On Robust Empirical Likelihood for Nonparametric Regression with Application to Regression Discontinuity Designs

Problem Statement

Empirical Likelihood (EL) is a powerful nonparametric inference tool that offers likelihood-based confidence intervals with desirable properties, such as respecting parameter constraints and possessing Wilks-type limiting distributions (asymptotically $\chi^2$). However, when applied to nonparametric regression and Regression Discontinuity Designs (RDD) using local polynomial smoothers, standard EL methods face a critical limitation: smoothing bias.

To achieve valid Wilks-type inference (where the EL ratio converges to a standard $\chi^2$ distribution), traditional approaches require undersmoothing (selecting a bandwidth $h$ smaller than the optimal rate for point estimation). This reduces bias but inflates variance, leading to overly wide confidence intervals in finite samples.

Alternatively, bias-corrected approaches attempt to estimate and remove the bias. However, the paper identifies a fundamental flaw in existing bias-corrected EL methods: they fail to account for the additional variability introduced by the estimation of the bias itself. Consequently, when the ratio of the estimation bandwidth ($h$) to the pilot bandwidth used for bias correction ($b$) converges to a positive constant ($h/b \to \kappa \in (0, 1]$), the standard Wilks phenomenon breaks down. The EL ratio converges to a scaled $\chi^2$ distribution rather than a standard one, leading to severe under-coverage and inflated Type I error rates if standard critical values are used.

Methodology

The authors propose a novel framework termed Robust Empirical Likelihood (Robust EL) to resolve the tension between bias correction and the preservation of the Wilks phenomenon under realistic bandwidth conditions ($h/b \to \kappa \in [0, 1]$).

1. Robust Weights Construction:
Instead of merely adjusting the EL ratio or the point estimator, the core innovation lies in constructing new robust weights that are fully data-adaptive. These weights simultaneously achieve bias correction and incorporate the variability of the bias estimator directly into the moment constraints of the EL formulation.

• Taylor-Expansion-Based Robust Weights ($W^\star$): Derived by re-expressing the bias estimator (based on a local quadratic fit) within the EL moment constraint. This effectively treats the bias estimator as a function of the resampling probabilities ($p_i$) rather than fixed constants.
• Difference-Based Robust Weights ($W^\ddagger$): Constructed similarly using a direct difference approach for bias estimation, ensuring the variability of the difference estimator is captured.

2. Robust EL Ratios:
Using these new weights, the authors define two robust EL ratio functions:

• $l_{TR}(\theta)$: Taylor-expansion-based Robust EL ratio.
• $l_{DR}(\theta)$: Difference-based Robust EL ratio.

3. Extension to RDD:
The framework is extended to both Sharp and Fuzzy RDD settings.

• For Sharp RDD, the method constructs robust weights for treated and control groups separately to estimate the treatment effect $\tau_S$.
• For Fuzzy RDD, the method handles the ratio of discontinuities in outcomes and treatment probabilities, constructing robust weights for all components ($Y$ and $T$) to estimate $\tau_F$.

4. Theoretical Framework:
The paper adopts the refined asymptotic framework of Calonico et al. (2014), where $h/b \to \kappa \in [0, 1]$. Under standard regularity conditions, the authors prove that both $l_{TR}$ and $l_{DR}$ converge to a standard $\chi^2_1$ distribution, regardless of whether $\kappa = 0$ (undersmoothing) or $\kappa > 0$ (bias-corrected regime).

Key Contributions

1. Methodological Innovation: The paper pioneers a bias-corrected EL inference framework based on fully data-adaptive robust weights. Unlike previous methods that adjust the ratio post-hoc, this approach integrates bias correction and its associated variability directly into the weighting scheme of the EL formulation.
2. Theoretical Validity: The authors establish that the Wilks phenomenon holds under the more general and practically relevant condition where $h/b \to \kappa \in [0, 1]$. This allows for the construction of valid confidence intervals using standard $\chi^2$ critical values without requiring undersmoothing.
3. Application to RDD: The framework is seamlessly extended to Sharp and Fuzzy RDD, providing the first robust EL procedures for these settings that maintain pivotal properties without complex scale adjustments or additional variance estimation steps.
4. Second-Order Analysis: The paper explores second-order asymptotic properties, noting that while the current method improves coverage error rates compared to original EL, further modifications might be required to match the optimal coverage error rates of Wald-type robust bias-corrected estimators.

Results

Simulation Studies:
Extensive simulations were conducted for nonparametric regression and Sharp RDD settings.

• Coverage and Size: The proposed Robust EL methods (TR and DR) consistently achieve empirical coverage probabilities close to the nominal level (95%) and correct empirical sizes across a wide range of bandwidth choices ($h$) and pilot bandwidth ratios ($b/h$).
• Comparison:
  • Original EL (Undersmoothing): Suffered from severe under-coverage when $h$ was not sufficiently small, and produced wider intervals due to high variance.
  • Conventional Bias-Corrected EL (TB/DB): Demonstrated significant under-coverage and inflated Type I errors when $h/b \not\to 0$, confirming the theoretical failure of the Wilks approximation in these regimes.
  • Robust EL vs. Wald-type (CCF): The proposed Robust EL methods performed comparably or superior to the robust bias-corrected Wald approach (Calonico et al., 2014) in terms of coverage accuracy, often yielding shorter confidence intervals.
• Computational Efficiency: While EL methods are computationally more intensive than Wald methods due to numerical inversion, the proposed algorithms (using Newton-Lagrange methods) constructed confidence intervals in less than 0.5 seconds for moderate sample sizes, making them practically feasible.

Real Data Applications:
The paper applies the proposed methods to two real datasets (details in Section 6 of the full paper, summarized as illustrative). The results demonstrate the practical utility of the method, showing that it provides reliable inference without the need for delicate bandwidth tuning or undersmoothing.

Significance and Claims

The paper claims to provide a "satisfactory solution" to a long-standing challenge in the EL literature: how to incorporate bias correction while retaining the distribution-free, Wilks-type calibration of Empirical Likelihood.

• Robustness to Bandwidth Selection: The primary significance is the elimination of the need for undersmoothing. Researchers can now use bandwidths that balance bias and variance optimally for point estimation (or slightly larger) without sacrificing the validity of the confidence intervals.
• Simplicity and Flexibility: The method avoids the cumbersome implementation of scale factors or separate variance estimation steps required by other bias-corrected EL approaches. It preserves the "likelihood-based" appeal of EL.
• Theoretical Rigor: By proving that the Wilks phenomenon holds under the refined asymptotic framework ($h/b \to \kappa$), the paper bridges the gap between theoretical optimality and practical implementation in nonparametric inference.

The authors modestly note that while their method achieves a coverage error rate of $O(nh^7 + h^2 + (nh)^{-1})$, which is an improvement over original EL, it does not yet match the optimal $O(nh^7 + h^3 + (nh)^{-1})$ rate of the best Wald-type estimators. They suggest that further methodological modifications could potentially achieve this optimal rate, leaving that as a direction for future research.