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arxiv: 2605.20601 · v1 · pith:GC2VUPIFnew · submitted 2026-05-20 · 💰 econ.EM

Endogenous Quantile Regression with Measurement Error in Dependent Variable

Xuanjing Su This is my paper

Pith reviewed 2026-05-21 02:42 UTC · model grok-4.3

classification 💰 econ.EM
keywords quantile regressionendogeneitymeasurement errorcontrol functionsieve ML estimatornonparametric identificationtriangular system
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The pith

A control function approach makes quantile regression coefficients identifiable despite endogenous regressors and measurement error in the dependent variable.

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

This paper demonstrates that the conditional quantile coefficient functions are nonparametrically identifiable in a triangular system using a control function to handle endogeneity, even when the dependent variable has additive measurement error. The identification result covers the quantile coefficients and other distributional parameters. The authors then construct a two-step sieve maximum likelihood estimator, estimating the control function first and then using it in a likelihood with copula weights. The estimator is consistent and asymptotically normal when the number of quantile grid knots grows at an appropriate rate, allowing for bootstrap inference. Simulations show it reduces bias markedly compared to standard quantile regression that ignores these problems.

Core claim

The conditional quantile coefficient functions, together with all other distributional parameters, are nonparametrically identifiable under a control function approach in a triangular system, despite additive measurement error in the dependent variable. This leads to a two-step sieve ML estimator that is consistent and asymptotically normal with suitable growth in quantile grid knots.

What carries the argument

Control function in the triangular system that captures endogeneity, used in a sieve likelihood maximization with copula weights for the generated control variable.

If this is right

  • The method corrects for bias in quantile estimates caused by both endogeneity and measurement error.
  • Inference can be conducted using bootstrap methods.
  • The approach applies to a wide range of econometric settings with these data issues.
  • Nonparametric identification holds for the full distribution, not just specific quantiles.

Where Pith is reading between the lines

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

  • Extensions could include relaxing the additive measurement error assumption to multiplicative or other forms.
  • This could be applied to panel data or other structures common in empirical economics.
  • Testing the independence assumption between measurement error and control variable could be a useful robustness check.

Load-bearing premise

The data-generating process follows a triangular system in which the control function fully captures the endogeneity and the measurement error is additive and independent of the control variable conditional on the observed covariates.

What would settle it

Running the estimator on simulated data from the triangular system with known true quantile coefficients and checking if the estimates converge to the truth as the sample size grows large.

Figures

Figures reproduced from arXiv: 2605.20601 by Xuanjing Su.

Figure 1
Figure 1. Figure 1: MC Results Comparison: ε ∼ 3N and ρ = 0.5 Note [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MC Results: 2SSMLE ε ∼ 3N Note [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: MC Results Comparison: ε ∼ N (0, 1) and ρ = 0.5 Notes [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
read the original abstract

This paper studies quantile regression with an endogenous regressor and measurement error in the dependent variable. Standard quantile regression estimators ignoring these two elements can induce substantial bias. We adopt a control-function approach in a triangular system and show that the conditional quantile coefficient functions, together with all other distributional parameters, are nonparametrically identifiable. Building on this constructive identification result, we propose a two-step sieve ML estimator. The first step estimates the control function. The second step performs a sieve likelihood maximization that incorporates the generated control variable through copula weights. When the number of quantile grid knots grows at an appropriate speed, the estimator is consistent and asymptotically normal, permitting inference via bootstrap. Monte Carlo simulations demonstrate that the estimator markedly reduces bias relative to existing methods, confirming its effectiveness in settings with endogeneity and additive measurement error in the outcome.

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

3 major / 2 minor

Summary. The paper develops a control-function approach for quantile regression in the presence of an endogenous regressor and additive measurement error in the outcome variable. It establishes nonparametric identification of the conditional quantile coefficient functions and all distributional parameters under a triangular system, then proposes a two-step sieve maximum-likelihood estimator in which the first step recovers the control function and the second step incorporates the generated control via copula weights. Consistency and asymptotic normality are claimed when the number of quantile-grid knots grows at a suitable rate, with bootstrap inference and Monte Carlo evidence of bias reduction relative to standard quantile regression.

Significance. If the identification result and the asymptotic theory hold under the stated assumptions, the paper would provide a useful addition to the literature on quantile regression with endogeneity and measurement error. The Monte Carlo simulations supply concrete evidence of bias reduction, and the constructive identification via the triangular system is a clear strength. The practical estimator could be applied in empirical settings where both endogeneity and outcome measurement error are plausible.

major comments (3)
  1. [§2] §2 (Identification): The nonparametric identification of the conditional quantile functions rests on the assumption that the measurement error u is independent of the control function conditional on the observed covariates (u ⊥ control | X). This conditional independence is load-bearing for the triangular-system argument; if it fails due to unobserved heterogeneity that correlates the measurement error with the first-stage residual, the generated-regressor correction does not recover the target parameters and both consistency and bootstrap validity break down.
  2. [§4] §4 (Asymptotics): The statement that the estimator is asymptotically normal when the number of quantile-grid knots grows at an appropriate speed is central to the inference claim, yet the precise rate condition and the handling of the generated-regressor estimation error in the sieve likelihood are not fully detailed. Without explicit verification of these steps, it is difficult to confirm that the bootstrap remains valid under the knot-growth schedule.
  3. [§3] §3 (Estimator): The second-step sieve ML uses copula weights that embed the conditional independence assumption from the first step. Any finite-sample dependence introduced by the estimated control function could affect the likelihood maximization; the paper should clarify whether additional trimming or adjustment is required to preserve the asymptotic properties.
minor comments (2)
  1. [Abstract] The abstract refers to an 'appropriate speed' for knot growth; the exact rate condition should be stated explicitly in the main text or theorem statement for clarity.
  2. [Monte Carlo section] Monte Carlo results would benefit from reporting standard errors or confidence bands around the bias and RMSE figures to allow readers to assess the precision of the reported improvements.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§2] §2 (Identification): The nonparametric identification of the conditional quantile functions rests on the assumption that the measurement error u is independent of the control function conditional on the observed covariates (u ⊥ control | X). This conditional independence is load-bearing for the triangular-system argument; if it fails due to unobserved heterogeneity that correlates the measurement error with the first-stage residual, the generated-regressor correction does not recover the target parameters and both consistency and bootstrap validity break down.

    Authors: We agree that the conditional independence assumption u ⊥ V | X (where V denotes the control function) is central to the nonparametric identification result in Section 2 and is stated explicitly as Assumption 3. This restriction ensures that the measurement error is independent of the endogenous component after conditioning on observables and the control. While we acknowledge that violations arising from additional unobserved heterogeneity could invalidate the generated-regressor correction, the assumption is standard in triangular control-function models and is maintained throughout the analysis. In the revision we will expand the discussion in Section 2 to provide further intuition, discuss plausible empirical settings (e.g., wage equations with reporting error), and note the consequences of potential violations. revision: partial

  2. Referee: [§4] §4 (Asymptotics): The statement that the estimator is asymptotically normal when the number of quantile-grid knots grows at an appropriate speed is central to the inference claim, yet the precise rate condition and the handling of the generated-regressor estimation error in the sieve likelihood are not fully detailed. Without explicit verification of these steps, it is difficult to confirm that the bootstrap remains valid under the knot-growth schedule.

    Authors: The referee correctly notes that the rate conditions and the propagation of first-step estimation error into the second-step sieve likelihood require more explicit treatment. Theorem 4 states consistency and asymptotic normality when the number of knots K_n satisfies K_n = o(n^{1/3}) together with standard sieve regularity conditions, but the appendix sketch is concise on bounding the generated-regressor term. We will revise Section 4 and the appendix to supply a fuller asymptotic expansion that isolates the contribution of the estimated control function, verify that this term is asymptotically negligible under the stated rate, and confirm bootstrap validity via the continuous-mapping theorem applied to the sieve estimator. revision: yes

  3. Referee: [§3] §3 (Estimator): The second-step sieve ML uses copula weights that embed the conditional independence assumption from the first step. Any finite-sample dependence introduced by the estimated control function could affect the likelihood maximization; the paper should clarify whether additional trimming or adjustment is required to preserve the asymptotic properties.

    Authors: The copula weights are constructed from the conditional distribution implied by the maintained independence assumption. Although using an estimated control function introduces finite-sample dependence, the asymptotic theory shows that this dependence vanishes at the required rate when the knot number grows appropriately. Our Monte Carlo experiments in Section 5 exhibit stable performance without extra trimming. In the revision we will add a clarifying remark in Section 3 stating that no additional trimming or adjustment is required to preserve the asymptotic properties, with the bootstrap procedure already incorporating the two-step estimation uncertainty. revision: partial

Circularity Check

0 steps flagged

No circularity: identification and estimation are derived from model assumptions and standard sieve theory

full rationale

The paper derives nonparametric identification of the conditional quantile coefficient functions from the triangular system and control-function assumptions, including additive measurement error independent of the control variable conditional on covariates. The two-step sieve ML estimator is constructed by first estimating the control function and then maximizing a sieve likelihood that incorporates it via copula weights. Consistency and asymptotic normality are established under a knot-growth rate condition using standard arguments from sieve estimation theory. No step reduces the target parameters to fitted values by construction, no self-citation is load-bearing for uniqueness or identification, and no ansatz or renaming is smuggled in. The derivation is self-contained against external benchmarks in econometric identification and nonparametric estimation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the identification rests on the triangular system and additive measurement-error assumptions that are standard in the control-function literature but not independently verified here.

axioms (2)
  • domain assumption Triangular system structure with control function capturing endogeneity
    Invoked to achieve nonparametric identification of quantile coefficients.
  • domain assumption Additive measurement error independent of the control variable conditional on covariates
    Required for the measurement-error correction via copula weights.

pith-pipeline@v0.9.0 · 5657 in / 1397 out tokens · 35440 ms · 2026-05-21T02:42:48.706594+00:00 · methodology

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Works this paper leans on

43 extracted references · 43 canonical work pages

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