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Two-way fixed-effects estimators in unbalanced panels are asymptotically normal but off-center; a single analytical correction removes both incidental-parameter and feedback bias without knowing which regressors or selection rules are prede

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2026-07-14 13:13 UTC pith:WL7HZYPW

load-bearing objection Solid, usable extension of Fernández-Val–Weidner to unbalanced panels with mixed selection and feedback bias; the math checks out and the jackknife failure result is the practical payoff.

arxiv 2607.10246 v1 pith:WL7HZYPW submitted 2026-07-11 econ.EM

Inference for Fixed Effects Estimators when Panels are Unbalanced

Daniel Czarnowske , Amrei Stammann This is my paper
classification econ.EM
keywords panel dataunbalanced panelstwo-way fixed effectsincidental parameter problemfeedback biasasymptotic bias correctionM-estimationselection process
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Unbalanced panels are the norm in applied work, yet large-N,T theory for two-way fixed-effects estimators has largely been written for balanced data. This paper derives the joint asymptotics when observations can be missing for deterministic design reasons, stochastic response reasons, or both, under only a conditional-mean restriction. The uncorrected M-estimator remains asymptotically normal, but its mean is shifted by two bias terms: the usual incidental-parameter bias from estimating the many individual and time effects, plus a feedback bias that appears whenever regressors or the selection process itself depend on past outcomes. The authors supply an analytical bias correction that subtracts sample analogs of both terms and restores centering at zero, without requiring the analyst to know which sources of feedback are present. Because selection can depend on unobserved effects, the panel is no longer homogeneous across subsamples, so jackknife corrections that rely on that homogeneity can fail; the analytical correction does not.

Core claim

Under joint N,T asymptotics with N/T oτ∈(0,∞) and the paper’s regularity conditions, the two-way fixed-effects M-estimator etâ satisfies √(NT)(etâ−eta^{0}) o N(− au^{1/2}W_∞^{-1}B_{α,∞}− au^{-1/2}W_∞^{-1}B_{γ,∞}, W_∞^{-1}Σ_∞ W_∞^{-1}). The non-zero mean consists of incidental-parameter bias plus a Nickell-type feedback bias that can be generated by predetermined regressors or by a predetermined selection process. The analytical correction etã=etâ+Ŵ^{-1}(T^{-1}B̂_α+N^{-1}B̂_γ) is asymptotically centered at zero with the same variance, and does not require knowledge of which components are predetermined.

What carries the argument

The asymptotic expansion of the profiled score for eta after concentrating out the two-way incidental parameters heta=(α',γ'), which isolates the two bias vectors B_{α,∞} and B_{γ,∞} (the former containing the double-sum feedback term) and the Hessian and variance matrices W_∞ and Σ_∞ that incorporate the selection indicators.

Load-bearing premise

The design must keep every unit observed for a positive fraction of periods and keep every pair of periods overlapping on a positive fraction of units; if the design splits the panel into disconnected blocks, the individual and time effects cannot be jointly identified and the expansion fails.

What would settle it

In a dynamic probit (or linear) panel whose entry/exit dates depend on the individual effect, check whether the analytical correction with small positive bandwidth restores near-nominal coverage while the split-panel jackknife still over-corrects and undercover; if the analytical intervals also fail, the claimed robustness to heterogeneous selection is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Applied researchers can use the same analytical correction whether or not they believe selection or regressors are predetermined; no case distinction is required.
  • Jackknife procedures that form time-wise or unit-wise subpanels become invalid once selection depends on unobserved effects, because those subpanels are no longer exchangeable.
  • Even models with only strictly exogenous regressors can suffer feedback bias if the selection process itself is predetermined; the correction still works.
  • The asymptotic variance already incorporates selection-induced heteroskedasticity, so conventional robust standard errors remain valid after debiasing.

Where Pith is reading between the lines

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

  • The same expansion strategy should extend, with extra bookkeeping, to three-way or interactive fixed-effects models that also suffer missingness, because the profiled-score logic does not rely on additivity per se.
  • Software that currently implements two-way fixed-effects bias corrections for balanced panels can be updated by simply replacing the balanced bias formulas with the weighted, selection-aware expressions given here.
  • If the design condition fails (e.g., staggered adoption with no overlap), practitioners will need either to drop disconnected blocks or to re-normalize the effects within connected components before applying the correction.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper derives the joint large-N,T asymptotic distribution of two-way fixed-effects M-estimators when the panel is unbalanced. Selection is allowed to be deterministic (Φ-measurable design), stochastic (response), or mixed, under only a conditional mean restriction rather than full conditional independence. The uncorrected estimator is asymptotically normal but centered at a non-zero mean that combines the usual incidental-parameter biases with a Nickell-type feedback bias that can arise from predetermined regressors or from a predetermined selection process. Analytical bias corrections are constructed as sample analogs of the population bias terms (with a truncated spectral-density estimator for the feedback component); the corrected estimator is centered at zero and retains the same asymptotic variance. The theory nests the balanced-panel results of Fernández-Val and Weidner (2016). Monte Carlo experiments for a dynamic probit with mixed selection confirm the bias expressions, the validity of the analytical correction for small positive bandwidths, and the failure of split-panel jackknife methods when the design induces subsample heterogeneity.

Significance. Unbalanced panels are the rule rather than the exception in applied work, yet the large-N,T fixed-effects literature has largely ignored missing observations. The paper supplies the first complete asymptotic theory for two-way M-estimators under a flexible mixed selection process and a weak conditional-mean restriction. The analytical correction is practically attractive because it does not require the researcher to know which regressors or selection components are predetermined, and because it remains valid when design-induced heterogeneity invalidates jackknife methods. The nesting of the balanced-panel case, the explicit characterization of feedback bias from selection, and the careful treatment of Hessian invertibility under incomplete overlap are genuine contributions. The Monte Carlo design (10 000 replications) is transparent and matches the theory. If the results hold, the paper will become a standard reference for inference with two-way fixed effects in unbalanced panels.

minor comments (4)
  1. Assumption 1(iv) allows δ ≥ 6 + ν rather than the fixed δ = 8 + ν of Fernández-Val and Weidner (2016). A short remark on how this affects the admissible bandwidth range (and the rate-optimal ς*) would help applied readers choose h.
  2. In the simulation design the intercept ρ0 of the response equation is calibrated so that average missingness is one-half. Reporting the resulting average response probability conditional on d_it = 1 would clarify how much of the missingness is stochastic versus design-driven.
  3. The correction factor ω_i(m) = |T_i| / ∑_t s_it s_i(t−m) is a natural adaptation for gaps; a one-sentence comparison with the balanced-panel factor T/(T−m) in the main text (rather than only in Remark 3) would improve readability.
  4. A few typographical slips remain (e.g., “canbeinducedbypredetermined”, “feedbackbiascanthereforearise”). A final proof-reading pass would eliminate them.

Circularity Check

1 steps flagged

No significant circularity: main asymptotic expansions and bias corrections are derived from FOCs/Taylor expansions under stated assumptions; self-citations supply only technical lemmas.

specific steps
  1. self citation load bearing [Online Appendix, Lemmas S.6–S.9]
    "Lemma S.6 … See Proof of Lemma 10 in Czarnowske and Stammann (2026a). Lemma S.7 … See Proof of Lemma 11 in Czarnowske and Stammann (2026a). Lemma S.8 … See Proof of Lemma 12 in Czarnowske and Stammann (2026a). Lemma S.9 … See Proof of Lemma 13 in Czarnowske and Stammann (2026a)."

    Auxiliary invertibility and spectral-norm bounds used to verify the implied regularity conditions of Lemmas 2–3 are justified solely by citation to the authors’ concurrent paper rather than proved in-line; the dependence is technical and non-central, so it raises the score only to 1.

full rationale

The load-bearing claims (Theorems 1–2) follow from the profile objective’s first-order conditions, Legendre-transform Taylor expansions (Lemma S.1), and martingale CLTs under Assumption 1; the bias terms Bα,∞ and Bγ,∞ and the analytical correction are sample analogs of those expansions, not quantities fitted to data and then re-labeled as predictions. Bandwidth h is an explicit free tuning parameter with an admissible rate range, not a fitted constant that forces centering. The design-overlap condition (Assumption 1(vi)) is an identification hypothesis, not a derived result. Self-citations to the authors’ concurrent work (Czarnowske & Stammann 2026a) appear only for proofs of auxiliary matrix-norm / invertibility lemmas (S.6–S.9) used inside the regularity verification; those lemmas are standard technical tools and do not close the central argument. The paper nests the balanced-panel results of Fernández-Val & Weidner (2016) by specialization, which is independent external support. No self-definitional loop, fitted-input-as-prediction, uniqueness import, or ansatz smuggling is present. Score 1 reflects only the minor technical self-citations.

Axiom & Free-Parameter Ledger

1 free parameters · 7 axioms · 1 invented entities

The central asymptotic claims rest on a standard large-N,T fixed-effects program plus six regularity conditions collected in Assumption 1. No new physical entities are postulated; the mixed selection process is a definitional partition of the observed indicator. Bandwidth h is the only free tuning parameter.

free parameters (1)
  • bandwidth h (or exponent ς) = h∈{0,1,2} in simulations
    Truncation lag for the spectral estimator of feedback bias; must satisfy h≍T^ς with ς∈(0,(κ−1)/(2κ)); rate-optimal ς* is given but finite-sample choice remains free.
axioms (7)
  • domain assumption Joint asymptotics N,T→∞ with N/T→τ∈(0,∞)
    Assumption 1(i); standard for incidental-parameter bias of order 1/T and 1/N.
  • domain assumption Conditional independence across i given Φ and α-mixing within i with polynomial rate φ>δ(δ−ν)/(2ν)
    Assumption 1(ii); used for CLT and moment bounds.
  • domain assumption Conditional mean restriction E[y_it | C_i^t]=g(π_it(β⁰,μ_it⁰)) with predetermined regressors and response
    Assumption 1(iii); identifies the score as a martingale difference.
  • domain assumption Four-times continuous differentiability and uniform δ-moment bounds on the envelope Ψ
    Assumption 1(iv); enables Taylor expansions of the profile objective.
  • domain assumption Uniform positive lower bound on response-weighted second derivative and generalized non-collinearity of residuals
    Assumption 1(v); guarantees local strict convexity and invertibility of W.
  • domain assumption Design overlap: inf_i T^{-1}∑d_it≥c_U and inf_{t≠t′}N^{-1}∑d_it d_it′≥c_O a.s.
    Assumption 1(vi); ensures connectivity so the two-way Hessian remains invertible.
  • ad hoc to paper Affine score form ∂ψ/∂π=w(π)(g(π)−y)
    Eq. (4); restricts the class of M-estimators to those whose score is linear in y, excluding quantile regression.
invented entities (1)
  • mixed selection process s_it=d_it r_it no independent evidence
    purpose: Unifies deterministic (design) and stochastic (response) missingness under a single asymptotic theory
    Definition 1; purely definitional partition of the observed indicator, not a new physical object.

pith-pipeline@v1.1.0-grok45 · 47280 in / 3085 out tokens · 28690 ms · 2026-07-14T13:13:29.571936+00:00 · methodology

0 comments
read the original abstract

We derive the asymptotic properties of two-way fixed effects M-estimators with missing observations in an asymptotic framework in which the numbers of cross-sectional units and time periods grow jointly. We allow the selection process to be deterministic (conditional on the unobserved effects and initial conditions), stochastic, or mixed, and we impose only a conditional mean restriction. The uncorrected estimators are asymptotically normal but not centered at zero, suffering from incidental parameter and feedback biases. Feedback bias can be induced by predetermined regressors in the outcome equation and by a predetermined selection process. We propose debiased estimators that handle both sources without requiring knowledge of which regressors or selection components are predetermined.

discussion (0)

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