Inference in dynamic models for panel data using the moving block bootstrap

Ayden Higgins; Koen Jochmans

REVIEW 2 major objections 5 minor 1 cited by

Inference in dynamic models for panel data using the moving block bootstrap

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read In a linear panel model with fixed effects and regressors that are not strictly exogenous, a cross-section moving block bootstrap reproduces the asymptotic bias of the within-group estimator, so reverse-percentile intervals and tests are…

desk verdict MBB validity for dynamic panels with n/m → c is a real new result with a careful proof; the stationarity assumption needs a boundary remark and the simulations need more substance, but the paper deserves refereeing. read the letter →

arxiv 2502.08311 v1 pith:AFCMPNER submitted 2025-02-12 econ.EM

Ayden Higgins , Koen Jochmans This is my paper · ORCID

classification econ.EM MSC 62G0962E2062P20

keywords paneldatafixedeffectsmovingblockbootstrapwithin-groupestimatorasymptoticbiasdynamicregressionreverse-percentileintervals

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 shows that a version of the moving block bootstrap, applied by resampling blocks of adjacent time periods jointly across all cross-sectional units, replicates the entire limit distribution of the within-group estimator in linear panel models with fixed effects. The key difficulty is that when the number of cross-sections n and time periods m grow at the same rate, the within-group estimator has an asymptotic bias that standard bootstrap schemes fail to reproduce. The paper proves that the bootstrap distribution of the within-group estimator, centered at the original estimate, is consistent for the limit distribution of that estimator. Consequently, confidence ellipsoids and hypothesis tests based on the reverse-percentile bootstrap are asymptotically valid without any explicit bias correction. This matters because applied work routinely faces regressors that are correlated with past and future errors, and existing inference methods require estimating or removing the bias explicitly.

What carries the argument

The key object is the moving block bootstrap applied in the cross-section dimension: divide the m time periods into p blocks of length q (with m = p q), draw p blocks with replacement from all possible contiguous blocks, and concatenate them to form the bootstrap sample, keeping the n cross-sectional units together. What makes it work is that the bootstrap replicates the bias term $b_{n,m} = -\frac{1}{n}\sum_{i=1}^n \sum_{\tau=1}^{m-1} \frac{m-\tau}{m} \left(E(z_{it}\varepsilon_{i,t+\tau}) + E(z_{i,t+\tau}\varepsilon_{it})\right)$, which appears in the limit distribution of the within-group estimator. The proof decomposes the bootstrap version of the normal equations and shows that within-block contributions estimate these autocovariances, while between-block contributions are asymptotically negligible, so the bootstrap bias equals the original bias.

What would settle it

Simulate a panel AR(1) model, $y_{it} = \alpha_i + \beta y_{i,t-1} + \varepsilon_{it}$, with $\beta = 0.5$, $n = m = 500$, and innovations whose variance doubles halfway through the sample period, so the stationarity assumption fails. Compute the moving block bootstrap reverse-percentile interval with block length $q = 20$ and check whether empirical coverage of the true $\beta$ stays near 95% across many replications; if coverage deviates substantially, the stationarity assumption is load-bearing. Alternatively, compare the bootstrap distribution's median to the analytic bias $-(1+\beta)\sqrt{n/m}$ in a stationary AR(1) with i.i.d. innovations: a mismatch for block lengths satisfying $q = o(\sqrt{m})$ would contradict Theorem 2.

Watch

Extended reading notes

Core claim

The central discovery is that, under standard moment and mixing assumptions and when the block length q grows with m but q = o(sqrt(m)), the moving block bootstrap distribution of sqrt(nm)(beta_hat* - beta_hat), conditional on the data, is consistent for the limit distribution of sqrt(nm)(beta_hat - beta). The within-group estimator remains consistent but its limit distribution is shifted by an asymptotic bias term of order sqrt(n/m). The bootstrap correctly recreates this shift because it resamples blocks of consecutive time periods, so within-block pairs (z_it, epsilon_{i,t+tau}) and (z_{i,t+tau}, epsilon_it) reproduce the autocovariances that make up the bias. The paper shows that the cross-block contributions vanish asymptotically, leaving exactly the same bias as in the original estimator. This generalizes earlier results that required n/m -> 0, where the bias would be asymptotically negligible.

Load-bearing premise

The proof requires the time-series processes $(x_{it}, \varepsilon_{it})$ to be stationary and strong mixing with coefficients decaying faster than a specific rate; the entire bias-replication argument treats the autocovariances $E(z_{it}\varepsilon_{i,t+\tau})$ and $E(z_{i,t+\tau}\varepsilon_{it})$ as constant over time $t$, so if the data are nonstationary the bias is no longer a common location shift and the moving block bootstrap cannot be expected to reproduce it.

Editorial extensions

If this is right

Reverse-percentile bootstrap confidence intervals and tests for linear contrasts of the slope vector are asymptotically valid without any bias correction, even when n/m converges to a positive constant.
The median of the bootstrap distribution converges to the asymptotic bias, so $\check{\beta} := \hat{\beta} - \text{median}_*(\hat{\beta}^* - \hat{\beta})$ provides a bootstrap-based bias-corrected estimator.
Studentized bootstrap inference using a block-bootstrap variance estimator also achieves asymptotic size control without bias correction.
The result extends the applicability of the moving block bootstrap from the setting where n/m -> 0, previously established, to the empirically relevant case where n and m grow at the same rate.

Reading between the lines

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

Beyond the linear model considered here, the same block-resampling mechanism plausibly reproduces asymptotic bias in nonlinear fixed-effect estimators whose bias is a sum of autocovariance-type terms; the paper conjectures such a generalisation, and the normal-means variance example it mentions supports this expectation.
If cross-sectional dependence is present, the bias expression is unchanged, so a similar block-bootstrap argument may extend to panels with weak cross-sectional dependence, though the current proof abstracts from it.
A direct test of the mechanism would be to check whether the finite-sample bias of the moving block bootstrap equals the analytic Nickell-type bias in an AR(1) panel for various block lengths; the bias should match even when the normal approximation is inaccurate.
The rate condition q = o(sqrt(m)) suggests practical guidance: choose block length q so that q is large enough to capture dependence but small relative to the square root of the time dimension, which can be examined in calibration experiments.

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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

The short version: this is a solid theory paper. The main result—moving block bootstrap validity for within-group estimation in dynamic panels when n/m → c > 0—is new and the proof is careful. If you work on bootstrap inference in panel models, you should read it.

What's actually new: Gonçalves (2011) proved MBB validity only when n/m → 0, where the asymptotic bias vanishes. Here they show that under n/m → c ∈ [0,∞), the MBB replicates the bias, so reverse-percentile intervals and tests are asymptotically valid without bias correction. Theorem 1 also gives a more general bias expression than Hahn and Kuersteiner (2002) or Chudik et al. (2018). The proof chain is long but credible; they use appropriate external results (Gonçalves and White, Fitzenberger, Hansen) and are explicit about where each is used.

Where the soft spots are: first, the stationarity assumption (Assumption 1(iv)) is load-bearing. The bias b_n,m is a common location shift only because autocovariances are constant in t; the bootstrap replicates that shift by resampling blocks. If the process is nonstationary—time-varying autocovariances, local-to-unit-root, or trending regressors—the bias becomes a time average and the whole bias-replication argument collapses. The paper doesn't claim to cover those cases, but it also doesn't tell readers this boundary exists, and a referee should ask for a remark or a robustness check near the boundary.

Second, the simulation is thin: a single AR(1) DGP, one table of average quantiles, and no code or implementation details. For a paper whose selling point is that applied researchers can avoid estimating the bias, the numerical section should be more convincing. A simple coverage-probability table across a few designs would help. The absence of code is a minor issue but worth noting.

The math and citations look clean. I didn't find a circular step; the bootstrap distribution is derived from primitive assumptions, and the external results are cited appropriately. The self-citations to Higgins and Jochmans (2024) are relevant and not padding.

Verdict: this deserves a serious referee. The central claim is likely correct under the stated assumptions, and the extension over Gonçalves (2011) is real. My main ask would be to acknowledge the stationarity boundary and beef up the simulations. If I were editing, I'd send it out.

Referee Report

2 major / 5 minor

Summary. This paper studies the moving block bootstrap (MBB) for linear panel data models with fixed effects when regressors are not strictly exogenous. Under asymptotics where the number of cross-section units n and the number of time periods m grow at the same rate, the within-group estimator is consistent but has an asymptotic bias. The authors show (Theorem 2) that a panel MBB, in which blocks of adjacent time periods are resampled with replacement jointly across all cross-sections, replicates the limit distribution of the within-group estimator under Assumptions 1–3 and q→∞ with q=o(√m). This yields asymptotically valid reverse-percentile confidence intervals and tests without explicit bias correction. The proof builds on mixing conditions and established MBB results for dependent processes. A small simulation study with a stationary AR(1) process provides supportive evidence.

Significance. The paper makes a valuable contribution by extending the applicability of the moving block bootstrap to the empirically relevant case where n/m → c ∈ [0, ∞), so that the asymptotic bias of the within-group estimator is of the same order as its sampling noise. This generalizes Gonçalves (2011), which required n/m → 0 and effectively ignored the bias. The proof of Theorem 1 is rigorous and complete. The central result of Theorem 2 is carefully argued and rests on a credible chain of established results. The paper is honest about the assumptions, and the proposed procedure is attractive in practice because it avoids specifying the feedback mechanism. The simulation evidence is consistent with the theory, though limited in scope. The paper also contributes a somewhat general expression for the asymptotic bias of the within-group estimator.

major comments (2)

[Assumption 1(iv); §3; proof of Theorem 2, Eq. (A.9)–(A.10)] The bias-replication argument relies crucially on stationarity of the time-series processes. In particular, the expectations E(z_it ε_{i,t+τ}) and E(z_{i,t+τ} ε_it) must be independent of t, so that the asymptotic bias is a common location shift across time. For many applied dynamic panels (e.g., with nonstationary regressors, time-varying coefficients, or local-to-unit-root processes), Assumption 1(iv) fails. The paper does not discuss this limitation, nor does it provide robustness evidence at this boundary. Since the entire mechanism of the reverse-percentile intervals depends on the bootstrap reproducing the same location shift, this is a load-bearing point. The authors should either relax the stationarity requirement, or explicitly state its role and add a caveat about settings where it is violated, possibly with a simulation illustrating the failure.
[§3, variance estimator consistency] The paper asserts, without proof, that the bootstrap variance estimators Υ̂* and Υ̂ are consistent, referring to Gonçalves (2011, Theorem B.1). This consistency is then used to claim validity of studentized reverse-percentile intervals and of inference based on the bias-corrected estimator. Because the present asymptotic framework allows n/m → c with c > 0, unlike Gonçalves (2011), it is not immediate that the cited arguments apply unchanged. The paper should provide a proof or at least a detailed verification of the conditions needed to apply those results in the current setting. As written, the studentized results rest on an unproven auxiliary claim.

minor comments (5)

[§3, definition of bootstrap residuals] The definition of ε̂*_{it} appears to contain a typo: it uses y_it and x_it (original data) rather than y*_it and x*_it (bootstrap data). This affects the definition of the variance estimator V and should be corrected for clarity.
[§3, notation for V] In the definition of Ω̂*_{n,m}, the object V_{̟p'} depends on i as well; using V_{i,̟} would be clearer and consistent with the subsequent formulas.
[§4, Table 1] The simulation section reports average bootstrap quantiles at selected normal percentiles but does not report standard errors of these Monte Carlo averages or, more importantly, the actual coverage of confidence intervals under the proposed procedure. Adding coverage probabilities would directly substantiate the inference claim.
[Theorem 2 statement] The statement uses a supremum over a without specifying the set (presumably R^d). Please define the domain explicitly, since the distribution function is multivariate.
[General] There are several typos and OCR artifacts in the appendix (e.g., missing parentheses, misaligned summation signs). A careful proofreading pass is recommended before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the moving block bootstrap consistency result is proven from primitive assumptions and does not reduce to a fitted value or a self-citation chain.

full rationale

Theorem 2 states that, conditional on the data, the moving block bootstrap distribution of sqrt(nm)(beta_hat* - beta_hat) is consistent for the limit distribution of sqrt(nm)(beta_hat - beta). This is derived in the appendix from Assumptions 1-3 together with q -> infinity and q = o(sqrt(m)), not imported from a fitted value or from the authors' prior work. The bias replication step is explicit: the proof shows that -1/n sum_i (chi*_i - chi_i) = b_{n,m} + o_{P*}(1), where b_{n,m} is the population bias defined in Theorem 1; the bootstrap is shown to reproduce the same population quantity, which is the intended content of a consistency theorem, not a definitional identity. The only self-citations (Higgins and Jochmans 2024; Dhaene and Jochmans 2015) appear in the introduction as context for existing bias-correction and parametric-bootstrap methods and are not load-bearing in the proof of Theorem 2, which relies on external tools (Kuensch 1989; Goncalves and White 2002-2005; Fitzenberger 1997; Hansen 1991; White 2000). The stationarity condition in Assumption 1(iv) is an explicit premise, so the fact that the bias formula depends on it is an assumption-boundary issue, not circular reasoning. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors, and no known result is repackaged under new coordinates. Score 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the primitive conditions in Assumptions 1-3 (stationarity, mixing, moment bounds, rank conditions), the rate condition on the block length, and a set of published auxiliary lemmas. No entity is invented and no parameter is fitted to data; the only user-chosen input is the block length q.

free parameters (1)

Block length q
User-selected tuning parameter in the moving block bootstrap; theory requires q→∞ and q=o(√m), with no data-driven rule provided. Simulations use q=5,10,20,25.

assumptions (5)

domain assumption Assumption 1: (i) uniformly bounded moments of order 2r for x_it and ε_it (r>2); (ii) moments of order 3r for x_it ε_it; (iii) independence across i; (iv) stationary strong mixing with mixing coefficients O(h^{-s}), s>4r/(r-2).
Invoked in the proofs of Theorems 1 and 2 to apply CLTs and moment bounds for dependent processes.
domain assumption Assumption 2: Σ_{n,m} and Ω_{n,m} are uniformly positive definite.
Ensures the within-group estimator is well-defined and the limit variance is non-degenerate.
domain assumption Assumption 3: n,m→∞ with n/m→c∈[0,∞).
This is the asymptotic regime in which the within-group estimator has a non-negligible bias; the bootstrap result is proved under this rate.
domain assumption Block length q→∞ and q=o(√m), with m=pq.
Required for the bootstrap to reproduce the bias and for the remainder terms in the proof to vanish; no data-driven choice is given.
standard math Published auxiliary results: Hansen (1991, Cor. 3), Gonçalves and White (2002, 2004, 2005), Fitzenberger (1997, Lemma A.1), von Bahr and Esseen (1965).
Used as black boxes for moment inequalities, bootstrap CLTs, and uniform consistency results; not re-derived in the paper.

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Cite this review

Pith. "Pith review of Inference in dynamic models for panel data using the moving block bootstrap." pith.science (2026). https://pith.science/paper/AFCMPNER

@misc{pith2026250208311,
  author       = {Pith},
  title        = {Pith review of: Inference in dynamic models for panel data using the moving block bootstrap},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AFCMPNER}},
  note         = {Machine review of arXiv:2502.08311}
}

read the original abstract

Inference in linear panel data models is complicated by the presence of fixed effects when (some of) the regressors are not strictly exogenous. Under asymptotics where the number of cross-sectional observations and time periods grow at the same rate, the within-group estimator is consistent but its limit distribution features a bias term. In this paper we show that a panel version of the moving block bootstrap, where blocks of adjacent cross-sections are resampled with replacement, replicates the limit distribution of the within-group estimator. Confidence ellipsoids and hypothesis tests based on the reverse-percentile bootstrap are thus asymptotically valid without the need to take the presence of bias into account.

Discussion (0). Continue with ORCID to comment.

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Reference graph

Works this paper leans on

24 extracted references · 21 canonical work pages · cited by 1 Pith paper

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if new.block note output fin.entry FUNCTION b...
[2]

Arellano, M. (2003). Panel Data Econometrics . Oxford University Press

work page 2003
[3]

Arellano, M. and S. Bond (1991). Some tests of specif ication for panel data: M onte C arlo evidence and an application to employment equations. Review of Economic Studies\/ 58 , 277--297

work page 1991
[4]

Arellano, M. and J. Hahn (2007). Understanding bias in nonlinear panel models: Some recent developments. In R. Blundell, W. K. Newey, and T. Persson (Eds.), Advances In Economics and Econometrics , Volume III. Econometric Society: Cambridge University Press

work page 2007
[5]

Gon c alves, M

Cavaliere, G., S. Gon c alves, M. . Nielsen, and E. Zanelli (2024). Bootstrap inference in the presence of bias. Forthcoming in Journal of the American Statistical Association

work page 2024
[6]

Chudik, A., M. H. Pesaran, and J.-C. Yang (2018). Half-panel jackknife fixed-effects estimation of linear panels with weakly exogenous regressors. Journal of Applied Econometrics\/ 33 , 816--836

work page 2018
[7]

Dhaene, G. and K. Jochmans (2015). Split-panel jackknife estimation of fixed-ef fect models. Review of Economic Studies\/ 82 , 991--1030

work page 2015
[8]

Efron, B. (1979). Bootstrap methods: A nother look at the jackknife. Annals of Statistics\/ 7 , 1--26

work page 1979

Show all 24 references

[9]

Fitzenberger, B. (1997). The moving blocks bootstrap and robust inference for linear least squares and quantile regressions. Journal of Econometrics\/ 82 , 235--287

1997
[10]

Gon c alves, S. (2011). The moving blocks bootstrap for panel linear regression models with individual effects. Econometric Theory\/ 27 , 1048--1082

2011
[11]

Gon c alves, S. and M. Kaf fo (2015). Bootstrap inference for linear dynamic panel data models with individual fixed effects. Journal of Econometrics\/ 186 , 407--426

2015
[12]

Gon c alves, S. and H. White (2002). The bootstrap of the mean for dependent heterogeneous arrays. Econometric Theory\/ 18 , 1367--1384

2002
[13]

Gon c alves, S. and H. White (2004). Maximum likelihood and the bootstrap for nonlinear dynamic models. Journal of Econometrics\/ 119 , 199--219

2004
[14]

Gon c alves, S. and H. White (2005). Bootstrap standard error estimates for linear regression. Journal of the American Statistical Association\/ 100 , 970--979

2005
[15]

otze, G. and H. R. K\

G\"otze, G. and H. R. K\"unsch (1996). Second-order correctness of the blockwise bootstrap for stationary observations. Annals of Statistics\/ 24 , 1914--1933

1996
[16]

Hahn, J. and G. Kuersteiner (2002). Asymptotically unbiased inference for a dynamic panel model with fixed ef fects when both n and T are large. Econometrica\/ 70 , 1639--1657

2002
[17]

Hansen, B. E. (1991). Strong laws for dependent heterogeneous processes. Econometric Theory\/ 7 , 213--221

1991
[18]

Higgins, A. and K. Jochmans (2024). Bootstrap inference for fixed-effect models. Econometrica\/ 92 , 411--427

2024
[19]

K\"unsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Annals of Statistics\/ 17 , 1217--1241

1989
[20]

Neyman, J. and E. L. Scott (1948). Consistent estimates based on partially consistent observations. Econometrica\/ 16 , 1--32

1948
[21]

Nickell, S. (1981). Biases in dynamic models with fixed effects. Econometrica\/ 49 , 1417--1426

1981
[22]

van der Vaart, A. W. (2000). Asymptotic Statistics . Cambridge University Press

2000
[23]

and C.-G

von Bahr, B. and C.-G. Esseen (1965). Inequalities for the r th absolute moment of a sum of random variables, 1 r 2 . Annals of Mathematical Statistics\/ 36 , 299--303

1965
[24]

White, H. (2000). Asymptotic Theory for Econometricians, Revised Edition . Emerald

2000

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if new.block note output fin.entry FUNCTION b...

[2] [2]

Arellano, M. (2003). Panel Data Econometrics . Oxford University Press

work page 2003

[3] [3]

Arellano, M. and S. Bond (1991). Some tests of specif ication for panel data: M onte C arlo evidence and an application to employment equations. Review of Economic Studies\/ 58 , 277--297

work page 1991

[4] [4]

Arellano, M. and J. Hahn (2007). Understanding bias in nonlinear panel models: Some recent developments. In R. Blundell, W. K. Newey, and T. Persson (Eds.), Advances In Economics and Econometrics , Volume III. Econometric Society: Cambridge University Press

work page 2007

[5] [5]

Gon c alves, M

Cavaliere, G., S. Gon c alves, M. . Nielsen, and E. Zanelli (2024). Bootstrap inference in the presence of bias. Forthcoming in Journal of the American Statistical Association

work page 2024

[6] [6]

Chudik, A., M. H. Pesaran, and J.-C. Yang (2018). Half-panel jackknife fixed-effects estimation of linear panels with weakly exogenous regressors. Journal of Applied Econometrics\/ 33 , 816--836

work page 2018

[7] [7]

Dhaene, G. and K. Jochmans (2015). Split-panel jackknife estimation of fixed-ef fect models. Review of Economic Studies\/ 82 , 991--1030

work page 2015

[8] [8]

Efron, B. (1979). Bootstrap methods: A nother look at the jackknife. Annals of Statistics\/ 7 , 1--26

work page 1979