# Literature

**Date**: 03/07/2026
**Domain**: social_sciences/economics
**Taxonomy**: academic/working_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Assumption L4 and Winsorization Boundaries**

Reviewers note that Assumption L4 requires the average quantile function to be strictly increasing, which is necessary for the Hadamard differentiability of the projection operator. However, income data often contains atoms or flat regions, and the empirical application winsorizes at the 95th percentile while reporting effects at that same threshold. Readers might note that income data often exhibits discrete support which challenges this smoothness requirement. It would be helpful to discuss the robustness of the inference procedure when the average distribution has atoms or flat spots, or to restrict reported results to quantiles strictly below the winsorization point to ensure the estimates reflect the underlying distribution.

**Bandwidth Selection and Finite Sample Performance**

The proposed method utilizes a global bandwidth derived from the local polynomial estimator's IMSE, justified by asymptotic equivalence. However, finite sample performance may differ due to heteroskedasticity across quantiles and the stability of derivative estimates required for the bandwidth choice. It would be helpful to include a sensitivity analysis comparing the proposed bandwidth proxy against a cross-validation bandwidth specifically tuned for the Fréchet estimator to confirm that the claimed advantages are not driven by the bandwidth choice.

**Validity of Empirical Quantile Convergence**

Assumption Q1 requires the within-group estimation error to vanish faster than the across-group rate, which depends on the number of observations per unit (n_i). In the CPS application, the number of families per state-year may be limited, potentially challenging this condition. It would be helpful to provide a diagnostic check in the empirical section, such as reporting the minimum n_i or the ratio of estimation rates, to assure readers that the asymptotic results for empirical quantile functions are applicable to the dataset used.

**Interpretation of Aggregate Effects and Test Power**

The conclusion describes an "equality-efficiency trade-off," yet the net effect is negative, which might be more precisely characterized as redistribution with an aggregate efficiency loss. Additionally, simulation results show slight undercoverage for small sample sizes, and the empirical application uses a similar number of state-years. Readers might note that the p-values for the nullity test are close to the significance threshold, indicating limited power. It would be helpful to discuss the power of the homogeneity and nullity tests in this context, as wide confidence bands suggest distributional effects may be difficult to detect precisely with the available data.

**Theoretical Scope and Simulation Clarity**

The paper develops theory for Fuzzy RDD but applies a Sharp RDD design in the empirical section, leaving the Fuzzy extension empirically unvalidated. Furthermore, ambiguity exists in the simulation ground truth regarding whether treatment effects are constant or distribution-valued. It would be helpful to either provide an empirical example where the Fuzzy design is relevant or clarify the motivation for the extension, and to clarify the DGP specification to ensure bias calculations in the simulations are interpreted correctly.

**Potential Violation of i.i.d. Assumption Due to Panel Structure**

Assumption L1(i) in Section 3.1 requires the units {(Y_i, T_i, X_i)}_{i=1}^n to be i.i.d. copies. However, Section 4.2.1 describes the empirical application using 356 state-year combinations from 1984 to 2010. Given that gubernatorial elections occur periodically (typically every 4 years), individual states appear multiple times in the dataset, creating a panel structure. Observations from the same state across different years are likely correlated due to time-invariant state characteristics, violating the i.i.d. assumption across i. The multiplier bootstrap procedure in Algorithm 1 (Section 2.6) draws multipliers independently for each unit i, which may lead to underestimated standard errors and confidence bands if serial correlation is present. It would be helpful to cluster the bootstrap multipliers at the state level to account for this dependence, similar to the clustering mentioned for the standard RD comparison in Table A-1.

**Potential Conflict Between Winsorizing and Strict Monotonicity Assumption**

Assumption L4 in Section 3.1 requires the average quantile function to be strictly increasing (\frac{\partial}{\partial q} E[Q_Y(q) \mid X=x] \geq \kappa > 0). However, Section 4.2.1 states that the income distribution is "winsorized at the 95th percentile to account for top-coding". Winsorizing creates a mass point at the 95th percentile for each unit, which implies the quantile function is flat for q > 0.95. Since the estimation grid extends to 0.95 + 10^{-6} (Section 4.2.1), the average quantile function may exhibit flat regions near the upper bound of the grid, violating the strict monotonicity requirement. This could affect the Hadamard differentiability of the projection operator used in the Fréchet estimator. It would be helpful to verify that the average quantile function remains strictly increasing in the estimation range or to adjust the grid to exclude the winsorized region.

**Status**: [Pending]

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## Detailed Comments (23)

### 1. LAQTE Distribution Interpretation

**Status**: [Pending]

**Quote**:
> distributions” in that it respects the intrinsic geometry of the underlying probability measures being averaged over. In particular, the distribution defined by the LAQTEs has the intuitive interpretation of being the unique distribution with the lowest possible cumulative “least-squares” cost of transporting its probability mass into each of the underlying distributions of the individual units. This is exactly analogous to the interpretation of the mean as the “central tendency” in the standard Euclidean setting, i.e. the unique quantity that has the lowest expected least-squares distance to all points. Note that one can 

**Feedback**:
The Local Average Quantile Treatment Effect (LAQTE) is defined as the difference $\tau^{R3D}(q) = m_1(q) - m_0(q)$. In optimal transport theory, the Fréchet mean is characterized by the average of quantile functions, not the difference. Since the difference of two non-decreasing quantile functions is not necessarily non-decreasing, $\tau^{R3D}(q)$ does not generally define a valid probability distribution. It would be helpful to clarify that the Fréchet mean interpretation applies to the average quantile functions $m_t(q)$ rather than the LAQTEs themselves.

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### 2. Lemma 1 Notation Inconsistency

**Status**: [Pending]

**Quote**:
> \eqqcolon\lim_{x\to 0^{+}}m(q)-\lim_{x\to 0^{-}}m(q)

**Feedback**:
In Proposition 1, the conditional average quantile is denoted as $m(x,q)$ to indicate dependence on the running variable $x$. In Lemma 1, the notation switches to $m(q)$, which obscures this dependence. It would be helpful to update the notation in Lemma 1 to $m(x,q)$ to maintain consistency with Proposition 1 and clarify the variable dependence.

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### 3. Assumption Label Typo

**Status**: [Pending]

**Quote**:
> satisfies the weaker continuity assumption 11 but still fails the continuity in quantiles.

**Feedback**:
The text refers to 'assumption 11', but the assumption is labeled 'I1' (Roman numeral one) in the text. This is a typographical error that could confuse readers regarding which assumption is being referenced. It would be helpful to correct '11' to 'I1' to ensure accurate cross-referencing.

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### 4. Typographical Error in Example

**Status**: [Pending]

**Quote**:
> Suppose he counterfactual distribution functions $Y^{t}$ are drawn from a class of normal distributions

**Feedback**:
The text contains a typographical error where 'the' is written as 'he'. This affects the readability of the example introduction. It would be helpful to correct 'he' to 'the' to ensure professional presentation.

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### 5. Mean-Quantile Identity Equation

**Status**: [Pending]

**Quote**:
> $E[Q_{Y}(q)]=\int_{[0,1]}\int_{\mathcal{Y}}Q_{y}(q)\,\mathrm{d}F_{Y}(y)\,\mathrm{d}q=\int_{\mathcal{Y}}\int_{[0,1]}Q_{y}(q)\,\mathrm{d}q\,\mathrm{d}F_{Y}(y)=\int_{\mathcal{Y}}E[y]\,\mathrm{d}F_{Y}(y)$

**Feedback**:
The equation equates $E[Q_{Y}(q)]$, which is a function of $q$, to a scalar integral over $q$. The correct identity relating the mean to the quantile function is $E[Y] = \int_{0}^{1} Q_{Y}(q) \mathrm{d}q$. It would be helpful to adjust the left-hand side to $E[Y]$ or integrate over $q$ on the LHS to correctly reflect the mean-quantile relationship.

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### 6. Indicator Variable Definition

**Status**: [Pending]

**Quote**:
> where $e_{0}$ is the first standard basis vector, $K(x)$ a kernel function, $\delta_{i}^{\pm}\coloneqq 1\bigl{\{}X_{i}\;\;\geqq\;c\bigr{\}}$, and $r_{p}(x)\coloneqq(1,x,x^{2},\ldots,x^{p})$.

**Feedback**:
The definition provided for the indicator variable $\delta_{i}^{\pm}$ assigns the same condition ($X_i \geqq c$) to both the positive and negative cases. In a Regression Discontinuity Design, the left-side estimator must use observations below the cutoff. It would be helpful to distinguish the indicators for each side (e.g., $\delta_{i}^{+}$ for $X_i \ge c$ and $\delta_{i}^{-}$ for $X_i < c$) to ensure the estimator functions as intended.

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### 7. Inconsistent Notation for Empirical Weights

**Status**: [Pending]

**Quote**:
> where $s_{+,in}^{(p)}(h)$ are the usual empirical weights for a local polynomial regression of order $p$ *(Fan and Gijbels, 1996)*, which I derive explicitly in Appendix A.2.

**Feedback**:
Equation (5) defines the estimator using weights denoted as $s_{\pm,\,in}^{(p)}(h)$, implying weights for both the left and right sides. However, the text below Equation (5) only explicitly describes $s_{+,in}^{(p)}(h)$. It would be helpful to update the text to reference the weights for both sides ($s_{\pm,\,in}^{(p)}(h)$) to maintain consistency with the equation.

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### 8. Precision of 'Closer' in Projection Inequality

**Status**: [Pending]

**Quote**:
> the resulting projected estimator will always lie closer to the true quantile function than the (unprojected) local polynomial estimator

**Feedback**:
The projection inequality in Hilbert space states $\|\Pi(x) - y\| \leq \|x - y\|$. Equality holds if $x$ is already in the set. Thus 'always lie closer' implies strict inequality, which is not guaranteed. It would be helpful to change 'always lie closer' to 'always lie at least as close' to correctly reflect the non-strict inequality of the projection theorem.

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### 9. Lemma 2 Norm Invariance Claim

**Status**: [Pending]

**Quote**:
> $\left\|\hat{Q}^{*}-Q_{0}\right\|_{L_{p}}\leq\left\|\hat{Q}-Q_{0}\right\|_{L_{p}}\quad\text{ for all }p\in[1,\infty].$

**Feedback**:
The estimator is defined via minimizing squared 2-Wasserstein distances, which corresponds to an $L^2$ projection. While the distance reduction property holds for $p=2$, it is not generally guaranteed for $L^p$ norms where $p \neq 2$. It would be helpful to restrict the claim to $p=2$ or provide a specific reference that proves the $L^p$ contraction property for this specific projection.

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### 10. Monotonicity Assumption Strength

**Status**: [Pending]

**Quote**:
> (1.1.3) (Monotonicity). $\lim_{x\to 0}P(T^{1}>T^{0}\mid X=x)=1$ and $P(\text{Indefinites})=0$.

**Feedback**:
The assumption specifies strict inequality with probability one, which implies every unit is a complier. For binary treatment, this forces the first-stage jump to be exactly one, contradicting the general fuzzy setup where the denominator is estimated and typically less than one. It would be helpful to align this with standard identification conditions (e.g., $P(T^{1}>T^{0}\mid X=x)>0$) to ensure the estimator applies to cases with imperfect compliance.

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### 11. Indefinites Set Definition

**Status**: [Pending]

**Quote**:
> Indefinites: $I=\{\omega:T^{1}(\omega)=T^{0}(\omega)\}\setminus\{\omega:T^{1}(\omega)=T^{0}(\omega)\in\{0,1\}\}$.

**Feedback**:
If the treatment variable is binary, the condition $T^{1}(\omega)=T^{0}(\omega)$ already implies the values are in $\{0,1\}$. Consequently, the set difference results in an empty set, making the definition of Indefinites vacuous. It would be helpful to clarify the definition to ensure it captures the intended group of units, or remove the exclusion clause if it is redundant.

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### 12. Non-standard LaTeX Command

**Status**: [Pending]

**Quote**:
> \eqoneqq \frac{m_+(q) - m_-(q)}{m_{+,T} - m_{-,T}}.

**Feedback**:
The command \eqoneqq is not standard LaTeX notation for equality or definition. This may cause compilation errors or ambiguity. It would be helpful to use standard notation like := or \equiv to ensure the document compiles correctly and is clear to readers.

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### 13. Confidence Band Centering in Algorithm 1

**Status**: [Pending]

**Quote**:
> 14: Construct uniform bands: $[\hat{\mathbb{G}}^{\mathrm{R3D},b}(q) \pm (\sqrt{nh})^{-1} \hat{c}_n^B(a,b;\lambda)]$ for $q \in \mathcal{T}^*$

**Feedback**:
The confidence band formula centers the interval on the bootstrap realization $\hat{\mathbb{G}}^{\mathrm{R3D},b}(q)$, which varies with the iteration index $b$. Confidence bands should be fixed intervals based on the observed point estimate. It would be helpful to correct the centering term to reflect the observed data (e.g., $\hat{\tau}(q)$) rather than the bootstrap iteration.

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### 14. Inconsistent Notation for Bootstrap Process

**Status**: [Pending]

**Quote**:
> where the critical values can be constructed by taking the $(1 - \lambda)$-th quantiles of $\left\{\max_{q\in [\underline{q},\overline{q} ]}\left|\hat{\mathbb{G}}^{\mathrm{R / F3D}'}(q)\right|\right\}_{b = 1}^{B}$

**Feedback**:
The notation $\hat{\mathbb{G}}^{\mathrm{R / F3D}'}$ introduces a prime symbol that is not defined in Algorithm 1 or the surrounding text. Algorithm 1 uses $\hat{\mathbb{G}}^{\mathrm{R3D},b}$ for the bootstrap process. It would be helpful to align the notation with the algorithm to ensure clarity and consistency for readers verifying the test statistics.

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### 15. Typo in Variance Integral Differential

**Status**: [Pending]

**Quote**:
> \mathrm{Var}(\hat{\tau}_{\oplus,s}^{\mathrm{R3D}}(q)\,\mathrmmathrm{d}q)

**Feedback**:
The differential element \mathrm{d}q is placed inside the variance function arguments, which is mathematically incorrect as variance is a scalar function of q. It should be placed outside the variance term to indicate integration. It would be helpful to correct the syntax to $\mathrm{Var}(\hat{\tau}_{\oplus,s}^{\mathrm{R3D}}(q))\,\mathrm{d}q$.

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### 16. L4 Assumption Inequality LaTeX Typo

**Status**: [Pending]

**Quote**:
> For each $x \in \mathcal{N} \coloneqq (-\epsilon, \epsilon) \setminus \{0\}$ for $\epsilon > 0$ and $q \in [a, b] \subset (0, 1)$, there exist constants $0 < \kappa \leq K & < \infty$ such that $\kappa \leq \frac{\partial}{\partial q} E[Q_Y(q) \mid X = x] \leq K$ for a.e. $q \in [a, b]$.

**Feedback**:
The ampersand `&` is a LaTeX alignment character and appears to be a typo in the inequality chain, which may cause rendering issues. It would be helpful to remove it to ensure the mathematical statement is clear and renders correctly.

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### 17. Dimension Mismatch for Basis Vector e0

**Status**: [Pending]

**Quote**:
> let $e_{0}$ denote the 0th standard basis vector of $\mathbb{R}^{p}$, $(1,0,\ldots,0)$

**Feedback**:
Readers might note that the polynomial vector $r_p(u) = (1, u, \ldots, u^p)$ has $p+1$ components. Therefore, the basis vector $e_0$ should belong to $\mathbb{R}^{p+1}$, not $\mathbb{R}^p$. It would be helpful to correct the dimension to $\mathbb{R}^{p+1}$ to match the polynomial space and ensure matrix dimensions in $\Gamma_{\pm,p}$ are consistent.

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### 18. Claim on Frandsen (2012) Uniform Convergence

**Status**: [Pending]

**Quote**:
>  quantile treatment effects in RDD to allow for distribution-valued outcomes. *Frandsen et al. (2012)* first developed the framework for quantile RD and derived uniform convergence results, though they did not derive uniform confidence bands.

**Feedback**:
Frandsen et al. (2012) is primarily known for pointwise asymptotic normality of quantile RDD estimators. Claiming uniform convergence results might be an overstatement of that specific work, as uniform inference is typically associated with later developments. It would be helpful to clarify the distinction between pointwise and uniform results in the foundational literature to improve accuracy.

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### 19. Claim on Clustering for Quantile RDD

**Status**: [Pending]

**Quote**:
> While the approach of clustering standard errors at the group level works for average treatment effect estimation, this does not carry through to quantile treatment effects due to their nonlinearity, motivating the new approach in this article.

**Feedback**:
Several papers have explored clustering in quantile settings. Stating it 'does not carry through' might overstate the novelty or misrepresent the capabilities of prior methods. It would be helpful to acknowledge specific attempts to adapt clustering for quantiles to position the contribution more accurately.

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### 20. Card and Krueger (2000) Design Characterization

**Status**: [Pending]

**Quote**:
> The running variable here is distance to the border $(X)$.

**Feedback**:
Readers might note that the seminal Card and Krueger (2000) study is widely recognized as a Difference-in-Differences design rather than a Regression Discontinuity Design using distance to the border as a continuous running variable. It would be helpful to clarify that while the R3D framework could theoretically be applied to a border-discontinuity version of this setting, the original study did not utilize a continuous running variable.

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### 21. Q-RDD Target Description Inconsistency

**Status**: [Pending]

**Quote**:
> The two population objects targeted are,
> 
> $\text{R3D}:\lim_{x\to 0^{\pm}}E[Q_{Y}(q)\mid X=x]$ $\text{Q-RDD}:\lim_{x\to 0^{\pm}}E[1(Z\leq z)\mid X=x].$

**Feedback**:
The provided population target for Q-RDD defines the conditional cumulative distribution function (CDF), yet the text initially claims Q-RDD estimates a 'quantile function.' While the CDF and quantile function are inverses, they are distinct statistical objects. It would be helpful to align the terminology to ensure readers understand the precise target of the Q-RDD estimator.

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### 22. Claim on Random Variation in Averaged Outcomes

**Status**: [Pending]

**Quote**:
> Indeed, while I1 is consistent with the common approach of averaging the outcome variable at the level of the aggregate unit and then estimating a standard RD, the continuity assumption in *Frandsen et al. (2012)* is not: there would be no random variation left in the averages.

**Feedback**:
The statement that averaging outcomes leaves 'no random variation' is factually incorrect, as unit-level averages still vary across units. Frandsen et al. (2012) requires continuity of the conditional distribution, which can be satisfied by these cross-unit variations. It would be helpful to clarify that within-unit variation is lost, but cross-unit variation remains, and specify that the restriction arises from the requirement of distributional homogeneity.

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### 23. Q1 Convergence Rate Description

**Status**: [Pending]

**Quote**:
> In particular, it balances the speed of the empirical quantile estimator $\hat{Q}_{Y_{i}}$ against the ratio of the within-group sample size $n_{i}$ to the across-group sample size $n$, requiring that together, they imply a within-group convergence rate that is faster than $\sqrt{nh}$, the across-group rate.

**Feedback**:
Readers might note that $r_{n_i}$ is defined as an error bound ($O_p(r_{n_i})$) which converges to zero, while $\sqrt{nh}$ diverges to infinity. Describing a shrinking error bound as 'faster than' a growing sequence is mathematically inconsistent. It would be helpful to clarify that the error bound should be $o((nh)^{-1/2})$.

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