results, with no fitted parameters or self-citation chain.

full rationale

The paper's central claim, Theorem 1, asserts that the modified wild bootstrap cdf F*_n approximates the true null cdf F_n uniformly. The proof proceeds by decomposing the bootstrap variance estimator, showing that it tracks the bootstrap conditional variances sigma*_i^2 up to o_p*(1), and then invoking the external conditional multiplier CLT, Lemma 2.9.5 of van der Vaart and Wellner (1996), for the resulting self-normalized weighted sum of i.i.d. bootstrap weights. The asymptotic normality of the original t-statistic t_n is taken from the prior external results of Cattaneo et al. (2018b) and Jochmans (2022), which are cited as established theorems and involve no overlap with the author. The adjustment factor a_n(beta0) is a deterministic function of the data under the null, not a parameter fitted to any target distribution; it is chosen to align the bootstrap variance with the Jochmans-type meat estimator, and the proof then verifies the needed remainder terms from stated moment and leverage assumptions. No prediction is obtained from a fitted input, and no load-bearing claim is justified solely by a self-citation or by a uniqueness theorem imported from the author's own prior work. The Monte Carlo results are simulations illustrating finite-sample behavior rather than estimates derived from the theory. A possible concern that the multiplier CLT's Lindeberg condition is not fully verified, and that the bound max_i sigma*_i^2 = O_p(1) may not follow from Assumptions 1-5, is a mathematical-rigor or correctness issue, not a circularity: the conclusion is not equivalent by construction to the assumptions, and the proof does not rename an input as an output. Therefore the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard bootstrap CLT machinery and on domain assumptions inherited from Cattaneo et al. (2018b) and Jochmans (2022). No free parameters are introduced by the method; the adjustment factor is a deterministic function of the data.

assumptions (5)

standard math Conditional multiplier CLT (Lemma 2.9.5 of van der Vaart and Wellner 1996)
Used in the proof of Theorem 1 to establish bootstrap normality of the t-statistic.
domain assumption Consistency of the Jochmans (2022) cross-fit variance estimator
The adjustment factor and the bootstrap variance estimator rely on the same estimator being consistent when q_n/n < 1. The paper cites Jochmans (2022) for this.
domain assumption Assumption 4: (min_i (M_n)_ii)^{-1} = O_p(1)
Ensures the leave-observation-out residuals u'_i are well defined and asymptotic negligibility of the diagonal terms in the variance decomposition holds.
domain assumption Assumption 3: approximation error terms satisfy eta_n + n(eta_n - rho_n) + n chi_n eta_n = o(1)
Controls misspecification of the linear model as an approximation to the conditional mean, needed for the original t_n asymptotic normality.
domain assumption Assumption 5: max_i |mu_hat_i(beta0)|/sqrt(n) = o_p(1) under null
Controls the conditional variance of the bootstrap variance estimator; needed for the proof of equation (7).

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Cite this review

Pith. "Pith review of Wild Bootstrap Inference for Linear Regressions with Many Covariates." pith.science (2026). https://pith.science/paper/NQGCPMOJ

@misc{pith2026250620972,
  author       = {Pith},
  title        = {Pith review of: Wild Bootstrap Inference for Linear Regressions with Many Covariates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQGCPMOJ}},
  note         = {Machine review of arXiv:2506.20972}
}

read the original abstract

We propose a simple modification to the wild bootstrap procedure and establish its asymptotic validity for linear regression models with many covariates and heteroskedastic errors. Monte Carlo simulations show that the modified wild bootstrap has excellent finite sample performance compared with alternative methods that are based on standard normal critical values, especially when the sample size is small and/or the number of controls is of the same order of magnitude as the sample size.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 30 canonical work pages

[1]

Boot, T. and J. W. Ligtenberg (2023): Identification- and many instrument-robust inference via invariant moment conditions, arXiv:2303.07822

arXiv 2023
[2]

Boot, T. and D. Nibbering (2024): Inference on LATEs with covariates, arXiv preprint arXiv:2402.12607

arXiv 2024
[3]

Cameron, A. C., J. B. Gelbach, and D. L. Miller (2008): Cameron(2008) Bootstrap-based improvements for inference with clustered errors , The Review of Economics and Statistics, 90, 414--427

work page 2008
[4]

Cattaneo, M. D., M. Jansson, and W. K. Newey (2018 a ): Alternative asymptotics and the partially linear model with many regressors, Econometric Theory, 34, 277--301

work page 2018
[5]

--- -.1pt --- -.1pt --- (2018 b ): Inference in linear regression models with many covariates and heteroscedasticity, Journal of the American Statistical Association, 113, 1350--1361

work page 2018
[6]

Mellace, and Z

Crudu, F., G. Mellace, and Z. S \'a ndor (2021): Inference in instrumental variable models with heteroskedasticity and many instruments, Econometric Theory, 37, 281--310

work page 2021
[7]

Davidson, R. and E. Flachaire (2008): Davidson-Flachaire(2008) The wild bootstrap, tamed at last , Journal of Econometrics, 146, 162--169

work page 2008
[8]

Davidson, R. and J. G. MacKinnon (2008): Davidson-Mackinnon(2008) Bootstrap inference in a linear equation estimated by instrumental variables , The Econometrics Journal, 11, 443--477

work page 2008

Show all 35 references

[9]

--- -.1pt --- -.1pt --- (2010): Davidson-Mackinnon(2010) Wild bootstrap tests for IV regression , Journal of Business & Economic Statistics, 28, 128--144

2010
[10]

--- -.1pt --- -.1pt --- (2014): Davidson-Mackinnon(2014b) Bootstrap confidence sets with weak instruments , Econometric Reviews, 33, 651--675

2014
[11]

Djogbenou, A. A., J. G. MacKinnon, and M. . Nielsen (2019): Djogbenou-Mackinnon-Nielsen(2019) Asymptotic theory and wild bootstrap inference with clustered errors , Journal of Econometrics, 212, 393--412

2019
[12]

Dov \` , M.-S., A. B. Kock, and S. Mavroeidis (2024): A Ridge-Regularized Jackknifed Anderson-Rubin Test, Journal of Business & Economic Statistics, 1--12

2024
[13]

El Karoui, N. and E. Purdom (2018): Can we trust the bootstrap in high-dimensions? The case of linear models, The Journal of Machine Learning Research, 19, 170--235

2018
[14]

Evdokimov, K. et al. (2018): Inference in Instrumental Variable Regression Analysis with Heterogeneous Treatment Effects, Tech. rep., Princeton University. Economics Department

2018
[15]

Finlay, K. and L. M. Magnusson (2019): Finlay-Magnusson(2019) Two applications of wild bootstrap methods to improve inference in cluster-IV models , Journal of Applied Econometrics, 34, 911--933

2019
[16]

Hall, P. and J. L. Horowitz (1996): Bootstrap critical values for tests based on generalized-method-of-moments estimators, Econometrica, 64, 891--916

1996
[17]

Horowitz, J. L. (2001): Horowitz(2001) The bootstrap , in Handbook of econometrics, Elsevier, vol. 5, 3159--3228

2001
[18]

(2022): Heteroscedasticity-robust inference in linear regression models with many covariates, Journal of the American Statistical Association, 117, 887--896

Jochmans, K. (2022): Heteroscedasticity-robust inference in linear regression models with many covariates, Journal of the American Statistical Association, 117, 887--896

2022
[19]

Kaffo, M. and W. Wang (2017): Kaffo-Wang(2017) On bootstrap validity for specification testing with many weak instruments , Economics Letters, 157, 107--111

2017
[20]

Kline, P. and A. Santos (2012): Kline-Santos(2012) A score based approach to wild bootstrap inference , Journal of Econometric Methods, 1, 23--41

2012
[21]

Wang, and Y

Lim, D., W. Wang, and Y. Zhang (2024 a ): A conditional linear combination test with many weak instruments, Journal of Econometrics, 238, 105602

2024
[22]

--- -.1pt --- -.1pt --- (2024 b ): A Dimension-Agnostic Bootstrap Anderson-Rubin Test For Instrumental Variable Regressions, arXiv preprint arXiv:2412.01603

2024
[23]

MacKinnon, J. G. (2023): Fast cluster bootstrap methods for linear regression models, Econometrics and Statistics, 26, 52--71

2023
[24]

(1993): Bootstrap and wild bootstrap for high dimensional linear models, The Annals of Statistics, 21, 255--285

Mammen, E. (1993): Bootstrap and wild bootstrap for high dimensional linear models, The Annals of Statistics, 21, 255--285

1993
[25]

Matsushita, Y. and T. Otsu (2024): A jackknife Lagrange multiplier test with many weak instruments, Econometric Theory, 40, 447--470

2024
[26]

Mikusheva, A. and L. Sun (2022): Inference with many weak instruments, Review of Economic Studies, forthcoming

2022
[27]

Moreira, M. J., J. Porter, and G. Suarez (2009): Moreira-Porter-Suarez(2009) Bootstrap validity for the score test when instruments may be weak , Journal of Econometrics, 149, 52--64

2009
[28]

(2023): An Identification and Dimensionality Robust Test for Instrumental Variables Models, arXiv preprint arXiv:2311.14892

Navjeevan, M. (2023): An Identification and Dimensionality Robust Test for Instrumental Variables Models, arXiv preprint arXiv:2311.14892

2023 arXiv
[29]

Nielsen, J

Roodman, D., M. . Nielsen, J. G. MacKinnon, and M. D. Webb (2019): Roodman-Nielsen-MacKinnon-Webb(2019) Fast and wild: Bootstrap inference in Stata using boottest , The Stata Journal, 19, 4--60

2019
[30]

van der Vaart, A. and J. Wellner (1996): Weak Convergence and Empirical Processes: With Applications to Statistics, Springer Science & Business Media

1996
[31]

Wang, W. and F. Doko Tchatoka (2018): Wang-Doko(2018) On bootstrap inconsistency and Bonferroni-based size-correction for the subset Anderson--Rubin test under conditional homoskedasticity , Journal of Econometrics, 207, 188--211

2018
[32]

Wang, W. and M. Kaffo (2016): Bootstrap inference for instrumental variable models with many weak instruments, Journal of Econometrics, 192, 231--268

2016
[33]

Wang, W. and Q. Liu (2015): Bootstrap-based selection for instrumental variables model, Economics Bulletin, 35, 1886--1896

2015
[34]

Wang, W. and Y. Zhang (2024): Wild bootstrap inference for instrumental variables regressions with weak and few clusters, Journal of Econometrics, 241, 105727

2024
[35]

(2024): Inference with Many Weak Instruments and Heterogeneity, arXiv preprint arXiv:2408.11193

Yap, L. (2024): Inference with Many Weak Instruments and Heterogeneity, arXiv preprint arXiv:2408.11193

2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Boot, T. and J. W. Ligtenberg (2023): Identification- and many instrument-robust inference via invariant moment conditions, arXiv:2303.07822

arXiv 2023

[2] [2]

Boot, T. and D. Nibbering (2024): Inference on LATEs with covariates, arXiv preprint arXiv:2402.12607

arXiv 2024

[3] [3]

Cameron, A. C., J. B. Gelbach, and D. L. Miller (2008): Cameron(2008) Bootstrap-based improvements for inference with clustered errors , The Review of Economics and Statistics, 90, 414--427

work page 2008

[4] [4]

Cattaneo, M. D., M. Jansson, and W. K. Newey (2018 a ): Alternative asymptotics and the partially linear model with many regressors, Econometric Theory, 34, 277--301

work page 2018

[5] [5]

--- -.1pt --- -.1pt --- (2018 b ): Inference in linear regression models with many covariates and heteroscedasticity, Journal of the American Statistical Association, 113, 1350--1361

work page 2018

[6] [6]

Mellace, and Z

Crudu, F., G. Mellace, and Z. S \'a ndor (2021): Inference in instrumental variable models with heteroskedasticity and many instruments, Econometric Theory, 37, 281--310

work page 2021

[7] [7]

Davidson, R. and E. Flachaire (2008): Davidson-Flachaire(2008) The wild bootstrap, tamed at last , Journal of Econometrics, 146, 162--169

work page 2008

[8] [8]

Davidson, R. and J. G. MacKinnon (2008): Davidson-Mackinnon(2008) Bootstrap inference in a linear equation estimated by instrumental variables , The Econometrics Journal, 11, 443--477

work page 2008