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arxiv: 2307.13475 · v3 · submitted 2023-07-25 · 💰 econ.EM · math.ST· stat.TH

Large sample properties of GMM estimators under second-order identification

Hugo Kruiniger This is my paper

Pith reviewed 2026-05-24 08:08 UTC · model grok-4.3

classification 💰 econ.EM math.STstat.TH
keywords GMM estimationsecond-order identificationlimiting distributionsexact identificationoveridentificationindirect inferenceasymptotic theoryJacobian rank deficiency
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The pith

GMM estimators under second-order identification have non-standard limiting distributions that differ between exact and overidentification.

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

The paper establishes that when a parameter vector is globally identified but locally underidentified at first order, the GMM estimator for the first p-1 components converges at the usual square-root rate while the remaining component converges at the slower fourth-root rate. It shows that an earlier condition for the limiting distribution of the slow component holds only in overidentified models, so the actual non-standard limit depends on whether the model is exactly identified or overidentified. These limits are derived from expansions of the GMM objective function taken to different orders. The same incompleteness affects earlier results on indirect inference estimators that rely on GMM auxiliary models, and the paper supplies the missing pieces along with optimal weighting matrices for both estimators.

Core claim

After reparametrizing the model so the expected Jacobian has a vanishing last column at the true value, the GMM estimator of the first p-1 parameters converges at rate T to the minus one-half and the estimator of the remaining parameter converges at rate T to the minus one-fourth; the limiting distribution of the scaled error for the slow parameter is non-standard, differs according to exact versus overidentification, and is obtained by expansions of the GMM objective of different orders.

What carries the argument

Higher-order expansions of the GMM objective function after reparametrization that sets the last column of the expected Jacobian to zero.

If this is right

  • The condition used by Dovonon and Hall for the limiting distribution is satisfied only when the parameter is overidentified.
  • The limiting distributions of the sign of the fourth-root scaled estimation error are different under exact identification and under overidentification.
  • The limiting distribution theories for indirect inference estimation under two scenarios with second-order identification are incomplete and can be completed with the GMM results.
  • Optimal weight matrices can be derived separately for the fast-converging and slow-converging parameter estimators.

Where Pith is reading between the lines

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

  • Applied researchers using second-order identification asymptotics should first verify whether their model is exactly or overidentified before selecting critical values or confidence intervals.
  • Finite-sample simulations that separately generate data under exact and overidentification could reveal how quickly the two limiting distributions emerge.
  • The distinction may affect the design of specification tests or the choice between GMM and other estimators when second-order identification is suspected.

Load-bearing premise

The first-order underidentification is caused exactly by the expected Jacobian having rank p-1 at the true parameter value.

What would settle it

A large-sample Monte Carlo experiment in which the distribution of the sign of T to the one-fourth times the estimation error for the slow parameter is identical under exact identification and under overidentification would falsify the claim.

read the original abstract

Dovonon and Hall (Journal of Econometrics, 2018) proposed a limiting distribution theory for GMM estimators for a p - dimensional globally identified parameter vector {\phi} when local identification conditions fail at first-order but hold at second-order. They assumed that the first-order underidentification is due to the expected Jacobian having rank p-1 at the true value {\phi}_{0}, i.e., having a rank deficiency of one. After reparametrizing the model such that the last column of the Jacobian vanishes, they showed that the GMM estimator of the vector comprising the first p-1 parameters, {\phi}_{1}, converges at rate T^{-1/2} and the GMM estimator of the remaining parameter, {\phi}_{p}, converges at rate T^{-1/4}. They also provided a limiting distribution of T^{1/4}({\phi}_{p}-hat-{\phi}_{0,p}) subject to a (non-transparent) condition which they claimed to be not restrictive in general. However, as we show in this paper, their condition is in fact only satisfied when {\phi} is overidentified and the limiting distribution of T^{1/4}({\phi}_{p}-hat-{\phi}_{0,p}), which is non-standard, depends on whether {\phi} is exactly identified or overidentified. In particular, the limiting distributions of the sign of T^{1/4}({\phi}_{p}-hat-{\phi}_{0,p}) for the cases of exact and overidentification, respectively, are different and are obtained by using expansions of the GMM objective function of different orders. Unsurprisingly, we find that the limiting distribution theories of Dovonon and Hall (2018) for Indirect Inference (II) estimation under two different scenarios with second-order identification where the target function is a GMM estimator of the auxiliary parameter vector, are incomplete for similar reasons. We discuss how our results for GMM estimation can be used to complete both theories. We also derive the optimal weight matrices for {\phi}_{1}-hat and {\phi}_{p}-hat, respectively.

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

0 major / 2 minor

Summary. The manuscript corrects Dovonon and Hall (2018) by showing that their non-transparent condition for the limiting distribution of the GMM estimator under second-order identification (with Jacobian rank deficiency of one) holds only when the parameter vector φ is overidentified. It derives the non-standard limiting distribution of T^{1/4}(φ̂_p − φ_{0,p}), which differs between exact and overidentification cases and is obtained via expansions of the GMM objective function at different orders; the sign of the normalized estimator has different limiting distributions in the two regimes. The paper also completes the corresponding limiting distribution results for indirect inference (II) estimation under two second-order identification scenarios and derives the optimal weight matrices for φ̂_1 and φ̂_p.

Significance. If the objective-function expansions and resulting distributions are correct, the paper supplies a necessary technical clarification for GMM asymptotics when first-order identification fails but second-order identification holds. The explicit dependence on exact versus overidentification, the completion of the II results, and the optimal weights are all load-bearing for valid inference in this setting. The work therefore strengthens the reliability of second-order identification theory in econometrics.

minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a short explicit statement of the reparametrization (last column of the Jacobian set to zero) and the precise orders of the objective-function expansions used for the exact-ID versus over-ID cases.
  2. Notation for the partitioned parameter (φ_1, φ_p) and the associated Jacobian columns is introduced in the abstract but should be restated once at the beginning of the main text for readers who start with the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive report, which accurately captures the paper's contributions in correcting Dovonon and Hall (2018) and completing the indirect inference results. The recommendation of minor revision is noted; no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results derive limiting distributions for GMM estimators under second-order identification by reparametrizing the model around a rank-(p-1) Jacobian deficiency (taken as given from the Dovonon-Hall setup) and performing explicit Taylor expansions of the GMM objective function to different orders depending on exact vs. overidentification. These expansions are standard asymptotic arguments and do not reduce to any fitted parameter within the paper, any self-citation chain, or a renaming of known results. The critique of the Dovonon-Hall condition is obtained by direct comparison of the orders at which the objective yields a non-degenerate limit; the II completion is likewise obtained by applying the same GMM expansions. No load-bearing step collapses to a definition or internal fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard regularity conditions for GMM asymptotics and the rank-deficiency setup from the cited paper; no free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard regularity conditions for GMM estimation and limiting distributions
    Invoked to justify the expansions and convergence rates described in the abstract.

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Reference graph

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