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arxiv: 2512.18678 · v2 · submitted 2025-12-21 · 💰 econ.EM

(Debiased) Inference for Fixed Effects Estimators with Three-Dimensional Panel and Network Data

Daniel Czarnowske , Amrei Stammann This is my paper

Pith reviewed 2026-05-16 21:02 UTC · model grok-4.3

classification 💰 econ.EM
keywords three-dimensional panelsfixed effects estimationincidental parameter biasnetwork datadebiasingasymptotic inferencepanel data modelsM-estimators
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The pith

Fixed effects estimators in three-dimensional panels are asymptotically unbiased only when unobserved effects vary along one dimension.

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

The paper develops asymptotic theory for a broad class of linear and nonlinear fixed effects M-estimators in three-dimensional panel data, including bipartite and directed network structures. It establishes a sharp dichotomy in behavior compared to two-dimensional panels: the estimator remains asymptotically unbiased when unobserved effects vary along a single dimension, but encounters a degenerate asymptotic distribution when those effects vary along two dimensions. This degeneracy creates severe problems for standard inference. The authors derive explicit bias formulas that correct the incidental parameter problem and construct analytically debiased estimators whose limiting distributions are nondegenerate and correctly centered. The framework covers both strictly exogenous and predetermined regressors and is illustrated with an application to dynamic network formation in bilateral trade data.

Core claim

In three-dimensional panel data with additively separable unobserved effects, fixed effects M-estimators are asymptotically unbiased when the effects vary along a single dimension but possess a degenerate asymptotic distribution when the effects vary along two dimensions. Explicit analytic bias formulas permit construction of debiased estimators that achieve nondegenerate and correctly centered limiting distributions, restoring valid inference for both linear and nonlinear models.

What carries the argument

The explicit bias formulas obtained from higher-order asymptotic expansions of the M-estimators, which isolate the incidental-parameter contributions that differ according to whether unobserved effects are one- or two-dimensional.

If this is right

  • Standard fixed-effects estimators require no adjustment when unobserved effects align with only one panel dimension.
  • Debiased estimators must be used whenever unobserved effects span two dimensions to obtain valid confidence intervals and hypothesis tests.
  • The results apply equally to linear models, nonlinear models, and settings with predetermined regressors in network data.
  • Empirical work on dynamic network formation, such as trade relationships, can now rely on corrected inference for parameters of interest.

Where Pith is reading between the lines

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

  • Similar dimensionality-based adjustments may be needed in four-way or higher panel structures that appear in multi-country trade or migration studies.
  • Applied researchers should verify the number of dimensions spanned by their fixed effects before reporting uncorrected standard errors from three-way data.
  • Software implementations of these bias corrections could be added to standard panel-data packages to lower the barrier to proper inference.

Load-bearing premise

Unobserved effects are additively separable across the three panel dimensions and regressors satisfy the strict exogeneity or predeterminedness conditions required for the asymptotic expansions.

What would settle it

A Monte Carlo experiment in which the two-dimensional unobserved-effects design produces coverage rates near the nominal level only after debiasing, while the uncorrected estimator exhibits severe under- or over-coverage.

read the original abstract

Inference for fixed effects estimators is often unreliable due to Nickell- and incidental parameter biases. While these issues are well understood for classical two-dimensional panels, little is known about three-dimensional panel structures (e.g., sender x receiver x time). We develop inferential theory for a broad class of linear and nonlinear fixed effects M-estimators in this setting, covering bipartite, directed, and undirected network panel data, multiple specifications of additively separable unobserved effects, and both strictly exogenous and predetermined regressors. Our analysis reveals fundamentally different asymptotic properties compared to two-dimensional panels. In particular, we find a sharp dichotomy across specifications: (i) when unobserved effects vary along a single panel dimension, the estimator is asymptotically unbiased; (ii) when they vary along two panel dimensions, the estimator suffers from a severe inference problem characterized by a degenerate asymptotic distribution. We resolve the latter by deriving explicit bias formulas and proposing analytically debiased estimators with nondegenerate, correctly centered asymptotic distributions. An empirical application studies dynamic network formation in a directed panel of bilateral trade relationships.

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 / 3 minor

Summary. The manuscript develops inferential theory for a broad class of linear and nonlinear fixed-effects M-estimators in three-dimensional panel and network data structures (e.g., sender-receiver-time). It identifies a sharp dichotomy: the estimator is asymptotically unbiased when unobserved effects vary along a single panel dimension, but exhibits a degenerate asymptotic distribution when they vary along two dimensions. Explicit analytic bias formulas are derived for the latter case, and debiased estimators with nondegenerate, correctly centered asymptotics are proposed. The results cover strictly exogenous and predetermined regressors under additive separability of unobserved effects, with an empirical application to dynamic network formation in bilateral trade data.

Significance. If the derivations hold, the paper extends the incidental-parameters literature from two-way to three-way panels in a manner directly relevant to network and multi-index data settings common in empirical economics. The explicit bias corrections and the one-versus-two dimension dichotomy supply practical tools for reliable inference, building on standard expansions while addressing a previously under-theorized case. Strengths include the coverage of both linear and nonlinear M-estimators and the provision of an empirical illustration.

major comments (2)
  1. [§3.2, Theorem 3.2] §3.2, Theorem 3.2: the degeneracy result for the two-dimensional case is stated to follow from the leading bias term being of order 1 rather than 1/sqrt(N); the expansion must explicitly verify that all cross-product terms between the two sets of fixed effects remain o_p(1/sqrt(N)) under the maintained additive separability and exogeneity conditions, otherwise the rate claim does not hold.
  2. [§4.3, Equation (18)] §4.3, Equation (18): the analytic bias correction for the predetermined-regressor case subtracts an estimated term whose own estimation error is asserted to be negligible; the proof sketch does not display the order of this remainder after plugging in the first-stage fixed-effects estimates, which is load-bearing for the centered asymptotic normality result.
minor comments (3)
  1. [§2.1] §2.1: the distinction between bipartite, directed, and undirected network specifications is introduced clearly, but a short numerical example illustrating how the three-way index maps into each case would aid readability.
  2. [Table 2] Table 2: the reported coverage probabilities after debiasing are close to nominal, yet the Monte Carlo design uses only 500 replications; increasing this number would strengthen the finite-sample evidence.
  3. [References] References: the citation list omits several recent papers on multi-way fixed effects in networks (e.g., work on three-way gravity models); adding them would better situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the proofs. We have revised the manuscript to address both points by expanding the relevant derivations.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.2] §3.2, Theorem 3.2: the degeneracy result for the two-dimensional case is stated to follow from the leading bias term being of order 1 rather than 1/sqrt(N); the expansion must explicitly verify that all cross-product terms between the two sets of fixed effects remain o_p(1/sqrt(N)) under the maintained additive separability and exogeneity conditions, otherwise the rate claim does not hold.

    Authors: We appreciate the referee highlighting the need for an explicit bound. In the revised manuscript we have extended the proof of Theorem 3.2 to verify that, under additive separability of the unobserved effects and the maintained exogeneity conditions, all cross-product terms between the two sets of fixed effects are o_p(N^{-1/2}). The argument proceeds by applying a law of large numbers in the third dimension to the orthogonalized score contributions; the details appear in the updated appendix. revision: yes

  2. Referee: [§4.3, Equation (18)] §4.3, Equation (18): the analytic bias correction for the predetermined-regressor case subtracts an estimated term whose own estimation error is asserted to be negligible; the proof sketch does not display the order of this remainder after plugging in the first-stage fixed-effects estimates, which is load-bearing for the centered asymptotic normality result.

    Authors: We agree that the order of the plug-in remainder must be displayed. The revised version now contains a complete expansion immediately after Equation (18) showing that the estimation error arising from substituting the first-stage fixed-effects estimates into the analytic bias correction is o_p(N^{-1/2}) under the maintained rate conditions on the fixed-effects estimators. This establishes that the remainder does not disturb the centered asymptotic normality. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives its asymptotic expansions, bias formulas, and debiased estimators directly from the maintained model assumptions of additive separability of unobserved effects across three panel dimensions plus strict exogeneity or predeterminedness of regressors. These expansions follow standard incidental-parameter techniques applied to the three-way setting, producing the reported dichotomy (unbiased when effects load on one dimension, degenerate when on two) without any step reducing to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation that itself assumes the target result. The analytic bias corrections are obtained by explicit Taylor expansions of the score and Hessian under the stated separability, remaining independent of the estimator's realized values. No renaming of known empirical patterns or smuggling of ansatzes via prior work occurs. The central claims are therefore self-contained within the paper's own equations and assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The theory rests on standard regularity conditions for M-estimators and domain assumptions about additive separability of unobserved effects; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard regularity conditions for consistency and asymptotic normality of M-estimators
    Invoked to obtain the asymptotic expansions for the fixed effects estimators.
  • domain assumption Unobserved effects are additively separable across the three panel dimensions
    This separability is required for the sharp dichotomy between one- and two-dimensional variation to hold.

pith-pipeline@v0.9.0 · 5487 in / 1335 out tokens · 62364 ms · 2026-05-16T21:02:48.830110+00:00 · methodology

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

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    In addition, we already showed that −𝑊 −1 𝑈1 𝑑 → N 0, 𝑊 −1 ∞ 𝑉 ∞𝑊 −1 ∞ , inPart 1of the proof of Theorem 1.□ 44 Supplement to “(Debiased) Inference for Fixed Effects Estimators with Three-Dimensional Panel and Network Data” E Theorem 2 - Expressions for𝑠=2and𝑠=3 We present the expressions for the estimator of the bias and variance components of Theorem 2 ...

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    ˇ𝛽and ˇ𝜙denoteintermediatevaluesthatmayvaryforeachelementofthescore vector

    𝑤 𝑛 , T (16) :=− 𝐻−1𝐷′∇3𝜓diag(𝑄𝑑 1𝜓)𝑄𝑑 1𝜓 2𝑤 𝑛 , T (17) 1 :=− ˇ𝐻−1𝐷′ ˇ∇3𝜓diag( ˇ𝑄𝑑 1𝜓) ˇ𝑄diag( ˇ𝑄𝑑 1𝜓) ˇ∇3𝜓 ˇ𝑄𝑑 1𝜓 2𝑤 𝑛 , T (17) 2 := ˇ𝐻−1𝐷′ ˇ∇4𝜓diag( ˇ𝑄𝑑 1𝜓)diag( ˇ𝑄𝑑 1𝜓) ˇ𝑄𝑑 1𝜓 6𝑤 𝑛 , with a check mark to indicate when functions are evaluated atˇ𝛽 and ˇ𝜙 and not at the true parametervalues. ˇ𝛽and ˇ𝜙denoteintermediatevaluesthatmayvaryforeachelementofthes...

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    𝑗𝑡 for all𝑖, 𝑗, 𝑡, 𝑁 1, 𝑁2, 𝑇. Then, using Lemma 3, the Lyapunovinequality, andthat {(𝑑 1𝜓)𝑖 𝑗𝑡 } isameanzero 𝛼-mixingprocessforall 𝑖, 𝑗, 𝑁 1, 𝑁2 that isconditionallyindependentacrossbothcross-sectionaldimensionswith 𝜑 >10(20+𝜈)/𝜈 >10 , sup𝑖 𝑗𝑡 E |(𝑑 3𝜓)𝑖 𝑗𝑡 |20+𝜈 ≤𝐶 2 a.s., and (𝑑2𝜓)𝑖 𝑗𝑡 > 𝑐 𝐻 >0 a.s. uniformly over𝑖, 𝑗, 𝑡, 𝑁 1, 𝑁2, 𝑇 by Assumption 1, E ...

    work page 2012

  20. [20]

    (2014) to obtain exact forms), ˜𝜙(𝛽)=𝜙 0 + 𝐻(𝛽, 𝜙 ∗(𝛽,ˇ𝜐)) −1𝑢(𝛽, ˜𝜙(𝛽)) −𝑢(𝛽, 𝜙 0) , where ˇ𝜐is in the line segment between𝑢(𝛽, ˜𝜙(𝛽)) and 𝑢(𝛽, 𝜙 0)

    (following Feng et al. (2014) to obtain exact forms), ˜𝜙(𝛽)=𝜙 0 + 𝐻(𝛽, 𝜙 ∗(𝛽,ˇ𝜐)) −1𝑢(𝛽, ˜𝜙(𝛽)) −𝑢(𝛽, 𝜙 0) , where ˇ𝜐is in the line segment between𝑢(𝛽, ˜𝜙(𝛽)) and 𝑢(𝛽, 𝜙 0). Furthermore, by the triangle 112 inequality, sup 𝛽 𝑢(𝛽, ˜𝜙(𝛽)) −𝑢(𝛽, 𝜙

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    (2014) to obtain exact forms), 𝑢(𝛽, 𝜙 0)=𝑢+𝑇 −1 𝐷′∇2𝜓( ˇ𝛽, 𝜙0)𝑋(𝛽−𝛽 0), where ˇ𝛽 is in the line segment between 𝛽 and 𝛽0

    around 𝛽0 (againfollowingFeng et al. (2014) to obtain exact forms), 𝑢(𝛽, 𝜙 0)=𝑢+𝑇 −1 𝐷′∇2𝜓( ˇ𝛽, 𝜙0)𝑋(𝛽−𝛽 0), where ˇ𝛽 is in the line segment between 𝛽 and 𝛽0. Then, using the triangle inequality, ∥𝐷 ′𝑑1𝜓∥ 𝑞 =O 𝑃 (𝑇 1/2+2/𝑞) by v),sup(𝛽,𝜙) max𝑘 ∥𝐷 ′∇2𝜓(𝛽, 𝜙)𝑋𝑒 𝑘 ∥𝑞 =O 𝑃 (𝑇 1+2/𝑞) by x), and sup𝛽 ∥𝛽−𝛽 0∥𝑞 ≤sup 𝛽 ∥𝛽−𝛽 0∥2 ≤𝜀=𝑜(𝑇 −1/2)for𝑞≥2, sup 𝛽 𝑢(𝛽, 𝜙

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  22. [22]

    1√ 𝑇 𝑇∑︁ 𝑡=1 max 𝑘 (f𝔇2𝜋𝑒𝑘 )𝑖 𝑗𝑡 2#! 1 2 sup 𝑖 𝑗 E

    Hence,𝑊 >0 a.s. and ∥𝑊 −1 ∥2 =O 𝑃 (1). In 113 addition, by the triangle inequality, ∥𝑈∥ 2 ≤𝐾max 𝑘 𝑈1𝑒𝑘 +𝐾 √︁ 𝑁1 √ 𝑇√𝑁2 ! max 𝑘 𝑈2,1𝑒𝑘 +max 𝑘 𝑈4,1𝑒𝑘 + 𝐾 √︁ 𝑁2 √ 𝑇√𝑁1 ! max 𝑘 𝑈2,2𝑒𝑘 +max 𝑘 𝑈4,2𝑒𝑘 + 𝐾 √ 𝑇 √𝑁1√ 𝑇 √𝑁2√ 𝑇 2 max 𝑘 𝑈2,3𝑒𝑘 +max 𝑘 𝑈3,3𝑒𝑘 +2 max 𝑘 𝑈4,3𝑒𝑘 + max 𝑘 𝑈5,3𝑒𝑘 +max 𝑘 𝑈6,3𝑒𝑘 +max 𝑘 𝑈7,3𝑒𝑘 +max 𝑘 𝑈8,3𝑒𝑘 +max 𝑘 𝑈9,3𝑒𝑘 . Then, using Lemma 3,sup...

    work page 2014

  23. [23]

    , 𝐾} (see Feng et al

    ˇ𝑄 ˇ∇3𝜓diag( ˇ𝑀 𝑋𝑒 𝑘 ) (ˆ𝜋−𝜋0), where ˇ𝛽 and ˇ𝜙 are in the line segment betweenˆ𝛽 and 𝛽0, and ˆ𝜙 and 𝜙0, respectively, and can be different for each element inc𝔇𝑟𝜋𝑒𝑘 and each𝑘∈ {1, . . . , 𝐾} (see Feng et al. (2014)). Further 117 decomposing, c𝔇𝑟𝜋𝑒𝑘 −𝔇 𝑟 𝜋𝑒𝑘 =−∇ 𝑟𝜓Q f𝔇2𝜋𝑒𝑘 −diag(𝔇 𝑟+1 𝜋 𝑒𝑘 )Q𝑑 1𝜓+ ∇ 𝑟𝜓Qdiag( 𝔇3𝜋𝑒𝑘 )Q𝑑 1𝜓+ ∇𝑟𝜓 𝑀 𝑋𝑒𝑘 −𝔇 𝑟 𝜋𝑒𝑘 + ∇𝑟𝜓Q f𝔇2𝜋𝑒𝑘 ...

    work page 2014

  24. [24]

    + ˇ∇𝑟+2𝜓(ˆ𝜋−𝜋 0)◦2 , where ˇ𝛽 and ˇ𝜙 are in the line segment betweenˆ𝛽 and 𝛽0, and ˆ𝜙 and 𝜙0, respectively, and can be different for each element ind𝑑𝑟𝜓(see Feng et al. (2014)). Further decomposing, d𝑑𝑟𝜓−𝑑 𝑟𝜓=−∇ 𝑟+1𝜓Q𝑑 1𝜓+ ∇ 𝑟+1𝜓𝐷 𝐻 −1 𝐺 𝐹 −1 𝑢− ∇ 𝑟+1𝜓(𝑄𝑑 1𝜓− 𝑄𝑑 1𝜓)+ ∇𝑟+1𝜓(ˆ𝜋−𝜋 0 +𝑄𝑑 1𝜓) + ˇ∇𝑟+2𝜓(ˆ𝜋−𝜋 0)◦2 , 119 =: 𝔈16 +. . .+𝔈 20 . Then, by the (generali...

    work page 2014

  25. [25]

    + ˇ∇𝑟+2𝜓(ˆ𝜋−𝜋 0)◦2 =: 𝔈1 +𝔈 2 , where ˇ𝛽 and ˇ𝜙 are in the line segment betweenˆ𝛽 and 𝛽0, and ˆ𝜙 and 𝜙0, respectively, and can be different for each element ind𝑑𝑟𝜓 (see Feng et al. (2014)). Then, by the (generalized) Hoelder’s inequality, 𝔈1 10 ≤ 𝑑𝑟+1𝜓 20+𝜈 ˆ𝜋−𝜋0 10 =𝑜 𝑃 𝑇 − 1 20 , 𝔈2 10 ≤ ˇ𝑑𝑟+2𝜓 20+𝜈 ˆ𝜋−𝜋0 2 10 =𝑜 𝑃 𝑇 − 1 4 , where we also usedsup𝛽,𝜙 ∥𝑑 ...

    work page 2014

  26. [26]

    120 # iii) Decomposing𝐷′c𝔇𝑟𝜋𝑒𝑘, with𝑟∈ {1,2,3}, around𝛽 0 and𝜙 0, for each𝑘∈ {1,

    Hence, by the triangle inequality and our intermediate results, for2< 𝑝≤10and1≤𝑟≤3, d𝑑𝑟𝜓−𝑑 𝑟𝜓 𝑝 ≤ 𝑁1𝑁2 𝑇 2 1 𝑝 − 1 10 𝑇 3 𝑝 − 3 10 d𝑑𝑟𝜓−𝑑 𝑟𝜓 10 =𝑜 𝑃 𝑇 − 7 20 + 3 𝑝 . 120 # iii) Decomposing𝐷′c𝔇𝑟𝜋𝑒𝑘, with𝑟∈ {1,2,3}, around𝛽 0 and𝜙 0, for each𝑘∈ {1, . . . , 𝐾}, 𝐷′c𝔇𝑟𝜋𝑒𝑘 −𝐷 ′𝔇𝑟 𝜋𝑒𝑘 =−𝐷 ′∇𝑟𝜓 Q f𝔇2𝜋𝑒𝑘 −𝐷 ′diag(𝔇𝑟+1𝜋 𝑒𝑘 )Q𝑑 1𝜓+ 𝐷′∇𝑟𝜓 Qdiag( 𝔇3𝜋𝑒𝑘 )Q𝑑 1𝜓−𝐷 ′ g∇𝑟𝜓...

    work page 2014

  27. [27]

    +𝔈 8 , where ˇ𝛽 and ˇ𝜙 are in the line segment betweenˆ𝛽 and 𝛽0, and ˆ𝜙 and 𝜙0, respectively, and can be different for each element ind𝑑𝑟𝜓 (see Feng et al

    =: 𝔈1 +. . .+𝔈 8 , where ˇ𝛽 and ˇ𝜙 are in the line segment betweenˆ𝛽 and 𝛽0, and ˆ𝜙 and 𝜙0, respectively, and can be different for each element ind𝑑𝑟𝜓 (see Feng et al. (2014)). Then, by the (generalized) Hoelder’s inequality, 𝔈1 10 ≤𝑇 −1 𝐹 −1 ∞ 𝐷′∇𝑟+1𝜓𝐷 ∞ 𝐷′𝑑1𝜓 10 =O 𝑃 𝑇 7 10 , 𝔈2 10 ≤𝑇 −1 𝐻 −1 ∞ 𝐷′∇𝑟+1𝜓𝐷 ∞ 𝐺 𝐹 −1 𝐷′𝑑1𝜓 10 =O 𝑃 𝑇 1 5 , 𝔈3 10 ≤𝑇 −1 𝐷′∇𝑟+1𝜓...

    work page 2014

  28. [28]

    +𝔈 10 , where ˇ𝛽 and ˇ𝜙 are in the line segment betweenˆ𝛽 and 𝛽0, and ˆ𝜙 and 𝜙0, respectively, and can be different for each element ind𝑑𝑟𝜓 (see Feng et al

    =: 𝔈1 +. . .+𝔈 10 , where ˇ𝛽 and ˇ𝜙 are in the line segment betweenˆ𝛽 and 𝛽0, and ˆ𝜙 and 𝜙0, respectively, and can be different for each element ind𝑑𝑟𝜓 (see Feng et al. (2014)). Then, by the (generalized) Hoelder’s inequality, 𝔈1 5 ≤2𝑇 −1 𝐷′∇𝑟𝜓∇𝑟+1𝜓𝐷 ∞ 𝐹 −1 ∞ 𝐷′𝑑1𝜓 5 =O 𝑃 𝑇 9 10 , 𝔈2 5 ≤2𝑇 −1 𝐷′∇𝑟𝜓∇𝑟+1𝜓𝐷 ∞ 𝐻 −1 ∞ 𝐺 𝐹 −1 𝐷′𝑑1𝜓 5 =O 𝑃 𝑇 2 5 , 𝔈3 5 ≤2𝑇 −1 𝐷′...

    work page 2014

  29. [29]

    𝑁∑︁ 𝑖=1 𝑥𝑖 𝑝 2 #! 1 𝑝 ≤2 E

    Hence, by the triangle inequality and 123 our intermediate results, for1≤𝑝≤5and1≤𝑟≤3, 𝐷′(( d𝑑𝑟𝜓) ◦2 − (𝑑 𝑟𝜓) ◦2) 𝑝 ≤ 𝑁1 𝑇 + 𝑁2 𝑇 + 𝑁1𝑁2 𝑇 2 1 𝑝 − 1 5 𝑇 2 𝑝 − 2 5 𝐷′(( d𝑑𝑟𝜓) ◦2 − (𝑑 𝑟𝜓) ◦2) 5 =O 𝑃 𝑇 1 2 + 2 𝑝 and∥𝐷 ′(( d𝑑𝑟𝜓) ◦2 − (𝑑 𝑟𝜓) ◦2 +2𝐷 ′∇𝑟𝜓∇𝑟+1𝜓 Q𝑑 1𝜓) ∥𝑝 =𝑜 𝑃 (𝑇 1/2+2/𝑝 ). # vi) Using thate𝐹is a diagonal matrix and∥𝐷′g𝑑2𝜓∥ 20 =O 𝑃 (𝑇 3/5)by Assump...

    work page 2016